Mastering Limits: From Basic Rules to Trigonometric Applications

Feb 06, 2026
•
4 min read
YouTube video ID: VSqOZNULRjQ
Source: YouTube video by Professor Leonard — Watch original video
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Introduction

In this lecture the instructor walked through the fundamental concepts of limits, demonstrated how to evaluate them for simple functions, introduced the key limit properties, and then extended the discussion to more complex cases such as rational functions, piece‑wise definitions, and trigonometric limits.
1. Basic Limits

Limit of a constant: (\lim_{x\to a} C = C). The graph is a horizontal line; approaching any point yields the same value.
Limit of the identity function: (\lim_{x\to a} x = a). The graph is the line (y=x); plugging (a) directly gives the limit.
These two facts let us evaluate many limits instantly.
2. Core Limit Properties

For functions (f(x)) and (g(x)) whose limits exist at (x=a): 1. Addition/Subtraction: (\lim_{x\to a}[f(x)\pm g(x)] = \lim f(x) \pm \lim g(x)). 2. Multiplication: (\lim_{x\to a}[f(x)\cdot g(x)] = \lim f(x) \cdot \lim g(x)). 3. Division: (\lim_{x\to a}\frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}) provided (\lim g(x)\neq0). 4. Power/Root: (\lim_{x\to a}[f(x)]^n = (\lim f(x))^n). This also works for radicals because a root is an exponent.
3. Polynomial Limits

Because polynomials are built from constants and the identity function using only addition, subtraction, multiplication, and powers, we can evaluate any polynomial limit by direct substitution: [\lim_{x\to a} P(x) = P(a)] The instructor demonstrated this with examples like (x^3-2x+7) at (x=2), obtaining (11) without tables or graphs.
4. Rational Functions and Domain Issues

When a limit involves a fraction, first check the denominator at (x=a).
If the denominator is non‑zero, substitute directly.
If you get a 0/0 form, try to factor and cancel common factors. A cancelled factor indicates a hole (removable discontinuity). After cancellation, substitute the value.
If the denominator still becomes zero after all possible cancellation, the point is a vertical asymptote (the instructor called this an "ASM toote"). In such cases the limit may be (\pm\infty) or may not exist.
Sign analysis: When a vertical asymptote remains, test values on each side of the point to see whether the function heads toward (+\infty) or (-\infty). This determines existence of the limit.
5. Piece‑wise Limits

For a function defined by different expressions on intervals, evaluate one‑sided limits at the break points: 1. Identify the left‑hand expression and compute (\lim_{x\to a^-}). 2. Identify the right‑hand expression and compute (\lim_{x\to a^+}). 3. If the two one‑sided limits are equal, the overall limit exists; otherwise it does not. The instructor illustrated this with a three‑piece function involving (1/(x+2)), (x^2-5), and (\sqrt{x}+13).
6. Trigonometric Limits

Three cornerstone limits were proved using geometry and the Squeeze Theorem: - (\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1) - (\displaystyle \lim_{x\to0}\frac{1-\cos x}{x}=0) - (\displaystyle \lim_{x\to0}\frac{\tan x}{x}=1) Key techniques: - Rationalizing (multiply by the conjugate) to eliminate square‑roots. - Using identities such as (1-\cos^2 x = \sin^2 x) or (\tan x = \frac{\sin x}{\cos x}). - Squeeze Theorem: Show a function is bounded between two others whose limits are known; the middle function shares the same limit.
7. Practical Workflow for Any Limit

Plug in the target value. If you obtain a finite number, that is the limit.
If you get 0/0, look for factoring, rationalizing, or trigonometric identities to simplify.
If a denominator still vanishes, determine whether the factor can be cancelled (hole) or not (vertical asymptote). Use sign analysis for the latter.
For piece‑wise definitions, handle each side separately.
For trigonometric expressions, aim to rewrite them in terms of the three basic limits.
8. Summary of Take‑aways

Limits of constants and the identity function are trivial.
All polynomial limits are found by substitution.
Rational functions require careful attention to zeros in the denominator; factor‑cancel or use sign analysis.
Piece‑wise limits hinge on matching left‑ and right‑hand limits.
Trigonometric limits reduce to the three fundamental limits via identities, conjugates, and the Squeeze Theorem.
By mastering these steps you can evaluate virtually any elementary limit without resorting to tables or graphing calculators.
The essential lesson is that most limits can be solved by simple substitution once you apply the basic limit properties, factor or rationalize to remove indeterminate forms, and use the three fundamental trigonometric limits together with the Squeeze Theorem for more complex cases.
Frequently Asked Questions

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Helpful resources related to this video

If you want to practice or explore the concepts discussed in the video, these commonly used tools may help.
Calculus Textbook Limits And Continuity Recommended
Provides clear explanations and practice problems for limits, helping students master substitution, factoring, and trigonometric limit techniques.
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Graphing Calculator Scientific
Allows quick verification of limit calculations and visualizes behavior near points of discontinuity, reinforcing classroom concepts.
Amazon →
Limit Practice Workbook Calculus
Offers extra exercises on polynomial, rational, piecewise, and trigonometric limits, giving hands‑on practice to solidify understanding.
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Full Transcript YouTube

we got to start with some Basics is that
1.2 or 2. 1.2 we're skipping the chapter
and I I was going we're in trouble
now yeah I'd like you to read the rest
of chapter one you know just kind of get
it and then we'll be on suck some
teachers do that I've heard anyway no
we're 1.2 my bad I I saw I just like the
number two so I thought I'd put it up
there on the board see what happens you
know we're going to have to start with
some
Basics some Basics that we know are
truths and then we'll be able to figure
out some properties use those properties
to discover how to compute some limits
so first little basic here
the limit as X approaches a of a
constant C or C is a
constant what's the picture what's the
graph of a constant show them with your
hands with the picture of a sure yeah
exactly what is it yeah what's the
picture of a
constant of of a function equal in a
constant yeah it's not this way right
it's this way that's a constant it
doesn't doesn't change slope wise the
slope is zero it just goes straight
across true so if I were to draw the
picture of
this you would agree that that would be
my function
right well here's the cool thing if I
take the limit at any point any point
whatsoever so for instance here's my a
here's my a I I just picked a random
point doesn't matter what it is okay
it's arbitrary what's the limit as we go
from the left and the right firstly is
it the same thing yeah it's not not
gapped or anything it's going to the
same thing uh what's the limit as we go
from the
right what's the what's the Y value as
we approach the a value c c how about
this way so the limit is what that means
is the limit of a constant is the
constant itself that's
cool where C's a constant that's nice
right that makes sense graphically we
just kind of interpreted so every time
we take a limit of just a number that's
a constant we just have the constant
that makes things nice for
us number
two how about the limit
of X as X approaches a so uh what's the
the graph of just X like yal X how's
that look line that passes through the
origin it goes through the origin great
because the slope would be one and the Y
intercept would be zero so this is just
if you show me with your hands how would
it look yeah just like that not like
this not like this but a
diagonal now I'm going to pick another
arbitrary point
a the question is if this is yal X just
our our diagonal line if you plug in a
to y x what do you get out of
it yeah
you get
out a true you plug in two you get out
two you plug in three you get out three
you plug in four you get out four you
with me on that so my question is as we
approach as X approaches a from the
right and from the left what's the Y
approach from the right and from the
left a it approaches a so from the right
our function the Y value is going
towards a true from the left the y-
value is going towards a does the limit
exist it's the same from left and right
and it's equal to how
much
cool so here's
one here's
two says the limit this is really
interesting it says the limit of a very
like X just just a simple function like
this is the same thing as if I were to
just plug that number in so if if I'm
taking the the limit of some function x
uh at at some point a the answer is just
a that's it that's kind of nice right
why because graphically we can see
that last
one amazingly with with this stuff uh
we'll be able to do everything else it's
kind of
cool this I'm just going to write it out
because we just covered it a second ago
this is why we covered a second ago
how much is the limit of 1 /x as X
approaches zero from the left hand side
it was on the board just a couple
minutes ago how much is that very good
negative
Infinity how much is the limit of 1x as
X approach to Z from the right hand side
what was
that now last thing we're going to do
firstly are you okay with these three
basics
okay these two this one we already knew
these two are kind of interesting
they're really unique and we're going to
use those to solve some limits here in
just a bit to evaluate some limits I
should say before we do that though I
got to tell you the properties of limits
we're going to leave at that for today
and I'll show you how to apply these
next
time
so so basic properties
we're going to assume that we have two
functions and that the limits exist for
each one so the limit of f ofx as X
approaches a
exists the limit of G of x as X
approaches
a exists notice notice please that these
limits are that the x is going
approaching the same exact value you see
that the a is the same it's got to be
the same for this really to work okay so
it's got to be the same approach M I
just made that word up I don't s got to
push same same number here's your
properties there's basically five things
we can do they're all quite interesting
first one
this is why both of those have to exist
is because we're going to do this if you
have a limit you're not going to see why
this is really interesting until next
time so make sure you come back on
Friday uh if you have a
limit where you have a function inside
of it added to another function no
matter what that function is you can
separate that function so for instance
you can do the limit as X approaches a
of f ofx
plus the
limit as X approaches a of G of
X that's legal you can separate limits
by addition it's kind of neat right by
the way this is why you had to have the
same letter here because we we have to
do that and you can also combine them
that's legal to do as well so these are
not oneway streets um let's see I'm
going to make this a little bit easier
for us I'm not write the second one I
just need you to know that this works
for addition and subtraction that than
nice it also works for multiplication
you can separate limits by
multiplication
so that's legal to
do you can also separate them by
division which means if you have limit
as x appr a of FX GX that's the same
thing as the limit of f
ofx over the
limit of G of x
that's true but you have to have
something what can't this equal you
can't have that so the
limit of G ofx cannot be
zero last one the one that I find the
most interesting this is really
cool I know we're a minute or two over
sorry about that uh if you have the
limit
raised to some power some function
raised to a power this is the coolest
one you're going to see why in next time
but you can actually do this this makes
it so you can do a limit of the
function and then raise it to the power
that's a neat one that means that you
can remove an exponent from your
function with limits that's very cool uh
what's what's fascinating about this is
it even works
with
this because that is an exponent right a
radical is an exponent you can remove a
radical I'm going to leave it like that
uh you can take a radical outside of
your
limit going go we're going to do many
examples of those things but have those
written down we'll start with that next
time well if you remember from last time
what we're talking about is limits and I
I think the last thing I did leave you
with was the properties of limits and
there's a few things that we can do with
our limits now now one thing we knew was
that from our basic limits our little
bases up there the limit of a constant
equals that constant that was real nice
also the limit of just a simple variable
like x equal whatever the X was
approaching that that number so those
are the two Basics I gave you and then I
gave you some properties basically with
our properties it says you can split up
limits by addition you can split up
limits by subtraction you can split up
limits by multiplication you can also do
it with division as long as the limit of
the denominator is not zero you can also
this is the coolest one I think anytime
you have a function taken to a power
inside of a limit you can take a limit
of the function and then take the power
afterwards so that means you can really
alleviate the problem of having an
exponent within any limit are you ready
to see how we can do that to to evaluate
that limit without doing those silly
like check a two in the middle and then
find out from the left and from the
right remember doing those right those
tables we don't want to do that anymore
we want to use the properties and our
basis for limits to be able to solve
this thing you ready to do it so here's
what we can do if we look at this thing
and say okay limit as X approaches two
of this function the first thing I know
is that I can separate limits by
addition and subtraction you with me so
what this says is okay well this can
become the limit as X approaches 2 of X
Cub minus the limit as X approaches 2 of
2x plus the limit as X approaches 2 of s
now a lot of people are going to ask do
I have to put the limit as X approaches
two every time yes yes you have to do it
it's like uh it's like asking someone to
take the absolute value you have to
write absolute value until you AB
actually take the absolute value of
something so you have to keep on writing
limit until you actually take the limit
if you don't well you're saying you've
taken the limit and that's what it was
so it's not a true statement so you have
to write the limit as you're doing these
problems now let's use those other rules
one other rule said that anytime this is
the cool one the one rule number
property five says anytime you have a
function to a power what I can actually
do is take the limit of just that
function and raise it to the the power
later that's true you can do that means
I can take the limit of x and then CU
whatever that happens to
be another thing said that I can
separate two limits by multiplication so
this would be the limit of
two times the
limit of
x plus that's already a constant we'll
deal with in just
a how many people are okay with this so
far do you see what we've done with
these properties we actually broken up a
pretty intense polom into limits of
constants and limits of just X terms
have you seen that what's interesting is
I want you to think about this can you
do this for any
polinomial you can always separate by
addition subtraction right you separate
term by term then can you always take
out exponents yes yes you can can you
always separate multiplication
absolutely so you could do this with any
polinomial you follow me on that one
well now check it out since we've broken
up into such simple components you can
now use those two B there's only two
Basics use those two Basics to actually
evaluate this
limit can you tell me what's the limit
of x as X approaches
2 it was one of those Basics that I gave
you I drew the the pictures of them it
said the limit of x as X approaches a
was a
it said the limit of C no matter what
was C you with me on that so limit of a
constant is the constant so limit of x
is a whatever it's approaching so you
all tell me now what's the limit of x as
X approaches two it's just two yeah two
and then I can Cube it minus oh how
about this one what's the limit of two
doesn't even matter what I'm approaching
well what's the limit of two as I'm
approaching
two the limit of a constant is the
constant so what's the limit of two two
it's two it never changes
two times uh how about this one what's
the limit of x as X approaches two it is
two plus what's the limit of seven no
matter what you're approaching what's a
limit of
seven you with me on the seven it's a
constant yeah so we had that those two
Basics limit of a constant is simply the
