Limits

The foundational idea of calculus — what a function approaches as its input gets arbitrarily close to a value.

Limit of f(x) = (x²−1)/(x−1) as x→1 — drag the slider to approach the hole

far ←→ close

As x approaches 1 from both sides, f(x) → 2 — even though f(1) is undefined

Definition

The limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ gets arbitrarily close to as $x$ gets closer and closer to $a$ — without $x$ ever actually equalling $a$ .

$\lim_{x \to a} f(x) = L$

The key insight: we care about what happens near $a$ , not at $a$ . The function need not even be defined at $a$ — the limit only asks what the function is approaching.

A limit where the function is undefined at the point

Find $\displaystyle\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$ .

At $x = 1$ , the expression is $\frac{0}{0}$ — undefined. But factor the numerator:

$\frac{x^2 - 1}{x - 1} = \frac{(x-1)(x+1)}{x-1} = x + 1 \qquad (x \neq 1)$

As $x \to 1$ , $x + 1 \to 2$ . So the limit is $2$ , even though the function has a hole at $x = 1$ .

Evaluate the limit

Evaluate $\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$ .

Related concepts

Calculus· Differentiation

DerivativesThe instantaneous rate of change of a function — defined as the limit of the difference quotient and interpreted as the slope of a tangent line.

Calculus· Series

Series & ConvergenceInfinite sums, tests for convergence, power series, and Taylor series — when adding infinitely many terms gives a finite and useful answer.

Limits

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