Table of limits

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

1 Limits for general functions
2 Simple functions
3 Logarithmic and exponential functions
4 Trigonometric functions
5 Near infinities

[edit] Limits for general functions

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

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

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

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

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

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

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

[edit] Simple functions

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

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

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

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

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

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