constant itself it's a horizontal line
the limits always be the same it's just
seven
can you add that together how much is
it 8 - 4 + 7 looks like 11 to me did you
get 11 as well that means we've just
calculated our very first limit without
doing tables and without looking at
graphs we're able to do
this now can I can I ask you do one
thing for me please just check this out
can you go ahead and just in your head
or off to the side take this number
two plug it in there and tell me what
you get can you do that for me what's F
of
two wait say that again 11 did y'all get
11 when you plugged in two wait that's
weird is that a coincidence no no it's
not a coincidence cuz here's what we can
do oh not two sorry
11 it says well if you can separate a
polom into its terms and then its
individual components raised to powers
and multiples like that essentially all
you're doing is plugging in that number
for those spots for your spots of X
right for your variables of X basically
you can do that here so here's what I'm
going to say you already determined I
already asked you this whether this
would work for every polinomial you said
that it
would it or wouldn't it can you do this
with every polinomial polom of things
that looks like that can you do it for
stuff like that absolutely you can
always separate them take Powers outside
break it up into a constant times a
variable that means that to find the
limit this an important thing to find
the limit of any polom all you have to
do is evaluate that function at the
point a whatever that is this is going
to work for every
polom so this idea works for every
polinomial is point a always given oh
yeah yeah and so it's never it won't be
it always be a given in the for
initially yes this number is always
given to you has to be
given works for any
polinomial in in plain uh well in
mathematics I'll say it in plain English
in just a bit here's basically basically
what all this stuff says it says that
the
limit as X approaches a of some
polom do you understand the notation the
limit of a a polinomial that's a
polinomial as X approaches some
number is equal
to P of a what's P of B
say that's exactly right says evaluate
the function at that point basically
here's what it says in English to find
the limit of a polom just plug in the
number that's what it says can youall do
that oh yeah that's great it says to
evaluate the limit of a polinomial plug
in the
value to find the
limit of polinomial
just plug in the
B which is a in our
case you want to see a few
more yeah hopefully you're not like nah
done get all the hard stuff just leave
like this you want the hard stuff don't
you
yes not really
yes limit of x 5th - 3x + 4 all to the 3
power as X approaches 2 that's what
we're asking here for now the question
is could you could you if if they wanted
you to could you do this whole process
and do the same thing yeah yeah
absolutely you can pull out that that
three that power three you see it you
can separate all the terms you can
separate out that power five you
separate the three and the X plug in
those numbers in those two spots and it
would be essentially just valuation of
that so how can we figure out this
limit just plug that number why don't
you do that on your own plug that number
in there substitute that in there
because we know that the limit is going
to equal whatever this function is with
evalu at the point AAL 2
was it 30 to 30 power or something like
that M
27
27,000 27,000 how many people got
27,000 20 is it
27,000 I hope so not everyone raise your
hand are you guys okay on how we got
those numbers I need to know if you're
not then I can re explain it but not
your head if you're okay on how we got
those numbers the 27,000
yes so do you see that what we're what
we're actually doing is saying okay I
know this limit is the same thing as
evaluation when I plug in the number
two and then Cube it 32
three yeah I know I do many twos 32 -
6 like 26 + 4 that's 30 30 to the thir
power and you get 27,000 you follow me
on that one okay and that's exactly what
that means hey you just evaluated your
first limit don't you feel
proud no oh well you're going to you're
going to you get a
plaque
sure eat a bunch of candy you get lots
of
plaque get it get
it plaque versus plaque oh it's so
whatever I
know but the way not all of our limits
go to two I'm just using two so you
know what about that now is that a
polom it can the answer is no no that's
a rational that's a rational function is
it a
polinomial but could you still do the
same basic thing and answer is yeah you
can because of I think it's rule number
four or your property number four three
or four one of those things which says
that you can actually separate your
division can't you provided that your
denominator is not what will that make
the denominator zero will this point
make that zero then you're fine then
you're fine you can do that so because
we can do this whole step this is the
limit as X approaches 2 of 4x^2 + 1 over
limit as X approaches 2 of x - 3 well
now look at it is that a
polinomial yes is that a
polinomial yeah they're both pols the
that's polinomial and that's a
polinomial so this says you can evaluate
that by plugging in two and you can
evaluate that by plugging in two the
only thing you got to check for is that
the denominator is not zero which we
already know right it can't be zero on
the denominator that would be a problem
so do you need to show me this step no
no just plug in the number if it works
then you're okay if you have zero on the
denominator then you're not
okay
okay do you get the idea though do you
see that we can separate division
provided that this number does not
evaluate that polinomial make it zero
then that's fine so we're going to
separate that we don't have to actually
show this part it's just I'm showing you
once that it is true because now it
makes it a polinomial it makes it a
polinomial and we know from here hey you
can evaluate a limit of a polinomial by
just substituting in that number so
therefore we can do it here uh have you
substituted that in yet
so it's going be 4 * 2^ 2 + 1 2 -
3 17
over7 notice how here I have to write
the limit here I have to write the limit
here I have to write the limit as soon
as you evaluate you don't write the
limit anymore do you guys see the
difference there so yes you must write
the limit until you actually evaluate it
once you do that you're done no more
limits you just have those numbers but
you have to write it up until that point
question um say that X was approaching
three would you still be able to you
would glad you ask that question can I
answer that in like 30 seconds okay okay
cool love those don't you love it like
when you're thinking about something and
you're going to make your next point
because you're a teacher you know that's
kind of cool you want to do then someone
goes hey what up with this you're like a
that's my next Point don't you love that
it doesn't happen to you it happens to
me all the time I love that
uh actually it's going to
be a minute and 45 seconds all
right do you have the time on that sheet
of paper
[Laughter]
maybe I'm not
OCD how about that one is
that a polinomial
please shake your head no no pols look
like this that's it that's pols no roots
no denominators that's pooms so where
this is definitely not a polinomial but
I want you to see what you can do do you
see that you could pull the cube root
out of it once you did that you could
separate by division like we did in this
problem so essentially anything will
work provided you plug the number in and
you don't have any domain issues so
basically if you plug this in it doesn't
make that zero you're fine it doesn't
make well actually that's a cube do even
matter cube roots you can plug in any
number that's fine so provided that
doesn't that's not zero if you plug in
one does this denominator go to zero no
then you're fine why don't you go ahead
and plug in one and see what we
have notice I have my limit here we
could do all this work right we could
separate the cube root we could do that
we could show that we could separate my
limits by division and show that as well
but you just need to know that we can
plug in a number into a rational or any
type of problem that we have provided
that we don't have that denominator zero
and don't run any into any domain issues
so if we plug in that value of one we
get the do we s the cube
root yeah don't forget about the cube
root how much does that give
you well if you're going to say it say
it how much it give you cube of six cube
root of six cube of
six pretty sure you were able to get the
cube root of
six okay I'm 10 12 seconds over now let
me show you this
problem firstly do you have any
questions on the whole polinomial idea
breaking that up any questions on what
we're allowed to do over here we're
allowed to uh evaluate limits by just
substituting numbers provided we don't
run into any problems problems would be
zero on a denominator or things that
don't validate our domain that would be
a
problem or sorry invalidate our that
don't invalidate
our I think this is now going to answer
your question
in every previous problem we were able
to just substitute in that number into
our limit because we knew we could break
it up into uh forms of pols or or
basically because they're pols Powers
with individual X's multiplied by
constants right and we could do those
limits that's why we were able to do
this stuff is because we could have
shown all that breaking it up and just
evaluated constant limits and limits
that just had x's and then taken the
powers
later the the only difference with this
one is if you try to do that if you
separate this by division the numerators
could work just fine because you're
going to get how much on the
numerator that's a polinomial right you
get zero is it okay to have zero on a
numerator sure yet on a numerator yes
now what would happen though if you try
to substitute in this two down
here zero the limit of your denominator
would be zero is that a good thing to
have that's a problem we can't ever have
that what do we
do the top is difference of squares the
top is a so wait a second though are you
saying
that if I do
this I know you all know a difference of
squares right what's it going to factor
as
are you saying that I should be able to
simplify those out of this
expression but wait a second wouldn't we
be simplifying out a domain issue you're
not
actually say that again say it not going
to actually ever get to the point two on
the gra so are we simplifying out a
domain issue not he's exactly right are
we actually getting to the point
two no so really this is kind of weird
but with a limit we're technically not
making anything any mistake here we're
not actually getting to two we're
actually getting that point so when we
look at this we go oh okay we could just
simplify that out we're not simplifying
out a domain problem because none exists
because we're not actually getting into
two you just can't evaluate the limit by
plugging it in because according to this
function yes you will at that point
that's the difference between pols that
don't have issues and things like this
that do that have domain problems if
you're trying to find the limit of this
sky as you're going two well there
there's a problem with that there's a
hole there right there there's do you
recognize firstly that there is a hole
and not an ASM toote here what was the
difference between a hole and an ASM
toote according to the
algebra oh you better know you got to
know that astes don't factor out say
that louder ASM totes don't factor out
you're right ASM tootes don't factor out
holes do factor out do you remember that
so if you can simplify it out that's a
hole if you can't you have some sort of
an ASM toote so this case yes we have a
hole that means that if you have a hole
in your graph can you actually find out
where the function is at that point you
can't even it doesn't even exist so we
have to do limits to find that out but
in order to to do that we do have to
simplify this this function we can
Factor we can simplify because we're
actually not getting to the two this is
okay it's okay because
x x never equals two never gets there
really close yes
so what we can do now is say all right
then this according to the limit is
exactly the same
thing as that is that now a
polinomial sure can you plug in two how
much you going to get four is
four that's kind of neat right oh this
should be amazing to you be like yeah we
can simplify this stuff out because
we're not actually getting there there
therefore we change these weird looking
uh holy functions not because they
happen on Sundays but because you know
we have and then we can just plug in the
number because now it's a polinomial and
we're good to go so if you can factor
and simplify then you can find the limit
does that make sense to you that's
awesome few more okay
we're going to make them a little bit
more advanced a little bit more advanced
Till by the end of the day we'll have
some pretty unique things going on
okay now before you get all crazy and
start factoring everything that you see
you might want to just check that point
and see if it's even a problem all right
because sometimes they might give you
that stuff and there's no issue it'd be
like this example this would have an
issue if we were going to the limit of x
equals or X is approaching three
wouldn't that be a problem but we're not
we're going to two so there was no
problem here so at least check and make
sure you actually have a problem before
you start factoring things out make
sense so try it do I have a problem if I
try to evaluate
at4 yes yeah I do I get Z over zero
right that means you should be able to
factor hey if you get 0 over Z I told
you this was the property of mathematics
if you get 0 over Z when plugg in the
same number that means x minus that
number is a root that means somewhere
over here I can guarantee you're going
to have an X+ 4 cuz x - -4 is x + 4 this
will be x + 4 * something this will be x
+ 4 * something and then we should be
able to simplify out that problem why
don't you try that right now factor that
I want you to simplify it and then see
if you can evaluate the limit of what
you get out of that
should be your quoti I suppose
did you
factor was there an X plus 4 like I
magically
predicted I love when that happens too
so if you were to factor this out by the
way have you noticed that when you're
factoring you still must have the limit
notation you see that on the board
you've got to have limit look at here's
a limit here's a limit here's a limit
when did I stop using
limit when I finally actually was able
to plug in the number do you see that
you must have a limit I will Mark you
points if you don't have those limits if
you go I'm going to write this limit but
not that one not that one I'm just going
to my answer mm that's not how limits
work if you say you don't have a limit
here you start Crossing stuff out and
you magically plug in you've made domain
issue if you're with limits that's okay
so you have to say I'm working with
limits here until you get down to this
point do you see the point all right so
we still have a
limit by the way that's going to be
annoying I know because you're going to
have lots of steps on some of these
limits but you need to do
it we're going to
factor we're going to factor
see anything that simplifies out of our
problem now some of you might be asking
well wait a
second don't I still have a domain issue
because if I plug in three I have a
problem don't I still have a domain
issue the answer is well yeah sure at
three but are we trying to figure out
what's happening around three I don't
care about three what point am I trying
to figure around yeah so I've gotten rid
of the problem cuz as soon as I cross
out this x + 4 now when I substitute in4
it's no longer an issue can you now
evaluate
at4 let's go ahead and do that notice
how I've written limit limit factored
limit still and now I can put well this
is going to equal 2 -4 - 3 I've now
evaluated my limit I don't write the
limit notation any longer because I I
don't have any more variables as soon as
you substitute it for your variable that
is your limit
we get -27 as
our how many people got 27 feel okay
with this is this understandable you
guys starting to get the the concept of
working with this stuff it's kind of
interesting isn't it what we can do I
hope it's interesting to
you oh how have it up
there
cool all right oh my gosh what are you
going to do first do you think
but before you factor don't get crazy on
factoring what are you going to do check
you're going to check it yeah check the
five because if it works well there's no
reason to factory that would just be a
waste of time probably confuse yourself
too like Wait nothing simplifies oh
no now if you try the five here do you
have a problem okay so if you do have a
problem then what do you
do now you factor so if you haven't
factored already go ahead and Factor
what's the top one factores
plus 2 -
5 how about the denominator what do we
