Limits and Continuity — AI Study Guide

Mastering Limits and Continuity

Limits are the foundational concept of calculus. The limit of f(x) as x approaches a is the value that f(x) approaches as x gets arbitrarily close to a (without necessarily equaling a). The formal epsilon-delta definition provides a precise mathematical meaning: for any ε > 0, there exists δ > 0 such that |f(x) - L| < ε whenever 0 < |x - a| < δ. This precision is needed to rigorously prove calculus theorems.

Limit laws allow algebraic computation of limits: limits of sums, products, and quotients equal sums, products, and quotients of limits (when each limit exists). Special techniques are needed for indeterminate forms (0/0, ∞/∞): factoring and canceling, multiplying by conjugates, and L'Hôpital's rule (taking the derivative of numerator and denominator when the limit has indeterminate form 0/0 or ∞/∞). One-sided limits (approaching from the left or right) must both equal L for the two-sided limit to equal L.

The Squeeze Theorem is a powerful limit-finding technique: if g(x) ≤ f(x) ≤ h(x) near a and lim g(x) = lim h(x) = L, then lim f(x) = L. It proves the important limits lim(sin x)/x = 1 and lim(1 - cos x)/x = 0 as x → 0, which appear repeatedly in derivative calculations. Infinite limits (f(x) → ∞ as x → a) and limits at infinity (f(x) → L as x → ∞) have related but distinct definitions.

Continuity at a point requires three conditions: (1) f(a) is defined, (2) lim f(x) as x → a exists, (3) the limit equals f(a). Types of discontinuities: removable (limit exists but ≠ f(a) or f(a) undefined — 'hole'), jump (one-sided limits exist but differ), infinite (limit is ±∞ — vertical asymptote), and oscillating (limit does not exist). The Intermediate Value Theorem (IVT) is a powerful consequence of continuity: a continuous function on [a,b] takes every value between f(a) and f(b).

Frequently Asked Questions: Limits and Continuity

What is a limit in calculus?

A limit is the value that a function f(x) approaches as x approaches some value a. Written lim_{x→a} f(x) = L. The function does not need to be defined at a, and the limit describes behavior near a, not at a. If f(x) approaches different values from the left and right, the two-sided limit does not exist. Limits are the foundation of derivatives (defined as a limit of a difference quotient) and integrals (defined as a limit of Riemann sums).

What is L'Hôpital's rule?

L'Hôpital's rule states that if lim_{x→a} f(x)/g(x) has the indeterminate form 0/0 or ∞/∞, then lim_{x→a} f(x)/g(x) = lim_{x→a} f'(x)/g'(x), provided this new limit exists. Apply L'Hôpital's rule by differentiating the numerator and denominator separately (not using the quotient rule). Can be applied repeatedly if the result is still indeterminate. Works for limits as x → a, x → ∞, or any other limit point.

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