Understanding Limits: The Foundation of Calculus

Introduction

The lecture kicks off by positioning limits as the core of calculus. Two fundamental goals are highlighted: 1. Find the slope (or tangent) of a curve at a specific point. 2. Determine the area under a curve between two points. These goals differentiate calculus from algebra and geometry, which cannot handle curved lines or irregular areas.

The Tangent Problem

• What is a tangent? A line that touches a curve at exactly one point.
• Why is the slope important? Knowing the slope at a point lets us write the tangent line using the point‑slope formula (y-y_1=m(x-x_1)).
• The difficulty: With only one point (P) on the curve, the slope is unknown. To create a line we need a second point.
• Solution: Introduce a movable point (Q) on the curve, forming a secant line (PQ). As (Q) slides closer to (P), the secant line becomes a better approximation of the tangent.
• Key idea: Limit – move (Q) infinitely close to (P) without actually coinciding. The slope of the secant approaches the slope of the tangent.

Formal Definition of a Limit

A limit describes what value a function approaches as the input variable gets arbitrarily close to a given number, without necessarily reaching that number. - Notation: (\displaystyle \lim_{x\to a} f(x) = L). - The limit cares only about the behavior near (a), not the actual value at (a).

Example: Slope of (y = x^2) at ((1,1))

1. Choose a generic point (Q) = ((x, x^2)).
2. Secant slope: (m_{sec}=\frac{x^2-1}{x-1}).
3. Factor numerator: (x^2-1=(x-1)(x+1)).
4. Cancel ((x-1)) (allowed because (x\neq1) while we are approaching 1).
5. Simplified expression: (m_{sec}=x+1).
6. Take the limit as (x\to1): (\lim_{x\to1}(x+1)=2).
7. Result: The tangent slope at ((1,1)) is 2, giving the tangent line (y-1=2(x-1)) or (y=2x-1).

One‑Sided Limits and Existence

• Right‑hand limit (\displaystyle \lim_{x\to a^+} f(x)): approach (a) from values greater than (a).
• Left‑hand limit (\displaystyle \lim_{x\to a^-} f(x)): approach (a) from values less than (a).
• Existence criterion: The (two‑sided) limit exists iff the right‑hand and left‑hand limits are equal.

Examples of Limits That Do Not Exist

1. Rational function with a hole: (f(x)=\frac{x-1}{x-1}) simplifies to 1 for (x\neq1) but is undefined at (x=1). The limit as (x\to1) is 1, even though the function isn’t defined there.
2. Function with different one‑sided limits: (g(x)=\begin{cases}3 & x>2 \ -1 & x<2\end{cases}). Right‑hand limit = 3, left‑hand limit = -1 → overall limit does not exist.
3. Reciprocal near zero: (h(x)=\frac{1}{x}).
4. (\lim_{x\to0^+} \frac{1}{x}=+\infty)
5. (\lim_{x\to0^-} \frac{1}{x}=-\infty)
6. Since the one‑sided limits differ, the limit at 0 does not exist.

Connecting Limits to the Area Problem

• Approximate the area under a curve by summing areas of many thin rectangles (Riemann sums).
• As the rectangle width (\Delta x) → 0 (infinitely many rectangles), the sum approaches the definite integral.
• This limiting process turns an impossible geometric area into a computable value.

Summary of Core Concepts

• Secant → Tangent: Move a second point infinitely close to the point of interest.
• Limit notation captures the “getting arbitrarily close” idea.
• One‑sided limits help diagnose existence and behavior near discontinuities.
• Infinity as a limit describes unbounded growth (e.g., (1/x) near 0).
• Area under a curve is found by a limit of Riemann sums, leading to integration.

Practical Takeaways for Students

• Always set up a table of values when first exploring a limit; it reveals the trend from both sides.
• Factor and cancel algebraic expressions before taking the limit, but remember the original domain restriction.
• Remember that a limit can exist even if the function is undefined at the target point.
• Use the limit definition to derive derivatives (slopes) and integrals (areas) later in the course.

The lecture concludes with a promise to teach systematic algebraic techniques for evaluating limits, moving beyond the trial‑and‑error table method.

Limits let us rigorously determine the slope of a curve at a point and the exact area under a curve—foundations that turn calculus from guesswork into a precise mathematical tool.