got
there ohus 5us 5 all
right did you all Factor it
correctly all right thank God good at
least we factored it correctly do you
see anything that
simplifies wait not not both x 5 one so
we
go here and here yes and what we get out
of this this is important example for
you to see you get x +
2 x is still approaching 5 x - 5 try to
plug in the five what are you going to
get you going seven over zero is that
still a problem oh man so wait a second
what do you do if you still have a
domain issue on this thing I've now
given you two cases I've us in cases
where the problem the domain issue is a
hole do you see that it's a hole and you
can get rid of a hole it's a removal
discontinuity and you can find the limit
of that it's very easy you factor it and
then you evaluate now I've also given
you another case where not only do we
have a hole here we actually have what
is that when you can't simplify out the
domain issue that is a vertical ASM
toote so now we have this issue what are
we going to deal how are we going to do
that well this is where you're going to
use what's called a sign analysis test
listen you you really do have to
comprehend that if you can get a zero
over zero like you did over here you're
going to be able to at least Factor it
and simplify something right sometimes
as I've shown you it will get rid of the
problem area and you can just substitute
in a number and be just fine that
happened here and that happened here
now' your have be okay with that one now
other times sure we got 0 over zero we
factored it we got rid of a what looked
like a problem area and it was that's
like a hole however can you get rid of
this x - 5 in any way whatsoever if you
can't what was that called again it's
ASM toote what you have to sign analysis
test here's what you do with a sign
analysis
test you put the numbers that will make
the numerator and the denominator equal
to zero put that on a number line what
numbers am I talking about
here five definitely five because we
want to find out what's happened at five
and that was our problem area so five is
going to have to go over here here
somewhere so far so good now what else
do we need now ne-2 is interesting we
don't really need it cuz really we're
looking at this area however we need to
know what happens between this range you
see what you're going to do is called a
sign analysis test and that means you're
going to find the sign in this interval
and the sign in this interval the only
reason why we need this
number is to make sure that you pick
something in this range because if you
don't if you pick something in this
range this is where the signs change
from positive to negative does that make
sense to you so you have to find
something in this little range and this
range over here it could be anything
over here but in this range it's got to
be between these two numbers that's why
we put the -2 we're not taking a limit
to2 we don't care really we don't care
we want to find out what happens at the
five is that clear for
you are you talking about like the trick
function sign no no this is easier okay
there's no trick function here okay
function I'm going okay see get there
where's these Tri functions coming from
I hate calculus no no trick functions
this is basic basic okay this is just
plugging in some
numbers sign as in I am signing okay
sign
like like a sign like sign oh I get it
sign like sin we're not
sinning we're signing here we want the S
let me WR that
sign analysis analysis is another one of
those things you can't really abbreviate
that well
anal got it the
sign
analysis that'll go viral on YouTube
that' be
awesome so again the reason why you have
to have the negative -2 is because your
sign will change in these two intervals
and we don't care what happens here but
we want to make sure we pick the correct
number for this interval we don't want
to be picking ne3 because it's going to
give us the wrong interpretation of what
happens around this now you have to know
what happens around the five and you
already said it once or twice before is
this a whole or an ASM toote as it's an
ASM toote because I can't get rid of it
you follow so you know for a fact it
looks like
this you know for a fact that you you
have an ASM toote right here right so
you're going to be in one of four
situations watch carefully you you are
either going to be going like
this true or like
this or like this or like that it's one
of those four situations and the sign
analysis will tell you what it is here's
how you do it I want you to take a
number in this interval what's a good
number in this interval pick six you
can't pick five you well obviously you
can't pick five pick six plug in six and
tell me what you
get plug it in not the original but no
no you know what what you've done here
is you just simplified out a whole it's
going to give you the same exact answer
so plug it in here what' you get you got
eight CU you did eight wait what' you
get oh right8
over one one so eight over one is
positive 8 that's what I'm looking for I
really don't care about the value I care
about positive or negative what you get
positive or negative make sure you know
how to deal with your signs okay so you
you got a
positive you know what that means if you
got a positive are you going to be going
down to negative infinity or up to
positive Infinity definitely positive
Infinity because you're doing positives
here you know it's an ASM toote already
somehow you're going to be looking like
that you okay with that so far okay now
try another number in this interval
again the reason why we have aa2 is to
stop us from going too far so what's a
number in
here pick a better number than four pick
zero zero is a great number interval so
plug in zero what' you
get 2 so is it positive or
negative am I going to be going up to
meet that other ASM toote at Infinity
I'm going to be
going I don't really care how the graph
looks honestly I don't I just really
care what the ASM tootes are what what
the function is doing around the ASM
toote because that's the limit is what's
it doing around the number do I need to
plug in a number over here pass -2 no
this is just to stop me so so you got to
tell me does the limit exist cuz we're
talking about five does the limit exist
at five or not no if it did they'd both
be going up or they both be going down
okay if they're like this then it
doesn't exist but this is how you show
that it doesn't exist you don't just
automatically go I can't do it does not
exist okay because you could be wrong
all right it happened
before it does not exist because you
showed it with your sign analysis test
so on your test when I give you a
problem very much like this which I will
do and I ask you to show your work or I
might even say show a sign analysis
that's what I'm talking about I need you
to prove to me prove to me that that
limit does not exist so if they say show
the limit doesn't exist that's it okay
if the limit does exist you'll be able
to simplify out the holes if it if it
doesn't exist well then you have to show
a sign analysis test show that you have
an ASM toote and it's going like that or
some other way that you you cannot make
those things go together how many people
understood that what we just talking
about good all right couple notes for
you so I'll write this out so you can
remember
later first
one if you get 0 over Z what that means
is you have a common factor at that
point X Plus x minus that point so if
you get 0 over 0 what you're going to
try to do is
factor and
simplify factor and
simplify 90% of the time that's going to
work just fine that's these examples
over here all right the other percent of
the time let's say you factored you
simplified you still have an issue so
that's number
two if you cannot simplify the problem
okay if you can't get rid of the problem
if you can't we'll say it this way if
you can't cancel the problem because
youall love the word cancel right if you
can't cancel the
problem that means it's not a hole if
you can cancel the problem it's a hole
if you can't cancel a problem that means
you have an ASM toote at that point what
you're going to need to do is evaluate
with a sign analysis so two two
situations holes cancel them great done
not holes those are ASM tootes sign
analysis will show you what that does
around the ASM toote if they're go into
the same thing it exists positive netive
Infinity if it's not then it doesn't
exist and that's the only way you can
show that for
us so if you cannot cancel the problem
on the denominator check with sign
analysis because the limit might not
exist
because a limit might not
exist how many people feel okay with
what we talked about so far
okay I'm going to show you a couple more
very unique examples we'll talk about
those um things you you
really have to know exactly what to do
uh when you see
them so I to make sure you see
them I've started writing in all caps
thanks to you you're
welcome my handwriting is so
atrocious that means bad
so condescending when I just did you
that's horrible um sorry um it was
really bad until I started writing all
all caps and now I have trouble even
writing limit like like
that also my dad was a carpenter and a a
draftsman and so they used he used all
caps all the time so I want to be like
him so I started doing that too he's
much better at
it all right now we got this this
situation
here limit x -1 over square root of x oh
my gosh well the first thing you try to
do is
what try to plug it in do you have an
issue now you have Z over zero true
unfortunately for us these aren't easy
to factor right you do you probably do
have some sort of factor out of that but
it's not easy to do polom are easy to
factor that's fine but with this that
that theorem really doesn't hold that
much water because it's not easy to
factor that
so okay what are we going to do any
ideas conjugate have you ever heard of
the conjugate the conjugate is those two
terms with a different sign in the
middle if we do that that's one way you
learn in your intermediate re course on
how to rationalize a denominator or
rationalize a numerator or basically
just rationalize out a square root so if
we do that that might help
us so don't forget the algebra that you
know sometimes we can rationalize if we
do we'll rationalize < TK
X -
1 oops sorry plus one over > x + 1 you
have to use the conjugate though other
if if you don't well then you're going
to just make your problem worse because
you're not going to get rid of a square
root it's going to be within a middle
term of your problem so you have to
alternate those signs you guys okay with
that you sure so use that conjugate now
let's
see of course we have parenthesis
here now I'm going to give you little
piece of
advice generally generally you don't
want to distribute in this case the
numerator if you're a rationalizer the
numerator you wouldn't want to
distribute the denominator because
ultimately you're trying to simplify out
something are you with me on that you're
trying to simplify that out so I'm not
going to I'm purposely not going to
distribute the numerator unless I
absolutely have to if I run into an
issue so right now I'm going to leave
this as x -1
and theun of x + 1 why because I'm
trying to simplify stuff that's why now
on the denominator you tell me uh when I
what do I have to do with this I do have
to foil or or distribute so that means
every ter times every term U tell me
what's otk of x * X uhuh now do you see
what happens and why we use the
conjugate if you distribute this we get
X we get plus < tkx we get minus rootx
what's going to happen with this that
that's why we use the conjugate and then
last you're going to get
what you get xus one do you see why we
don't distribute the
numerator if you distribute the
numerator you got to you got to mess of
crap up there right you're going have to
refactor it's not going to be easy if
you don't distribute it the factoring is
obvious the simplification is obvious
what are you going to
simplify well at least I hope it's
obvious is it obvious to you what you're
going to cross out all right good wo
yeah kind of
obvious all right well our limit as X
approaches one notice I'm still writing
limit is the what what what do we have
left are you going to have an issue oops
that doesn't look like an
X are you going to have an issue if you
plug in
one it's not negative it's positive one
so we're okay we have no denominator
anymore because we rationalized it it
went away because we were able to
simplify it if we plug in one what are
you going to
get that's your
limit how many people feel okay with
this so far would you like to try one on
your
own let's do that I'm going to give you
a little bit more complicated of one not
too
bad and if you get stuck on it no big
deal that's fine but I want you to at
least think about it while you're here
this will be where we end
today so limit as X approaches Z of < TK
1 + x - 1/ X are you going to have an
issue here yeah I mean straight up we're
going to zero and that's over zero
that's a problem so what could you
possibly
do run
away yeah done no we're not going to
just leave the problem we can't plug in
zero the only thing you can do do here
is what do you
think can we use that same idea but only
in reverse let's try that go ahead and
do that multiply by the conjugate
conjugates have to have different signs
they have to that's got to be the same
thing on the numerator denominator it
has to otherwise you're not multiplying
by one and if you're not multiplying by
one you're changing the problem you
can't change the
problem also one more thing I need you
to look up here at the board when you do
this that sign doesn't change it's only
the thing after the square root so this
stays the
same did you all multiply by exactly
that do you see how that is the
conjugate we have the square root
whatever that sign is and that whatever
that constant is that's what we have
here the square root the different sign
same constant same exact thing why you
why you need the same exact thing in
case you're wondering well Mr Leonard
why don't you change that sign as well
what you're trying to do is multiply
this in such a way that you actually
eliminate the root so when you multiply
this one times this one the whole entire
root goes away right the only time you
can do that is if the roots are
identical so you can't have different
signs otherwise they're not
identical hey which one aren't we going
to distribute here the numerator or the
denominator denominator don't distribute
the
denominator by the way I'm saving your
lives here if you distribute the
denominator you have to factor it again
that waste time so literally I'm saving
your time there go your life nice
right I know I'm such a nice guy today
so on the denominator I know I'm going
to have X and then the < TK of 1 + x + 1
I'm not going to distribute that the
numerator yes you're going to distribute
the thing you're trying to rationalize
so if you were to distribute this once
youall help me along here when I
distribute this what's the first
expression I'm going to get
1 + x very good okay and then
what so I'm going to getun
x 1 + x positive < TK 1 + x negative
that's going to be gone that's what the
conjugate does for you and lastly I'm
going to
get so this is 1 + x -
1 anything else we can do with that
like term anything else yeah combine
like terms what do you get if you
combine like terms on the
numerator that's kind of nice cuz one
and negative one and this is why we
didn't distribute the denominator
because if you look at that that's what
we're trying to simplify out right
trying to get rid of that 99% of the
time this works out for
you I wrote it back it doesn't matter
now that you see the X and X as a factor
on the denominator those things are gone
what's on your numerator
please oh zero is not on your numerator
what's on your numerator come on now
people one one yeah when you cross
something out you don't get a zero
you're actually factoring that out
saying X overx is one so you have a one
up there uh by the way please don't make
the the intermediate algebra mistake of
doing
this a lot of people do this when
they're just beginning they go oh yeah I
cross everything out therefore I
have that is that true no no you don't
have that you actually have
this if you forget that what you're
going to end up is end up with is the
reciprocal of the answer that you
actually want that's not
good hey now can you plug in zero and be
okay yeah even though you still have X
on the denominator look what happens
what's 1+ 0
one sare root of one plus
two said
two I was one step ahead of
myself I'll just write on the board
there you go
one2 would your ra have feel okay with
our our limits good deal now you know we
went a long way today we now know how to
evaluate and compute these limits uh
next time I'll show you how to do some
peace wise limits I'll show you some
trigonometric limits and then we're off
to a fun start did you have fun
today you have fun
today all right so welcome back we're
talking about limits uh we're going to
