table of limits

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

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:}

= L_1 \pm L_2

= 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

\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.

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

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

\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

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