CalculusCalculus 1

Limits Study Guide

Limits describe what a function approaches near an input, even when the function is undefined there. They are the foundation of continuity, derivatives and integrals.

Key concepts

Direct substitution

If the function is continuous at the point, the limit is just the function value. Always try this first.

Indeterminate forms

A result of 0/0 means more work is needed: factor and cancel, rationalize with a conjugate, or use a known special limit.

One-sided limits

Approach from the left or the right separately. The two-sided limit exists only when both agree.

Limits at infinity

For rational functions compare degrees: a bigger denominator degree gives 0, equal degrees give the ratio of leading coefficients, a bigger numerator degree gives no finite limit.

Definitions

Limit: The value $f(x)$ approaches as $x$ approaches $a$ .
Continuous at a: $\lim_{x\to a} f(x) = f(a)$ .
Vertical asymptote: A line $x = a$ where $f(x)$ grows without bound as $x \to a$ .

Formulas to know

Special trig limit

\lim_{x\to 0} \frac{\sin x}{x} = 1

Exponential limit

\lim_{x\to 0} \frac{e^x - 1}{x} = 1

Definition of e

\lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^n = e

Worked examples

Factor and cancel

$\lim_{x\to 2} \frac{x^2 - 4}{x - 2}$ gives $\frac{0}{0}$ by substitution.
Factor: $\frac{(x - 2)(x + 2)}{x - 2} = x + 2$ for $x \ne 2$ .
Substitute: $2 + 2 = 4$ .

Conjugate

$\lim_{x\to 0} \frac{\sqrt{x + 9} - 3}{x}$ gives $\frac{0}{0}$ .
Multiply by $\frac{\sqrt{x + 9} + 3}{\sqrt{x + 9} + 3}$ to get $\frac{1}{\sqrt{x + 9} + 3}$ .
Substitute: $\frac{1}{6}$ .

Common mistakes

Writing that a 0/0 limit equals 0 or does not exist without simplifying.
Forgetting to check both sides for piecewise or absolute-value functions.
Using degrees instead of radians in trig limits.

Practice questions

1. $\lim_{x\to 1} \frac{x^2 - 1}{x - 1}$
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2
2. $\lim_{x\to\infty} \frac{2x + 1}{5x - 3}$
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$\frac{2}{5}$
3. $\lim_{x\to 0} \frac{\sin 3x}{x}$
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3

Keep practising

Limit Calculator: worked steps for your own numbers.
Flashcards for this topic: review the key facts with spaced repetition.
Derivative Rules cheat sheet: every formula on one printable page.
Step-by-step solver: hints one step at a time, or check your own answer.

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