start talking about peace wise limits
now now for us we found out that when
we're taking a limit usually we can
cancel out a problem or if we can't
we'll use a sign analysis test because
we have an ASM toote hopefully you
practice that in your homework were you
okay with that idea so generally what
we're trying to do is plug in the number
plug in the where X is approaching if it
works great that's your limit if it
doesn't work well then you have to
factor do something to find that limit
typically factoring we'll be able to
simplify that a little bit or we found
out we can rationalize denominators and
numerators if we have square roots we do
things like that to work around that
limit now what we're going to talk about
today is some some different aspects to
that we're going to talk about peie wise
limits I'll show you what you can do
with those things it's going to be kind
of nice just go follow me on this then
we'll talk about trigonometric limits
and that'll that'll end our day we have
a lot to talk about on trigonometric
limits so piecewise limits here's
basically the idea when you're talking
about pie wise limits and peie wise of
course means different functions all
matched together that have different
directions for each part of it you have
seen PE wise functions before PE wise
limit says what's the limit as we're
approaching that interchange basically
does it exist does it not exist what
what what does it happen to be so our
idea is we're going to have to take some
one-sided limits because each piece is
different right we're going to take
one-sided limits and see if they match
up if they match up great limit exists
if they don't match up then no the limit
doesn't exist are you with me on this
I'll give you a real nice way to do this
a graphic organizer hopefully this will
help you out and then of course if
there's questions man ask so let's let's
talk about this our idea is we're going
to find some one-sided limits and we're
going to see if they are equal
let's start off with an example let's
say that I give you
this I say that your function is
actually made up of three parts
the first part says you're going to do
1/ x + 2 if x is less
than2 I say you have a different range
you have x^2 -
5 if we're between -2 and pos3
and lastly our last little step we're
going to be the < TK of x + 13 if x is
greater than
3 now because we have a peie wise limit
we basically have in this case three
different functions that means we can't
just look at this thing Hammer at it one
function and find out what is the limit
uh of of any particular
place what I'm going to ask you for is
can we find the limit at two and three
as X approaches two and three why two
and three well if you look up here
that's the only place that could po
possibly have an issue right is -2 that
would have an issue whether everywhere
else it's continuous the limit's going
to exist no problem same thing here
that's continuous everywhere only
problem would be maybe at the end points
of two and three here the only problem
could be at three with that function so
we're going to talk about the limits as
X approaches -2 and X approaches pos3
are you with me on that so that that's
our idea here what we're going to have
to do is find our left side limit for
each function and a right side limit and
see if they meet up somewhere if they
meet up at those points now here's the
way that I like to do this first thing I
like to do is draw a representation of
your of your graph of your number line
basically just like that break it up
into the places where your pie wise
functions broken up what are those key
points what numbers delineates our
function from the next piece of the
function where does one function start
and one function stop
basically two that's a key point right
so we're going to have to have -2 on
here somewhere what's another key point
for
us let's call this one function one and
function two and function
three can you tell me what function is
in this range over here what what
function takes over for this
interval function three because it says
for any X is bigger than three I'm Now
function three does that make sense to
you so I know that I'm looking at
function three
here what interval or sorry which
function takes over in this
interval function two for
sure and that leaves this one with F1
let's make sure X is less than -2
function one takes over do you
understand that this is how actually our
graph should look should be function one
then function two then function three
and I'll draw this graph for you at the
end uh just so you see what this really
does look like you with me still so far
though so here's the idea if we want to
find the
limit as X approaches -2 and the limit
as X approaches -3 we don't have to find
one side limits for all three functions
just the two functions that are
approaching that number number so for
instance if I want to find the limit as
X approaches -2 if I'm going from the
left if I'm going from the left which
function am I going to
use function what from the left I'd be
using function one does that make sense
to you now from the right as we're
approaching -2 which function am I using
from the
right function two yeah that's right so
so around this function I'm going to
have a one side limit from the left
using F of one uh sorry the first
function then I'm GNA have a one side
limit from the right using the second
function does that make sense to you
it's a way to picture this how about
around three we want X to approach three
how about you guys over here what
function are we going to use as we
approach three from the left function
function two that's right that's in the
interval how about from the right
function function three we're not using
function one over here are we we're not
using function three over here we're
just looking around that that number
what functions we have so now because we
have this gives you a pretty good idea
of what your graph looks like right this
is how you make up your one-sided limits
now okay so I'm going to
take a limit as X approaches -2 from the
left a limit as X approaches -2 from the
right and I'm going to see if those two
things are equal if those are
equal then we'll have a limit as X
approaches -2 in
if not well then we
won't do you all understand the idea so
far are there any questions so far on
the
idea okay so let's go ahead let's try
this um what's the specific function I'm
going to use for this one this is -2
from the left hand side you all said
that was function which one function
what's function one one
now write
that this one by the way is going to be
the
hardest rest a reason how about going
from -2 from the right hand side what
function would we be
using isn't this kind of nice you just
look at that can't you kind of cool look
at that one uh Now function two what's
the actual function
two so write that
so we said for this range from the left
Function One this range from the right
function two we're going to see what
these are if they're the same thing then
our limit will exist if they're not the
same thing will our limit exist okay so
we have that idea down now how about we
set the other ones as well let's start
talking about the as we approach three
so we're going to need again a limit as
X approaches three from the
left three from the
right if they are the same we'll have a
limit at
three let's fill out the functions
though let's fill up the functions as we
approach everybody three from the left
hand side which function are we
using and function two again you said
that was
x^2 - 5 so we'll write that
how about from the right hand side what
function are we
using function three what's function
three I'm assuming your mumbling was
sare otk of x + 13 was that what you
were mumbling I 13
okay now here's the deal I'm going to
work this side this these ones first
then this one cuz that one's a little
bit more difficult let me pait something
to you what I'm going to say is that if
a limit exists a one-sided limit
definitely exists would you agree to
that if a limit overall exists then a
one-sided limit will certainly exist
agreed so a limit existing is stronger
than a one-sided limit existing does
that make sense to you well here's the
cool part about that then and this a lot
of you were asking how do I find one
side limits check it out if you know for
sure that a limit exists then the one
side limit will exist plugin three
ignore the from the left plug in three
what do you
get you get four guess what that's your
limit why because does the limit of
three sorry does the limit as X
approaches 3 exist for that function
absolutely without equivocation because
that right there's a polinomial right
and you know with polinomial is you can
just plug in a number unless you have a
problem well if the limit at three
exists the limit from the left certainly
has to it must that's a weaker statement
plug in three you're going to get four
does that make sense to
you let's try it
next again if a limit exists a one side
limit will also exist at the same number
so ignore the the Plus for a second can
you plug in three without a problem
there what are you going to get you're
going to get four saying the limit
exists therefore the limit from the
right well it has to exist if the limit
exists the right and left side limit for
sure exists they have to go to that same
number that's going to be four now
you're able to answer your wasn't that
nice and easy just plug them in that's
it just plug them in if you can just
plug them in nice and easy does this
limit
exist
why yeah I don't care that they're not
the same function they go to the same
value limit is four
ra you have you're okay with that one so
far kind of nice right one side limits
aren't so bad P wise functions they can
get kind of messy we find that over here
but not too too bad do you have any
questions on this because I'm going to
erase it because I need the room any
questions on that okay so again if a
limit exists a one- side limit will
exist find out your one- side limits if
they're the same then your limit at that
point for sure you got
it now let's start over
here can I plug this into that function
yeah absolutely that's a polinomial so
ignore the the right plug in -2 how much
are you going to
get
one absolutely negative one there's the
issue okay here's the issue uh ignore
from the left can I plug in -2 aha so
what you're saying is that the limit at
-2 you don't know whether that exists or
not right you don't know what's that
mean can you can can you cross out any
part of this can you factor and cross it
out
okay so think back what do you use if
you can't cross out a problem what do
you
use the thing you couldn't abbrev that's
right the sign analysis thing you have
to use that something you can abbreviate
it it's just going to look funny in your
paper but you have to use a sign
analysis here so if you have a sign
analysis at -2 what we care about what's
Happening Here what is happening at -2
do you have a hole do you have an ASM
toote which
one folks you all should be a telling
that you all should know that at this
point if cannot cross out your problem
what is it it's an ASM toote so what do
you have as you approach
-2 it is certainly an ASM toote yes it's
we just want to know is it going upwards
or is it going
downwards what could you do to find that
out any value any value to the left over
here so plug in -3 if you plug in3 to
this function are you getting a positive
or A negative negative definitely
negative so am I going upwards or
downwards like that right so here's what
you're just finding out what does this
limit I don't care from the right look
why don't I care about this why don't I
care is this the same function on this
side of that -2 is that the same
function over there no no that's we've
already taken care of that one I don't
care about that I just care about this
so what is this limit equal can you tell
me yeah as we go towards -2 from the
left hand side it is going like that
that's going to towards
where so let me recap just a little
little bit what you're doing you're
breaking up your interval you have three
different functions in this case you set
up your limits around those breakoff for
your your intervals you use left side
limits you use right side limits and you
see if they're the same some of them are
going to be easy like what I showed you
over here others of them you might have
to do a little bit of work these ones
you can just plug in that one you can
just plug in this one if you can't cross
anything out and eliminate the problem
you've got to use a sign analysis test
don't get stuck on that okay you're
going to have something like this on
your test don't get stuck on what do I
do oh no I can't plug in -2 what now
sign analysis test if you don't know
what to
do you don't got to throw things jeez my
goodness
you're in
luck it
surved you just got that yeah now you
just got that with some nice marks on it
congratulations don't just get a new
motorcycle
okay anyway uh so this one yes you plug
this in this one well you can't cross
anything out you have to do a S analysis
test do what you know don't get stuck on
it if you can't cross anything out sign
analysis that's that's the only two
things you got okay so we have this we
were able to plug in number for the left
of our interval that's the side we were
wanting went to negative Infinity do
does the limit exist for this this is1
that's netive Infinity last time I
checked those things aren't exactly the
same so this does not
exist but that folks is how you check
you don't check by plugging in numbers
okay randomly you check by showing your
work like this this is showing your work
does that make sense to you I don't want
little tables or anything like that see
look the numbers are not the same I
don't care I want you to show me what
this is now the way this graph
looks if you want to see why this
doesn't exist uh what this thing is is a
parabola part of
Parabola that's symmetrical around the
y- axis and
intersects it's going
from -2 non- inclusively to three
inclusively so if you plug in -2 what we
get
1 looks like that and three if we plug
in three uh we get out of that
four that's not really accurate that
part can't cross three and end at three
but whatever then from there on out we
have the square root of X+ 13 which
starts at four and takes off something
like that
and then we have this one which at this
point I'm sorry this one this one over x
+ 2 if you graph that that's some sort
of descending function that goes like
that at that point does the limit exists
as we approach -2 no this is a rough
sketch by the way does the limit exist
two absolutely because that was that was
closed off by the way it's a function
because that's not equal to and that is
that's okay we don't fail that part uh
this part would have had the open c
circle around it but it's filled in by
that point so we have this Parabola
limit exists there absolutely limit
exists here no netive infinity and a
value Rich have you're able to follow
that feel okay with it good deal are you
ready for some
trigonometry
no I know it's supposed to be
trigonometry Tuesday but we're we're day
early so it's all good was that your
drinking party idea drinking party IDE I
don't have drinking parties
koola limits of tree
functions okay there's a couple things
you have to buy in on for me to do this
properly and the first thing you got to
buy in on is that s and cosine are
continuous everywhere continuous means
you can draw it without lifting your
pencil off the paper can you draw s and
cosine such that they are continuous do
you ever lift your pencil off the paper
no so they are continuous so first thing
this is going to come up later in our
class but I want to say it now s and
cosine are continuous everywhere
tra
tra so because we can do that because
there are no problems we can use
something that we use for pols you see
pols were continuous everywhere if you
think about any polinomial I'm not
talking about a rational function okay
I'm not talking about denominators I'm
not talking about roots that we have
over here this is a rational function
clearly it's not continuous everywhere
-2 fails this is not not a continuous
ever because it doesn't even exist for
part of it okay so we're not talking
about that what we're talking about is
continuous functions like this one
that's continuous everywhere you're not
going to have a problem no holes no ASM
tootes agreed s and cosine we just said
are the same thing therefore if we have
S and cosine are continuous everywhere
we can apply the same logic and say then
the
limit of s of X as X approaches a is
what do you
think it's not a it's not a you wouldn't
say this you wouldn't say the limit as X
approaches 5 of x^2 - 5 is 5 that's not
true what would you do to find it plug
in you would plug it in so to find the
limit of this what what we do we plugged
in right it wasn't -2 it's what we got
after we plugged it in so it's not a
what is it a s of a absolutely
likewise because cosine is also
continuous everywhere we can do the same
thing
oops yeah that's true it says that you
