The Ultimate Table of limits - American History Information Guide and Reference

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This is a table of limits for common functions. Note that a and b are constants with respect to x.

Contents

1 Limits for general functions

2 Simple functions

3 Logarithmic and exponential functions

4 Trigonometric functions

5 Near infinities

Limits for general functions

$\mbox{If }\lim_{x \to a} f(x) = L_1 \mbox{ and }\lim_{x \to a} g(x) = L_2 \mbox{ then:}$

$\lim_{x \to c} \, [f(x) \pm g(x)] = L_1 \pm L_2$

$\lim_{x \to c} \, [f(x)g(x)] = L_1 \times L_2$

$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L_1}{L_2} \qquad \mbox{ if } L_2 \ne 0$

$\lim_{x \to c} \, f(x)^n = L_1^n \qquad \mbox{ if }n \mbox{ is a positive integer}$

$\lim_{x \to c} \, f(x)^{1 \over n} = L_1^{1 \over n} \qquad \mbox{ if }n \mbox{ is a positive integer, and if } n \mbox{ is even, then } L_1 > 0$

$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \qquad \mbox{ if } \lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0 \mbox { or } \lim_{x \to c} |g(x)| = +\infty$ (L'Hopital's Rule)

Simple functions

$\lim_{x \to c} a = a$

$\lim_{x \to c} x = c$

$\lim_{x \to c} ax + b = ac + b$

$\lim_{x \to c} x^r = c^r \qquad \mbox{ if } r \mbox{ is a positive integer}$

$\lim_{x \to 0^+} \frac{1}{x^r} = +\infty$

$\lim_{x \to 0^-} \frac{1}{x^r} = \left\{ \begin{matrix} -\infty, & \mbox{if } r \mbox{ is odd} \\ +\infty, & \mbox{if } r \mbox{ is even}\end{matrix} \right.$

Logarithmic and exponential functions

$\lim_{x \to 0} \log_a x = -\infty$

$\lim_{x \to -\infty} a^x = 0 \qquad \mbox{ if } a > 0$

Trigonometric functions

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

$\lim_{x \to a} \sin x = \sin a$

$\lim_{x \to a} \cos x = \cos a$

$\lim_{x \to n^{\pm}} \tan (\pi x + \frac{\pi}{2}) = \mp\infty \qquad \mbox{ for any integer } n$

Near infinities

$\lim_{x\to\infty}N/x=0 \mbox{ for any real N}$
$\lim_{x\to\infty}x/N=\begin{cases} \infty, & N > 0 \\ \mbox{does not exist}, & N = 0 \\ -\infty, & N < 0 \end{cases}$
$\lim_{x\to\infty}x^N=\begin{cases} \infty, & N > 0 \\ 1, & N = 0 \\ 0, & N < 0 \end{cases}$
$\lim_{x\to\infty}N^x=\begin{cases} \infty, & N > 1 \\ 1, & N = 1 \\ 0, & N < 1 \end{cases}$
$\lim_{x\to\infty}N^{-x}=\lim_{x\to\infty}1/N^{x}=0 \mbox{ for any } N > 1$
$\lim_{x\to\infty}\sqrt[x]{N}=\begin{cases} 1, & N > 0 \\ 0, & N = 0 \\ \mbox{does not exist}, & N < 0 \end{cases}$
$\lim_{x\to\infty}\sqrt[N]{x}=\begin{cases} 1, & N > 0 \\ (-1..1), & N = 0 \\ -1, & N < 0 \end{cases}$
$\lim_{x\to\infty}\log x=\infty$
$\lim_{x\to0^+}\log x=-\infty$