AP®MathsCollege BoardCalculus ABStudy GuidesUnit 1: Limits & ContinuityUnit 1 OverviewUnit 1 Summary

Unit 1 Summary (College Board AP® Calculus AB): Study Guide

Written by: Roger B

Reviewed by: Jamie Wood

Updated on 13 April 2026

Limits & continuity summary

Key definitions

The limit of a function $f$ at the value $x = c$ is $\lim_{x \to c} f (x) = L$ if
- if $\lim_{x \to c^{-}} f (x)$ and $\lim_{x \to c^{+}} f (x)$ exist
- $\lim_{x \to c^{-}} f (x) = \lim_{x \to c^{+}} f (x) = L$
A function $f$ is continuous at the value $x = c$ if
- $f (c)$ and $\lim_{x \to c} f (x)$ exist
- and $\lim_{x \to c} f (x) = f (c)$
A function $f$ has a removable discontinuity at the value $x = c$ if
- $f (c)$ and $\lim_{x \to c} f (x)$ exist
- and $\lim_{x \to c} f (x) \neq f (c)$
$y = c$ is a horizontal asymptote for the graph $y = f (x)$ if $\lim_{x \to \infty} f (x) = c$ or $\lim_{x \to - \infty} f (x) = c$
$x = c$ is a vertical asymptote for the graph $y = f (x)$ if $\lim_{x \to c^{-}} f (x) = \pm \infty$ or $\lim_{x \to c^{+}} f (x) = \pm \infty$

Key theorems

Squeeze theorem
- Let $f$ , $g$ and $h$ be functions defined on an open interval including $a$ such that
  - $g (x) \leq f (x) \leq h (x)$ for all $x$ in the interval (except possibly $a$ ), and
  - $\lim_{x \to a} g (x) = \lim_{x \to a} h (x) = L$
- Then $\lim_{x \to a} f (x) = L$
Intermediate value theorem (IVT)
- If $f$ is a continuous function on the closed interval $[a, b]$
- and if $d$ is a value within the closed interval created by $f (a)$ and $f (b)$
- then there is at least one number $c$ between $a$ and $b$ such that $f (c) = d$

Key formulas

The limit of a constant function: If $k$ is a constant then
- $\lim_{x \to a} k = k$
The limit of a multiple of a function: If $k$ is a constant and $\lim_{x \to a} f (x) = L$ , then
- $\lim_{x \to a} (k f (x)) = k L$
The limit of a sum or difference of functions: If $\lim_{x \to a} f (x) = L$ and $\lim_{x \to a} g (x) = M$ , then
- $\lim_{x \to a} (f (x) \pm g (x)) = L \pm M$
The limit of a product of functions: If $\lim_{x \to a} f (x) = L$ and $\lim_{x \to a} g (x) = M$ , then
- $\lim_{x \to a} (f (x) \cdot g (x)) = L \cdot M$
The limit of a quotient of functions: If $\lim_{x \to a} f (x) = L$ and $\lim_{x \to a} g (x) = M$ with $M \neq 0$ , then
- $\lim_{x \to a} \frac{f (x)}{g (x)} = \frac{L}{M}$
The limit of the power of a function: If $\lim_{x \to a} f (x) = L$ and $n$ is a real number, then
- $\lim_{x \to a} {[f (x)]}^{n} = L^{n}$
The limit of a composite function: If $\lim_{x \to a} f (x) = L$ and if the function $g$ is continuous at $x = L$ , then
- $\lim_{x \to a} g (f (x)) = g (L)$

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