don't nothing bad happens
basically that would be something bad
happening
there we go that looks a little bit
better what about
tangent
tangent so my question is is tangent
continuous everywhere draw tangent with
your hands how's it go
look like you're
dancing yeah you doing I should have
used that I went out dancing the other
night I should have used my my SC curves
it would have been the hit of the
party probably
not maybe probably not okay so we got to
do a little bit more work since it's not
continuous everywhere we're going to
have to break this down a little bit so
the
limit of tan X as X approaches he let's
figure this out tell me what is
tangent use an identity what's
tangent sure so we know for a fact that
the limit of tangent is the same thing
as the limit of
s over
cosine
true very true very
true now there was one property of
limits that said you can separate limits
by division remember that property kind
of cool let's do it
okay okay so limit of s limit of cosine
do we now know what the ignore the
bottom do we know what the limit of s x
is as X approaches a sin a sure so this
says okay I've got sin
a what's the limit of cosine as X
approaches
a what's s over
cosine Aha and you go yay what's the one
thing I've screwed up
on be Z say say that again can't be zero
why can't cosine be
zero that was the one thing that one
little condition about that property if
you remember it said you can do that
provided that your denominator when you
split that up doesn't ever equal zero
and so we'd say sure the
limit the limit of tan X as X approaches
a is T
A except at certain points at certain
points and that's where cosine would
equal zero uh that's specifically is the
reason why you get all those Asm tootes
on your tangent line is because tangent
is s over cosine so naturally the sign
over cosine and cos equals z you have an
ASM toote remember the ASM toote idea if
you can't cross it out you get an ASM
toote right that works all time so
that's why we get those ASM tootes going
on and those ASM tootes occur
at X cannot equal or we will have a a
discontinuity uh because right here oh
let me draw some purple so you see it
from the
different cosine X cannot equal zero it
does you have a problem where that
happens is uh every plus and minus < / 2
so piun / 2 3K 2 things like that X or a
or either one cosine X can't equal zero
at those points you could say x can't
equal these things
so or cosine cosine
a really doesn't matter say cos a a
little more
sense cosine a can't equal zero does you
get a problem
so tangent would even be continuous at
at these points
um plus orus pi/ 2 because when you put
those into cosine you get zero plus or
minus 3 Pi 2 and so
on and you can really you can see this
in the domain of tangent if you look at
the domain of tangent that's again
that's why you get those ASM toes every
pi/ 2 remember tangent power two astop 3
power two asmt and so on and so on and
so on and negatively as
well you guys all right with this so far
this is our Basics you guys okay with
our Basics start ramping it up a little
bit you're like no no this is fine let's
just stick with this good we need to
practice this a
lot too
bad said
disappoint a lot in this
car there you go that looks messy enough
didn't it how about that yes
sweet do
this okay uh what in the world let's
just give up call it a day you ready out
screw this stuff math just watch
football can't do math and football
they're oxymorons mathematical
football
oons brain injuries and football that
goes together not
math work okay what can you do with this
well I'll tell you something uh you can
this is interesting but if a function is
continuous you can treat it like a
composition is co sign continuous
everywhere so what you can do with this
is kind of cool what you can do is say
all right s cosine continuous
everywhere you can do a
composition cosine is continuous
so by composition
actually reverse composition look what
you can do you can say instead of taking
a limit of cosine of something you can
actually do this it's very much like
removing an exponent from your limit you
can say I don't want to deal with that I
want to deal with cosine of my
limit that's
legal why would you rather deal with
that have you seen things like this
before yeah actually I think I've given
you that exact same example so far in
this class you've seen things like that
can you do this limit so as long as you
can do this limit and don't forget about
your cosine you'll be
fine it's interesting to think that the
angle now is a limit that's weird
right so cosine of what are you going to
do with that limit what are you going to
do first you're going to try to before
you factor you're going to try to try to
plug it in I mean don't may as well just
try that if you plug in one though
you're going to get 0 over 0 what does 0
over 0 tell you is factoring going to
work or not yes 0 over Z tells you that
so you're going to factor it probably
going to work now we've seen one case
where it doesn't where it was a double
root okay that that happens but
factoring is something you should try
first so we'll have the
limit X goes to one of well I know
youall can Factor we got x + 1 we got x
-
one what can you do now
sure what was that discontinuity that we
just talked about was that a hole or an
ASM toote in this particular
case hole ohle sure because you can
cross it out and there's no more problem
left that was a hole now um other
question let's see oh I had one more was
on my mind ah forgot it cross it
out do you have any problems left
notice how we have to keep writing limit
until we actually evaluate the limit
that's important part of
it what can you do can you plug in the
number and be okay so then what we have
right
here is cosine of I know I can just plug
in the one 1 + 1 what's cosine of 1 + 1
cosine of two
done cosine of
two we can do one plus one I like one
plus
one how do you feel okay with this one
so we can treat that like a composition
no we just insert the one with the
initial problem why do we have to pull
the cosine out can you just let keep
writing it the same way and come to the
exact same answer yeah would that be
still an acceptable way of writing it I
don't know okay just showing you you can
do it cuz a lot of people go what do I
do with that you go well can you still
Factor it can you still cross stuff out
and the answer is well technically no
because it's inside of cosine but as
soon as I move the cosine outside see
what we're dealing with now is a limit
and now it's not a domain problem
because you're not actually approaching
the one so yeah then you can does that
make sense okay so moving the cosine out
is just a matter of semantics basically
yes yes but important semantics
nonetheless so can you do it yes you can
do it does that answer your question
all right would you like to travel
more answer is yes most of the time
yeah unless I'm really
tired never came
down one could always
wish I don't even drink coffee oh man if
I drink coffee you guys would be in
trouble if I drink coffee that day it's
like
crazy yeah so basically do you actually
have to pull out the cosine yeah you can
you can that's the reason why this is
allowed to do it do you have to show me
that I don't really care um it what it
basically says is if you can plug in the
number you can do it that's that's what
cosine being continuous everywhere says
to do also says that you can still cross
stuff out as you're going through your
problem because it's a composition it's
continuous everywhere you can pull that
out does that make sense to you
so can we try to plug in a number here
is that okay sure as long as we don't
run into an issue IE you you plug in
something that's not even in the domain
like a negative root or you have a
denominator that's zero that would be
the only issues you have do you have an
issue if you plug in pi over2 here no
actually if you look at this that's a
polinomial right plus a function that's
continuous everywhere that means you
could split that up by addition we knew
limit rules for that you could do this
one by itself just plug in Pi / 2 you
can plug in pi2 to cosine because it's
continuous everywhere not a problem so
that says that just like other limits
that we're dealing with try plug it in
the problem first if you can no problem
so that would be
3 Pi / 2^
2 plus
cosine pi/
2 uh what's all this going to give
you not as a decimal
oh come on what's pi over 2^
2 you can say pi^ squ it's
okay Pi s over 4 very good so this is
going to give
you 3 pi^ 2 over four are you following
me on that okay what's that going to
give you don't say that loud I want
youall to think about it you need to
know
that I said don't say loud and hold it
in zero notice how at this point I don't
have to write limit anymore I actually
inserted the pi/ two you guys see
that that's your answer so once you once
you're no longer dealing with variables
you don't have to keep writing limits as
soon as you evaluate your limit at at
that point no more limits but what if
you keep doing step after step you know
because like I I write it out I do this
part of the math and this part of the
math and I keep continuing on but once I
put in once there's no longer variables
then once you evaluate the limit normal
once you are able to plug in that number
no more
limits you've done the limit right
yeah over and over and over again well
you should you need to write it I had to
write it here I had to write it here
here but not here because I was able to
plug in one now what if we say we write
it too much are we on it there's no more
limit here so if we wrote limit in
there the whole thing is you got to know
what a limit actually is right you got
to know that a limit is what does the
function do as you're get into that
number you got to know that as you're
getting to that number it's PO number
consider everywhere you can actually
plug it in and that is your limit okay
you say this is still a limit of
something no it's
not that is the answer to your limit
question but it's that was a good
question though I'm glad you brought
that up because maybe other people were
confused about that that's a good
question remember for something a little
bit more
interesting you know what interesting
means m mathematics right harder harder
scarier scarier yeah it's a little bit
scary I'm going to show you something
it's kind of
cool it is not my Darth Vader mask was
pretty
cool no I'm
lying my parents once told me that I had
a stormtrooper mask but I've never been
able to find it so I know they're lying
they like oh yeah we bought that for you
it's it's like when you're like growing
up and you ask your parents have I ever
been to Disneyland like yeah but really
it was just a park down the street with
some guy named
Greg same
thing that's Mickey Mouse no that's
Uncle
Greg
okay
okay let's try
that let's try that oh okay let's go
through the process of finding limit
right right the first thing you try to
do is what yeah you try to plug it in so
you plug it in what's s of
Zero no sign of Z is not one sign Z is
zero over how much can you factor
sign you ever been able to factor sign
in your life no no no so can you factor
out anything and cross anything out can
I just cross out the x and x and get sin
yay it doesn't work okay sign has to
have something next to so if you do that
you have sin in mathematics that's not
acceptable you can't do that so we're
basically stuck there's no way to do
this Pro unless you want to just plug in
endless numbers there's no really way to
do this but we're going to find a way to
do it you ready to find a way to do it
some trigonometry here's our idea I'll
see if I remember how to do this in a
while first thing we're going to do oh
yeah we're going to bound this we're
going to make up a triangle
so let's here
what I'm going to
do is take part of my kind of a unit
circle not really make it
better that's
better I'm going to take my unit circle
what I'm going to do is take an
arbitrary angle I can't be specific
because of course you want this to work
all the time
so I'm going to draw that now what I'm
going to do from this I'm going to make
up two
triangles I'm going to drop a
perpendicular right from
here I'm also going to drop a
perpendicular it goes like that
okay so let me let me Define a couple
things
firstly we're going to call X our angle
second thing what do I need I need
this I need
that now what I'm going to tell you is
if we if we need to solve for that let's
see this is going to be
our t
x how far is this distance right here
from here here to here
why so would you agree
that stick with me here folks are you
okay that X is our angle forly I'm I'm
calling this length y Okie doy and and
this this is one from from here to
here that's
one give me the relation sh of tan X
then s over cosine yes in the specific
instance with a large
triangle tangent is opposite over
adjacent true what's the
opposite y over one one because that is
the unit
circle well interesting then y = Tan
x so this equals
Tan
x oh let's see one more
thing we also
need this distance I we need that
distance right there so let's
say what do we want to do this
coordinate is what's the coordinate on a
unit circle
well that would be for specific angle
what's the what is is it cosine or
cosine jeez I'm asking you come on you
got to know
this agree okay yes sinx is typically y
for inside Triangle
what that what that means is I I'll try
not to screw this up too much for
for that means that our our X for this
inside Triangle is cosine X do you
agree that means that our our Y is
actually sinx right now also what we're
going to be doing is looking at this
this
not this well this one gives our height
but also this one
that triangle right
there what we're going to be doing is
comparing the areas of these triangles
so let let me recap just in in 30 30
seconds what we've done we've gone ahead
we made up two actually three triangles
this one just gave us the
height that gave us the height this
would be cosine but more importantly
this one this height is sinx do you all
agree with that one furthermore if I
call this y y is equal to tan X because
of the relationship of X and one you
guys okay with that one okay now we're
ready to compare the triangles so we got
we've actually have three things going
on I need you see all three things first
we have a big triangle you see the big
triangle yes we have a smaller triangle
that's this one so we have big triangle
we have small triangle and then we have
sector do you agree with that so what we
want to do is consider the areas
we have big
triangle we have
sector and we have small
triangle can you tell me how you find
the area of a triangle it's area of a
triangle or base times height over two
yes one half Bas let's look at the big
triangle what's the uh what's the base
this is
one times what's the
height instead of Y let's say Tan
x and to find the triangle I divide by
two I need to show a hand see if you're
okay with the area of the large triangle
base time height over two we have basee
1 height is y yes but y = Tan x I'm
going to use Tan x cuz I want everything
in the same variable all in x's and then
divide by two uh let's do this small
triangle then we'll talk about the
sector small triangle that's this one
right here not this too small one but
this one right there are you with me on
that one
it's that
one and that's the big
one okay and then we'll talk about the
sector in a moment uh what's the base of
my smaller
triangle what's the height the height of
my smaller
triangle sin x not cosine X right cosine
X would be the sin x would be the height
of the smaller triangle because that is
actually a point on the unit circle do
you believe me so that would be sin
x and then well oh shoot sorry I'm off
point 1 * sin
x over two
are you okay with big triangle and small
triangle you all right with that one do
you know how to find the area of the
sector what are we talking which which
one of these we talking this is the
sector High
the are the sector minus the small
triangle sure how do you find the area
of
sector use a big triangle M there's
actually a formula for it it's this it
is your
radius times your angle
radius angle
two you ready for the math
magic ready you have to buy into one
thing you have to agree
that the area of the sector is in
between the areas of the big triangle
and the small triangle would you agree
with that it goes big triangle then
sector than small triangle true so what
we're saying here is that the area of
the small
triangle is less
than the air
dector Which is less than the area of
the big
triangle agreed or
not what can I do to the twos
If I multiply all those inequalities by
two which is legal what happens to the
twos twos are gone TW are gone so what
I'm going to do is change this into
s x is less than x is less than Tan
x some basic algebra basic
algebra you okay with that so
far now here's the cool part what I'm
going to do I'm going to divide each of
these three things by
sinx where sin x is positive it's in the
positive quadrant that's okay we're
talking about a positive angle right
here what that means is that we're not
going to change around these
inequalities so then this does
that you still okay with that so far we
divide everything by sin x how much is
Tan x over sin
x say it again 1/ cosine 1/ cosine
because Tan x is s over cosine yes
divided by sign that means you flip
multiply you're going to be getting rid
of those signs so you're going to get
one
over
cosine we're almost
done here's do I'm going to reciprocate
each of those fractions I'm going to
reciprocate the run the run the one the
X over sin x and the one over cosine X
what that's going to do is also flip
around my inequalities if you
reciprocate that it's going to flip your
inequalities does that make sense to you
that's a mathematical truth so I'm going
to
have one still because the reciprocal of
one is one is greater than sin
XX which is greater than cosine X over
one do you see this thing
anywhere you see it right in the middle
do you see how it's squeezed between two
functions it's squeezed between one and
cosine X does that make sense now here's
the theorem it's called The Squeeze
theorem The Squeeze theorem says if the
limit of this goes to something and the
limit of this goes to something then the
thing in the middle has to go to that
same limit if these are equal does that
make sense to you says I'm trying to
take the limit as we we go to zero
agreed so then the
limit of
one what's the limit of the number one
as X approaches zero can you tell me
that what's the limit of one doesn't
matter what we're going to right the
limits
one tell me what's the
limit cosine X as X approaches
Z Now what do you get of cosine of0
what's cosine of 0
cosine Z is it's not zero one it is
one here's what that says check it out
this is by The Squeeze
theorem it says that right now if I take
the
limit sin
XX as X approaches zero it was squeezed
between two functions whose limit was
one as we went to zero says the limit of
one is one says the limit of cosine as
we approach zero is also one what does
this limit absolutely have to be if it's
between those two functions the limit
has to be between those two functions
only they're going to the same number
what's that limit has to be one it's
squeezed it's saying the right bound is
one the left bound is one can it be
anything different than one no no it has
to be one Tech technically I'm supposed
to have these throughout the whole
thing and that's the big punch line for
this part right now we're not done with
no so again what we just did we learned
we learned one thing about this we
learned last time that the
limit as X approaches zero of sin x
overx was equal to what was it equal to
do you remember okay and again for the
for the last time we did not prove the
squeeze theorem we used the squeeze
theorem to prove this thing we bound it
between two functions whose limit was
one therefore this was in the middle of
it that limit had to be one as we
approach zero so this equal one that's
number one thing you got to remember you
got to remember that one that's going to
be in your head all right so memorize
that thing when X approaches zero of
this function you get one if x
approaches anything else besides zero
well it doesn't matter you can just plug
that number in okay that's fine now
there's a couple of them that that we
might also want to know is there a
relationship for cosine firstly and is
there a relationship for tangent we're
going to look at those today then I'll
show you how to put everything together
and do any limit well that's reasonable
of some trigonometric functions you
ready for it yes it's no Monday excuses
this time it's not Monday it's Monday
Monday no no it's Tuesday I got you
Tuesday
and Wednesday Friday it's Friday okay I
can understand but no no no it's Tuesday
today we're rolling so number one
thing another number one thing can we
find the
limit as X approaches
zero of this
1 - cosine X over
X well let me ask you the question can
you just plug in the
zero if you do plug in the zero what do
you
get you get z z what's cosine of Z
folks you didn't know that cosine of 0
is one so on the numerator you get 1 -
one that would be zero right and of
course if you plug in zero X would
become zero you get 0 over zero that's a
problem can you factor one minus cosine
X and cross out the X on the bottom
nothing we can do with that we're GNA
have to find some way around it just
like we did with this one are you ready
for it yes it's not going to be nearly
as time consuming or mentally consuming
as this one was okay this one's actually
quite basic here's what I'm going to do
what I'm going to do is find some way to
to see some sort of an identity in here
so so what we're going to try to do is
work around this 1 - cosine X and one
thing I can
do perhaps multiplying by the conjugate
would would help us remember what the
conjugate
is just changes what about that yeah
that's it so in our case the conjugate
would be one + cosine X over 1 + cosine
X okay
well let's keep this going then I've got
a limit as X approaches
zero the denominator is
now x * 1
+ cosine
X you guys okay on the
denominator I haven't distributed
anything I'm just putting those together
for right now now could you tell
me on the numerator on the numerator
let's do a little work here what am I
going to get If I multiply 1 - cosin x *
1 + cosin x what would you what would
you get out of
that okay try it on your own if you if
you can't do it in your head try it on
your own distribute it foil it
out you know you've got this
right you know you've gotve done that uh
the first thing you should be getting is
a one yes then you're going to get plus
Co sin x and you're going to get minus
cosine X do you see that what's going to
happen to those middle terms they're
gone so we we're going to have one and
then lastly you're going to do cosine x
cosine X is that going to be a plus or
minus minus
what cosine not just cosine x
cosine oh now here's the
deal using your massive knowledge of
trig identities some of you have and
some of you don't have right now to be
the Brut honest truth uh what is 1 -
cosine 2 x equivalent to using the
Pythagorean
identity oh you do have that knowledge
that's awesome very good if you know
that that's right because listen if sin
s+ cosine s =
1 if that's true which it is true and if
I try to make this out of this thing
that would be subtracting cosine right
from both sides if I do that
we make up another
identity we make up the identity 1 - c^
2 X is equivalent to sin^2 X therefore
this
thing is equivalent to
s^ s
x can you follow so
far now we're going to do a little bit
of algebraic manipulation I have gone
from this I multiply by the conjugate
I've used the Pythagorean identity in a
unique way to get sin^2 X I haven't done
much with this at all I just said x * 1
+ cosine X you follow so far now because
this is a limit I can split up limits by
multiplication true so I'll show you
every step by step watch what we could
do here with our sin squ
x I know that I could make this limit of
sin
x * sin x
is that not still sin^2
X
okay again as X approaches zero and then
this thing I'm not going to distribute
that I'm going to not distribute it for
a very specific reason check this out
this is the same thing as X
Time 1 +
cosine X so I haven't really changed
that at all whatsoever you still okay
with this you
sure raise your hand if you are if you
all right with that good okay basically
I'll just wrote here's the idea look at
because we can separate limits by
multiplication check out what we can do
I can group together because it just
fractions this
thing and this thing do you see it does
that thing look familiar yeah so if we
group it then this is going to
be the limit as X approaches
zero of sin x
x
x time the
limit as X approaches
Z of sin
x over
1+ cosine
X very brief recap very brief we
multiply by the conjugate what that does
is creat a sin s x for us the sin s x
can be split up into sin x * sin x the x
* 1 + cosine X we don't change that at
all what we do is we group it in a
certain way so that we can use a
previously known identity that we have
proven that's okay we're using something
we've already proven so now we have a
limit of x sin xx and a limit of sinx
over 1 plus cosine X it's okay to split
up limits by multiplication I taught you
that property are you still okay with
this proof so far all right can you tell
me how much is the limit as X approaches
Z of
sinx beautiful we just did that right we
just did that that's fantastic so this
is one times now what I want you to do
is I want you to try to evaluate this
limit at
zero are you going to have a problem
with
that are you going to have a problem why
not why don't you have a problem
now what's sign of
zero good is it okay to have zero on the
numerator yeah sure does it to make zero
on the denominator no cuz you have 1
plus how much 1 plus 1 that's two you
have 0 over
two this is 0 over two I
agree so how much is 0 over two what's 1
*
0 don't say one how much is one time
zero
Z what that says to us is we've just
proved something else we now have this
identity that's identity the first one
we did number one this is the second
one the limit as X approaches 0 1 -
cosine XX is z
interesting you know maybe you can
remember it this way well I do at least
I remember it this way so something
involving s gives you one something
involving cosine gives you zero that's
kind of interesting right because
usually it's if s gives you zero cosine
gives you one right if cosine gives you
One S gives you zero it's almost the
same idea here the S one is giving you
one the cosine one is giving you zero
it's kind of cool right same sort of
relationship is how I remember at least
would you like to do the tangent
one tangent one's even a little bit
quicker actually much quicker
let's try to find Tan x
overx now could you just plug in the
zero what's tangent of
zero what's tangent of
zero zero because sign is zero and
cosine would be one that would be zero
but now you have over Z so you have 0
over Z again so we do need to manipulate
it but here's what we can do how much is
T as the identities say it is so over
cosine so we could make
this
that are you okay with that one so far
if we do a little bit of work with this
just a little little bit remember this
is like
uh X over one you're going
to instead of dividing reciprocate and
multiply true so this would be the same
thing as the
limit X approach zero
of sin x over cosine X time
1/x still true still true now all in
going to do because of the commutativity
of multiplication and what we can do
with fractions I'm going to
Interchange this with that is that okay
with you just going to flip them why
does it matter the order in which we
multiply and this is fractions right so
I can make everything one fraction true
so that means I could make it one
fraction commute the denominators and
then split it back up again so that
means I okay that's the
limit X approaches zero
of sin
XX * 1 over cosine X I've just switched
those two things and that's fine you can
do that because multiplication is
commutative and you can make one
fraction out of that by
multiplication see anything
interesting have you seen that we're
trying to get this thing all the time
because we know what that is do you see
the sinx overx we can still split up the
limits so this would be the
limit X approaches 0 of
sin XX * the
limit X approaches 0 of 1
over cosine
X interesting interesting hey now that
you have this memorized I hope you do
what's the limit as X approaches zero
don't read the board right is right up
there just kind of hopefully you
memorize it already limit as X
approaches Z sin XX what is that so this
one is one times can you plug in the
zero now sure what's cosine of zero it's
one yeah what's one over one
okay what's 1 *
1 you said one a lot today haven't you
that means the
limit X approaches Z of tan
x x = 1
just like that just like that one did
interesting like it so far there's
basically three things we need to know
these are three limits we need to know
of we need to know the limit as X
approaches 0 sin XX 1 we know the limit
as X approach 0 1 - cos x x that equal 0
and tan XX = 1 again if you know those
limits you can break down all the ones
I'm going to give you into those three
identities if you can do that that this
is is very easy to master but it
involves some thinking outside the box
box in certain cases would you like to
see some examples of how to do that okay
let's start over here just memorize
those three properties of of limits all
right those are the proofs that we just
did also I'll say it's got to fit those
pretty darn perfectly in order to be
true
all right so let's start with some
examples you're going to see some
interesting mathematics this is kind of
thinking outside the box we're trying to
make these things fit in the format that
I just gave you okay one of those three
formats of of
limits we'll start
there now firstly is this exactly like
the limit that I gave you what's
different about this two is a problem
two is a problem what we need to do
somehow is we need to make the inside
angle the same thing as the denominator
so for instance if I have a 2X inside my
angle I need a 2x on the bottom of my
fraction you with me on that you got to
have that otherwise it's not exactly the
same you can't do that limit I'll show
you how to do this in just a second so
since we have this maybe we can do
something a little bit special the only
thing we can really do is multiply by
one right that that's it otherwise we
change the value of the limit but maybe
we can multiply by one in a special way
for instance what if I said you know
what I want to multiply by one but I'm
going to make it 2 over two is that
legal to do okay let's see what we can
do with this thing now you can put the
twos anywhere you want to provided you
don't change this a lot of people when
they're first starting out in
trigonometry hope hopefully this is not
you they go well let's just pull the two
out front wouldn't that be easy two time
no you can't do that that you can't do
that right that that is 2x you can't
change that unless you use an identity
to do that the double angle or the half
angle or what whichever ones that you
you can manipulate those the only ways
you can change your sign of of the
inside of the angle okay you can't just
pull out that two it doesn't
work but we can choose to put these twos
anywhere we want so what I'm going to do
is I'm going to say all
right let's be smart about
this now this two maybe I put it in
front of my sign
but this two let's make that
2x you okay with that so
far you see where the twos
went now here's something cool do you
remember that anytime I have
a function multiplied by a function I
can split off the limit of that function
we we've done that a couple times right
here right we did that but with a
constant the limit of of a constant is
always just that constant let me explain
that again when you have a limit of a
function times a function you can break
it
up the limit of a constant is always the
limit I'm not teaching anything we
haven't had before what that says in
plain English is you can always pull out
a constant to the front of a limit
because the limit of a constant is that
constant you are you with me on that so
basically pull the two out that's what
you can do that two right here I can
bring that up
front so two
times
this now some of you might be thinking
well wait Mr Leonard why why didn't you
also pull this two out why didn't you do
that well if I if I had what's 2 over
two then I'm right back to where I
started right that would be silly I'm
doing this on purpose purp so I can make
these two things look identical and
multiply it by the number that's
basically just left over raise your hand
if you can follow that feel okay with it
all right now here's a cool deal how
much is s of 2x over
2x you know how much is sign of limit of
uh s of X overx as X approaches zero one
one now here's why this is also one you
could actually make a substitution check
it I'll show this only one time just so
you see it once
but let's make a dummy substitution like
U = 2x you okay with that tell me
something as X goes to zero does U also
go to zero yes yes it does because if
you plug in zero here you get a zero
there as well right that means you can
make the substitution for the limit so
if I make this substitution then this
limit now becomes U is going to
zero s of well instead of 2x I put U
instead of 2x I put U do you see how
that now fits our identity perfectly you
can do that provided your your b or your
your variable still goes to the same
spot it's still going to zero because
when 2x is zero U is also zero it's
still going to the same spot does that
make sense to you so we can do this this
go oh yeah this is one therefore this is
one that's our substitution that's kind
of
neat so then we have 2 * 1 what's our
limit how much two two
two can you almost see it though in the
original problem s of 2xx how much is it
not one it's two what's inside the angle
besides the X two interesting isn't
it are you ready to make these a little
bit more advanced a little bit more
advanced start build them up a little
bit this was very basic very very basic
we're going to start incorporating some
other ideas in here so how about just
little by little though don't worry
now but I got to warn you I'm going to
cut out some of the steps I've already
covered in the class so for instance I'm
not going to ever show you the squeeze
theorem for sin x over anymore CU we've
already done that I'm probably not going
to show you this whole routine for
getting that answer anymore does that
make sense to you probably not going to
show that to you anymore um maybe one
maybe once more but after that I'm just
going to assume that you you can see
that and then you can get there on your
own is that
fair I hope so cuz that's what I'm going
to
do oh well this is kind of nasty does
this look like any of our identities so
far we've only got three of does it look
like any of them you got to make it fit
one you got to make it fit one of those
which one's it's closest to the tangent
one the cosine one or the S one it's
closest to the sign one but I need to
have s over an X
true okay is there anything we could do
specifically can I
multiply by one one something numerator
denominator that's exactly the same
that's going to give me something over X
something over X
stunk job remember that you can
incorporate new variables as well
provided that you multiply or you divide
and gives you
one I want this thing over X and this
thing over X that's what I want can you
make it happen
here x overx close very close so we know
we're going to need to incorporate
another X you agree with that right
somewhere the x is going to have to
happen oh okay so if I listen If I
multiply by just an X I get X sign right
I don't want that I want overx how we
get overx say it again how about 1X is
that legal to
do is this still
one okay let's see what that does to our
problem if you multiply s of 5x * 1X
you're going get
S of
5xx are you okay with that algebraic
step what's this one going to
be so we're we're multiplying in such a
way that we can find some resemblance to
an ident identity that we already have
here we've chosen to multiply 1X over 1X
which is legal to do because this is
basically just one it's just one we're
doing it so that we get sin x overx you
seeing the point here you sure now are
you going to be able did were you able
to think of that on your own no but now
you can use it right now you can use it
in some of your problems hey using what
we found out here can you tell me what
the limit of this thing is going to
be why is it
five could I do the same thing here um
remember I can separate limits by
division right so I write limit over
limit I can do that so if I multiplied 5
over five like I multiply 2 over two
ultimately I'm going to have five times
this limit where the 5x and 5x is the
same I'm going to have 6 times this
limit where the 6X and the 6X are the
same I'm going to have that if you want
to see it for the last time this is what
it would be you'd say oh okay let's do 5
over five let's do 6 over
6 I'm going to do a couple steps at once
right now I'm going to break those
limits up I'm going to say okay I
want the
limit 5 sin
5x over 5x I want limit
6 sin 6X over 6X can you make it that
far you guys are right with that
one yes
no what can you do what can you do with
this five and with that six what can you
do with them outside the limit you can
because they're constants and we know
that the constants really don't affect
the limit because you can separate by
multiplication and you'd be okay this is
all multiplication that's great so this
would be now
5 * the
limit s of
5x over 5x as X approaches 0 notice how
I'm still writing the limit all over 6 *
the limit as X approaches
0 S of
6X over
6X do you see what we've done we
basically put two ideas together we put
this idea that I need sinx over X we put
this idea that I can multiply by some
constants and manipulate them in order
to get my exact identity out of that now
I'm not going to show you the
substitution again basically I want you
to know that a substitution is possible
in these cases how much is the S of 5x
over
5x how much one this is this is one
right here this limit is one whenever
your angle matches your denominator with
that sign and your limit is going to
zero that angle or that limit is one how
much is this limit right here so you
have 5 * 1 you have 6 * 1 what's your
answer
56 we have that made sense to you you
okay with that one all right good you
want to try a few
more it's kind of fun I like them teach
you a little bit about trigonometry
won't
it or at least
refresh your memory about
someon and learn some things you can do
with limits which are pretty cool
[Applause]
oh let's see about
this does this look similar to one of
our our limits that we already knew how
to do is it exactly the same though
what's different about it it's got X
squar now again can you change the
inside of your limit or I'm sorry the
inside of your sign can you change the
inside of your sign so basically can I
change the x s never can't change that
thing unless you use identity that's the
only time you can do that you can't just
say I think I'll pull an X out and put
it there no you can't no no you don't
what you want to do is make it so that
if you haven't noticed this in the
previous two examples make it so this
angle matches your denominator what do
you need to multiply by to make your
angle match your denominator cuz that's
what we did here multiply to make your
angle equal to your denominator multiply
to make your angle equal to your
denominator that's where the five came
from what do you need to multiply by X
overx would work
so keeping with what we know this x I
can't do much with that besides put it
out in front of my limit so right here
I'll do okay I'm going to put that out
in
front but these two yes that's what I
want because what what I want I want
that angle and that denom to be exactly
the same you all right with this so far
now because limits are separable by a
multiplication I can say well this is
just multiplication I can separate off a
limit of x so this would be a limit of x
as X approaches Z times the
limit sin
x^2 x^2 as X approaches 0
hey tell me something what's the limit
of x as X approaches zero it's not x z
zero because you plug that in you get
zero times what's the limit of sin x^2
X2 as X approaches zero one yeah because
you could make that same substitution
like we well I had on the board earlier
for a dummy variable equals X2 right and
then as x s goes to zero so does that
variable and so does this uh this
variable it would be like a u you would
be going to zero as these two things
also went to zero that matches our our
identity perfectly this goes to
one and as I've said anytime your angle
matches up with your denominator you can
do
that so you pull out
that absolutely because it's not a
constant it's not a constant yeah the
only reason that that's a great point
the only reason why we were able to pull
this constant out and leave this
constant is because if you did this
watch this if you did the
limit as X approaches zero of that
constant what's the limit of
two no matter what you're going to right
that's why we can do
it okay let's continue on with this
what's 0 *
1 our limit of x2x sin x^2 x as x 0 is0
we'll try um try a few more of these I
really want to make sure you get the
handle on this because there's a lot of
different combination and problems we
can do so I'm going to give you as many
as I can in the time that we have that
way you feel a little bit more
comfortable on your homework you ready
for them so again some other examples
[Applause]
hey by the way is this the same thing as
this this the same as that s of x s is
that same thing as Sin s x what's sin s
x
mean
good maybe you can see where this is
going
so that's
true you're always trying to break it
into limits you know limits you
understand what's the limit of sin
x/x this is going to give you one what's
s x what's the limit of sin x as X
approaches zero it's zero because when
you plug in the zero you get zero so
this is ultimately going to
be 1 * 0 or 0
this limit if I I'm not showing you a
step right here I'm not actually
breaking them up if I were to break them
up the limit of this would be one limit
of this would be zero 1 * 0 is
0 that one
now we we're going to talk about this
I'm not going to prove it to you why it
doesn't exist you can you can uh get
there on your own or or you can look I
think the book has a proof in it it's
pretty good I'm going to do this
graphically just to show you what this
looks
like now of course sign is an
oscillating function right goes up and
down up and down up and down one overx
as you get close to zero goes to
Infinity right it goes there very fast
so what happens with this
graph it looks like this sure it tops
out at one and it pops out at negative
one because sign is bounded in those
ranges this looks something like this
starts off nice it goes down it goes up
and down and up and down and up and down
and as it get closer it goes faster and
faster and faster and faster and faster
and faster faster till a bunch of crap
happens it's by the End by the time it
gets to zero it's like that does it ever
get to zero no you tell me can you plug
in zero no then it never gets there it's
undefined at that point you can't do it
cannot do it so it just goes nice and
slow and then faster faster faster
faster faster F until it gets nowhere
the other side looks like
this as we get closer to zero are you
approaching a certain point or is it
just continuing to oscillate like
that it's not actually reaching a point
it's not gonna it's not gonna ultimately
go oh I'm gonna I'm gonna level out and
make zero or I'm gonna level out and
make it's not doing that as you get
closer it's still oscillating even
faster and faster and faster by the end
it's just crazy all right you can't even
you can't see it it's going so fast so
is the limit going to exist answer is no
no it doesn't
exist that one does not exist sign of
one re does if you ever see that just
sign of one of Rex that doesn't exist
okay so question this is an interesting
question if that doesn't
exist Does that exist
does that
exist oh as we
approach let me get my white cat and
smoke it did you see that commercial
with that creepy guy with the white cat
I don't know what that was yeah it was
but it was like on the Super
Bowl I think it was CEO you know who CEO
iseven
I Was Made I was forced to watch The
Voice last night that's the only forced
forced
yeah not like handcuffed or anything but
like it was on and of course if it's on
I have to watch it so it is anyway does
that exist let me show you something
again we're going to use the squeeze
theem are we going to prove the squeeze
theem no no we're going to use the
squeeze the check this out this is
interesting uh do you agree that
oh by the way I did make a mistake on
the previous squeeze theorem using the
squeeze theorem um I was supposed to
have the equal signs underneath it I
neglected that for some reason I don't
know why I did that but with your
squeeze theem you do have to have those
symbols
not those
symbols I think I set it towards the
very last bit of the proof I said you
know what I'm supposed to have these
everywhere but I don't know if you
caught
that let's use some of our knowledge
about sign
to create some bounds on sign forget the
one /x part for a second what's the
maximum value you can attain with
sign what's the highest sign goes one
what's the lowest sign
goes so you agree that's true
right true okay we'll check this out I'm
now going to create this in here by
multiplying all three of these sections
by X is that legal to do sure
yeah so if I multiply by
X actually I think I yeah I do I want
the absolute value of x because I don't
want to change those signs so I'm going
to make sure I multiply by the absolute
value of
x okay so here's a slightly better
interpretation of what's actually going
on in the problem we're working on right
here we're trying to work on the limit
as X approaches zero of sin 1X * X now
of course we know that sign is bound
between 1 and - 1 s doesn't get any
bigger than one doesn't get any less
than 1 so we can go naturally make this
an equality any sign function is between
-1 and 1 including s of 1 /x so we have
that down can't get bigger than one
can't get less than one now the way we
make the jump from here to here which in
class I have absolute value around this
X and I said we can get there by
multiplying this absolute value of x and
this absolute value of x and this
absolute value of x which is true but it
doesn't quite give us back exactly what
we want which is X without the absolute
value how we get there is just by a
little bit of little rational thought so
we say if we multiply by X in all three
spots what could potentially happen is
if I don't have these absolute values I
could get a negative number over here
which wouldn't be an upper bound so what
we're saying is let's make this one
absolute value of x that way we know
that that's always going to be positive
no matter what I plug in I have positive
X well that's certainly going to be
bigger than x s of 1/x if this is true
then this has to be true where X is
positive also if we let X be negative
we're going to have to have a lower
bound now if I don't have this absolute
value what that says a negative times a
negative could actually get me a
positive I don't want that to happen I
want a legitimate lower bound so that's
where the absolute value is coming from
that says all right well then x s of 1x
certainly has to be bigger than that
negative number if this is true then
this one also has to be true and that
makes our double inequality here from
there it's a simple matter of limits we
say well we know the limit as X
approaches zero of the absolute value of
x that's going to zero we know that
that's that's that V curve as we're
going from both sides the height that
function is getting down to zero so that
limit exists it's zero same thing the
limit of negative absolute value of x
that's negative absolute value of x
we're going up to a height of Zero from
both sides that limit zero so we we have
the limit of this function is zero the
limit of this function is zero since
we've now squeezed our function we
wanted to find out this one since we
squeeze that between two functions whose
limit is zero as we approach Zero by The
Squeeze theorem we can now draw that
conclusion that's how we get the limit
as X approaches Z of x s of 1
/x equals
z does the limit only exist because you
have an X in front of the S one /x yeah
okay and which is
interesting because that little that X
right there in front of it allows you to
bound it between two numbers and then
use that fact to say the limit of this
goes to something Li of this goes to
something the problem is if you look at
this one just thetive
one sin
1/x less than one
where's this limit
go where's this limit go are they the
same then you haven't squeezed it that's
the problem we need that X to say okay
by multiplying by well the absolute
value of x which which is a small point
but multiply by absolute value of x the
limit of the absolute value of x yes
that is going to zero and the absolute
value of x yes that is going to zero
that is what actually squeezes it that's
the point without the X you can't do it
it fails here it doesn't work here if
works here this is another one that we
can use limit of x sin x or sin 1X that
does exist it actually equals zero it's
an interesting thing just by having that
X in
there so our whole idea for this is
you're trying to break up these limits
into the identities that you the limit
identities that we can work with the
ones we've already know that's the sinx
overx goes to one and the 1 - cosine X
overx goes to zero and the Tan x overx
goes to one again that's what we're
trying to do um I think I have three or
four more examples would you like to see
some other things that we can do okay
I'm gonna I'm going to have to kind of
paraphrase these for you because I just
want to give you the the head start on
them some of these are are in your
homework as a matter of fact of think
but I want to give you some key points
so that you can do these on your
own so I'll be moving kind of quickly uh
follow along if you're not quite getting
to see see me in the math lab I'll be
there often or my office hours or check
out these videos again later on see that
you can follow these on your own you
with me on that so A couple of things
that we can do this is basically just
practice at this point you've learned
everything we need to know
okay while this looks kind of nasty I
want you to see that we can actually do
a lot of fun things with this firstly do
you see how we have a few cosiness up
there we're probably going to be moving
towards the cosine limit that we we
learned about so if we do that well what
I want out of that limit is the oneus
the cosine true overx I've already got
the overx but I don't have the one
however what I can do with that number
two can I separate that into 1 + 1 is 1+
1 the same thing as two sure so then
this is really check out what I'm going
to do I'm say this is really 1 + 1 I'm
going to reorganize the stuff then and
say this is the limit as X approaches Z
of 1 - cosine
3x + 1 - cosine
4X all over X
does that work for
you is that still the same exact thing
just reorganized it now what's cool
about fractions is that if you have a
common denominator you can add them
right and subtract them what it also
means you can pull them apart by that
common denominator so what that says is
I can actually make
this 1 - cosine of
3x/ x well there's one of them but now I
have the common denominator plus 1 -
cosine of
4X
x wouldn't those be different limits
though and then you're adding them
together we are adding two different
limits right now we're not there
yet but that's a good point we can split
up those limits by addition right so we
now have two different limits
oh
now what can we do
now what can we
do we can do that what we want is the
angle to be the same same as our
denominator so if we do that if we do
the 3 over3 thing and the 4 over4 thing
move those numbers out front don't
affect this one minus cosine of that
that uh that angle then we'll be okay
then that's going to allow us to do that
substitution just like we did for the
sign did you follow that so we want 1
minus cosine of an angle over the same
angle basically down on the denom that's
what we're looking for here so if I
multiply this by
* 3 over 3 * 4 over 4 all this becomes
is 3 * the
limit X goes to Z of 1 - cosine of
3x over
3x plus 4 * the
limit 1 - cosine of
4X that's hard to draw
4X so we do the same exact idea that we
did for our sign only here can you tell
me what's the limit as X approaches Z of
1us cosine of some angle over that same
angle how much is that that's the zero
so we have 3 * 0 plus 4 * what's that
one going to be we do that substitute
again just like we we learned for signs
so 3 *
0 4 * 0 what's the
answer I know
right all that time spent and nothing
zero you should should have seen the
looks on my my math C kids when I I did
a whole problem using rational functions
took about 10 minutes and then the
solution didn't work out because it made
the denominator equal zero and they're
like wait there's no solution like yeah
isn't that exciting they're like I hate
you oh
well okay other things you can do this
is very similar uh Now understand that
we were able to break up these fractions
because we had that denominator we can
do the same thing here so we could
actually make this limit as X approaches
Z of x^2 /x - 3 sin x
overx you guys okay with that one yeah
we can do things like that we can split
up our limit using our algebra how much
is x^2
x so we'd have a limit as X approaches Z
of x - 3
* sin x/ X tell me what happens here
tell me what happens where does The X go
to Zero by the way the reason why we can
do that is because we're not actually
getting to zero right we can actually
cross out the X and X that's okay so
what's the limit of x is X approaches
zero this is zero minus what's uh what's
the limit of
three is the three going to change now
you you know I'm skipping steps here
right you know that technically what
we're supposed to do is is separate
these by subtraction then separate these
by multiplication but we found out that
earlier with our pols if you can plug it
in and you know the identities then
you're going to be okay so here this
goes to zero this three that's going to
stay at three what's the limit of sin XX
as X approaches zero what's my
answer answer
is3 would you ra have be okay with this
one we'll practice a couple more next
time I want to show you two other things
we can do and that'll wrap it up so
we're still learning how to do some
problems involving trigonometric limits
so when we deal with these limits our
basic idea is you want to get it down to
one of those three identities that we
knew for limits uh namely sin x/x as X
appr to 0 goes
to oh
boy one one very good okay somebody else
one minus cosine x/ X as X goes to Zer
goes
to zero and tan
XX oh good all right you got that one
that's that's the one we don't use that
often either oh nuts okay so basically
we're trying to get those things in
these problems some way or another now
often times you'll have to use some sort
of an identity to do some of these
things so for instance when I look at
this this top example t^2 over 1 -
cosine 2 T if I plug in the T I have a
problem because I have 0 over 0 do you
see the0 over Z that's a problem but is
there a way I can manipulate this in
such a way to get rid of the 1us cosine
s t that right there at this point for
you who have taken math 2 our
trigonometry you should that should be
going oh yeah I know what that is what's
1us cosine squ T do you
know sin Square t very good so I could
change
that now at this point there's a couple
ways to go about doing this problem
one option is do you see how I could
break this up T * t s * s do you see
that you could have t over sin * T over
s you with me you can break each of
those up as different limits each one of
those limits would be going to one I I
didn't say this to you but it should
maybe have been apparent to you from the
squeeze theorem that this
limit is also equal to one
was that apparent to you as well that's
the same thing that's the same thing the
reason is because when you do the
squeeze theem instead of reciprocating
if you left it alone you have that
already that's that's in there when we
proved it with the squeeze theorem also
I can show it to you this way um do you
remember that if you take out an
exponent or you reciprocate a fraction
you can actually have a negative
exponent some of you zoning out already
let me let me show you what I mean maybe
let's make more sense if I flip these
around
but make it netive
1 that's the same thing as that do you
remember that reciprocation of a
fraction changes the sign of your
exponent if that's to the first Power I
can reciprocate those fractions make to
negative first Power yes I can pull out
a negative I pull out any exponent right
so I could have a limit of sinx overx
that's one 1 to the 1 still gives you
back one either way that's one so in our
case up here we could do it that way
that I explained earlier we could break
this up
as T sin T * T sin
T each of those is going to give us a
limit of one are you with me on that one
* 1 our our ending limit this should
equal one but I want to show you the
different way to do
this that's one option do you see the
option that you do that the other option
is well if you did something similar
with that exponent I could say that this
is the
limit as T approaches
zero since the numerator squared and the
denominator squared I can break that
exponent out to the top you with me on
that
one sure okay also if I change the sign
of that exponent I can reciprocate my
fraction do you follow that one so this
says all right well then why don't you
just make it
limit T approaches zero
of sin t/ T that's something we want but
now since I reciprocated the fraction I
have it to the negative2 power what we
can do with the ne2 power is pull that
exponent outside of our limit
now we're home
free do you need me to recap here a
little bit Yeah okay
recap still get a kick out of that every
time trig identity youall need to know
that one taking an exponent out it's
very similar to this if you have x^2
over y^2 that is X over y squ
agreed then this is similar to this if I
reciprocate the fractions y
/x then that's just -2 that's exactly
what I have done here and then there
true properties of limits say that if
you have a function raised to some power
it's the same thing as a limit of that
function raised to that power so
basically says I can raise the limit to
the power instead of raising the
function to the power and then taking
the limit so in other words that's
possible pull out the exponent take the
limit first first and then do the
exponent you okay with that one that's
the recap now what's the limit as T
approaches
zero of sin t/ T so you all agree that's
one
right what's
one to the2 power one it is just one
yeah anything to one to any power gives
you one so this is going to give you one
you okay with that one
well that was
fine any questions on the first one is
it only one if it's approaching zero or
can it be approaching another number
that's a good question if it's
approaching any other number besides
zero well then okay the only difference
on this one for like right right here if
it's approaching number besides zero
well then s is actually a number right
so you can plug that in and be just fine
you wouldn't have to do any of this uh
the only time that really matters is if
the S of T is zero which happens at Z
and then pi and then 2 pi again for this
example here it would really matter
unless T goes to zero so that that's our
our situation so if T any if T is
anything else besides zero you don't
even have an issue remember I said you
should always plug in the number first
to make sure you even have an issue so
if this was like going to one or well
let's make it better Pi / 2 you go let's
let's do pi over 2 do I need to have a
limit Identity or anything no you just
do s of pi over two over pi over two and
that will actually work out for you be 1
over pi over 2 or two over Pi that would
be your answer so you wouldn't have to
do any of this stuff if you have a
number besides zero are you with me on
that it all goes back to you should try
your limit before you start doing fan
this is fancy stuff try your limit
before you do fancy stuff plug in the
number if you don't have a problem then
that's your limit if you do have a
problem then do fancy
stuff fancy
stuff my fancy
hands okay does that answer your
question though yeah all right let's try
the next one now let's practice this
then are you going to have a problem if
you plug in
zero What's the numerator become zero oh
good we got that one what's the
denominator
become this is pi over two right pi over
2us 0 is pi over 2 right what's cosine
of pi/
2 I'll give you a hint start with the Z
and rhymes with
Hero Zero yeah it's zero you get zero
over zero that's a problem so to answer
your question again would you could you
if you if that wasn't zero if it was
something else could you potentially
just plug in the number without doing
any of this fancy stuff where about to
do absolutely but now that we have 0
over Z this is where you have to use
those limits and identities to get
something you know what's the things you
know you know the limit of sin x/x as X
approaches zero is 1 you know the limit
of 1 - cosine XX as X approaches Z is z
and you know that uh the limit of tan XX
as X approaches 0
is one again that's the only things you
know so if you can't do the limit
directly you try to make it into one of
those things you follow me on that okay
so what can we do someone who's really
good at identities tell me what that
is I'll take care of the numerator for
you all right here you go boo do the
denominator what's cosine of 12 Pi minus
X it's a special little
thing half angle
identity
yeah oh man you guys are killing me you
guys killing me know your double angles
your half angles your pythagoreans and
the tangent ones okay know what secant
cosecant cotangent mean and you'll be
okay those are the basic ones you need
to have those down what this says if you
want to check in the back of your book
cosine of 12 pi x or minus your angle
whatever that that happens to be
whatever variable you have
that's equal to sinx that's it this
is
sinx did that make your problem
easier much easier right if you try
messing around with this one you're like
I have no idea what what's Mr Leonard
going to do there this is crazy stuff
but that's actually just an identity do
you know how much this is limit of x
over sinx as X approaches zero it is one
yes we actually just talked about that
one one and done cool yeah okay so do
you sometimes need some identities to do
these problems yes you do if you be good
at identities in this class not a
professional otherwise you'd be teaching
but relatively okay okay you need to at
least know them and how to how to use
them that's why I had you do that that
identity section right so that was heads
up on you should do this now most of you
skipped those problems don't problems uh
mostly you skip those problems but you
you should have the idea that yeah we're
going to be using this in class and it's
somewhat helpful unless you just want to
spend massive amounts of time banging
your head literally against the wall and
pulling your hair out I don't want you
to go bald so learn your learn your
identities and then you get a little bit
easier okay now our last one that we're
going to do in dear again I can't give
you every example but I've given you
quite a few I think you'll agree what
can we do with that what do you
think any ideas
we did something with 1us cosine earlier
in the class maybe that would work
conjugate conjugate conjugate yeah let's
conjugate it why don't you multiply by
the conjugate right now I'm going to
walk you through this one multiply by
the conjugate of 1 minus cosine Theta do
them both the numerator denominator
uh again what's the
conjugate uh of this that I'm talking
what how do you find the conjugate
basically change change the sign okay so
what is the conjugate specifically for
us you got to do it on both the
numerator and denominator
you okay so
far we don't necessarily have to
distribute the numerator I'm going to
leave that alone in case we absolutely
have to later what I do need to do is
distribute the denominator you follow so
let's try that so we're going to have
the
limit th approaches
Z Theta
squ 1 + cosine thet can you tell me how
much this is going to give you when you
do your your distribution what are you
going to
[Music]
get one hey we've seen this already how
much is 1 - cosine squ
still okay so
far are you okay so far with that give
me a little head knot if you are yeah
okay good do you see anything we can do
now what do you think do just see
anything looks familiar in this
problem what looks familiar maybe that
we've done earlier in this class that
you
Coulda squ you see theeta Square sin squ
right we just did that it's right here
we've done that problem so what I want
you to do right now is break that
up I want this the Theta squ over sin
squ times the remaining
well we'll talk about that in a second
right now all I care about is that you
see that this right
here is going to go together do you see
it's going to go
together and it's going to be multiplied
by 1
+ cosine th do you all have that in your
paper right
now you seen how to if you didn't get
that you seen how to get that
yes no it's okay if it's a no but I need
to know it's a no it's it okay or
not
yeah not so much this is like uh Co this
is like over one right if you multiply
numerator time numerator and denominator
time denominator it will give you back
that thing therefore we can split it up
uh in the opposite
manner okay now what can you do with
limits that are being multiplied
together can you split them up by
multiplication can you split them up by
do that now make two limits out of it
limit of this one time the limit of this
one do that for me
also one little piece of information
when you're doing these limits you have
a lot going on make sure you're putting
the parentheses around the argument of
your limit just to make sure that you
know I'm taking the limit of this whole
thing not just a limit of one and then
adding cosine Theta does that make sense
to you make sure you see that did you
make it this far okay now since we've
already proven this we already know how
much this limit is how much is this
limit we can redo it if you really want
to right pull out the two reciprocate it
-2 uh 1 to the second power is still one
so this thing is one now can you
evaluate this limit at theta equals 0 is
that okay to do now sure you have no
denominator that's fine plug in the zero
and do that on your
own notice how I'm not writing limit
anymore this limit is equal to 1 time
this limit is equal to 1 + cosine of 0
since Theta is
zero notice how you still need par
because you're multiplying one *
something that's
okay how much is cosine of
Z it is how much one cosine of 0 is 1 so
what we have in here is 1 + 1 so 1
* 1 + 1 what's our limit equal to limit
is
two there are a couple different ways
you could have done this you could have
broken this up independently done a few
different things this is probably in my
opinion the most concise at least one I
found it's most concise so do that and
then we can do any limit as long as
we're finding uh at least some
identities that we've dealt with before
inside of our limits and using those
things how many people understood what
we talked about so far our section how
about this story are you guys all right
with this any questions that we have
before we
continue all right finally done with
Section
1.2 feel good it's good for me this
thing was long my goodness