List of limits

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

Limits for general functions

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

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

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

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

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

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

$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \qquad \text{ if } \lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0 \text{ or } \pm\infty$ (L'Hôpital's rule)

Limits of general functions

$\lim_{h\to 0}{f(x+h)-f(x)\over h}=f'(x)$

$\lim_{h\to0}\left(\frac{f(x+h)}{f(x)}\right)^\frac{1}{h}=\exp\left(\frac{f'(x)}{f(x)}\right)$

$\lim_{h \to 0}{ \left({f(x(1+h))\over{f(x)}}\right)^{1\over{h}} }=\exp\left(\frac{x f'(x)}{f(x)}\right)$

Notable special limits

$\lim_{x\to+\infty} \left(1+\frac{k}{x}\right)^{mx}=e^{mk}$

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

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

$\lim_{x\to 0} \left(1+x\right)^\frac{1}{x}=e$

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

$\lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}=e$

$\lim_{n\to \infty }\, 2^{n} \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\text{...} +\sqrt{2}}}}}_n= \pi$

$\lim_{x \to 0} \left( \frac{a^x - 1}{x} \right) = \ln{a}, \qquad \forall~a > 0$

Simple functions

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

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

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

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

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

$\lim_{x \to 0^-} \frac{1}{x^r} = \begin{cases} -\infty, & \text{if } r \text{ is odd} \\ +\infty, & \text{if } r \text{ is even}\end{cases}$

Logarithmic and exponential functions

$\lim_{x\to1}\frac{\ln(x)}{x-1}=1$

or

$\lim_{y\to0}\frac{\ln(y+1)}{y}=1$

$\mbox{For } a > 1: \,$

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

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

$\lim_{x \to -\infty} a^x = 0$

$\mbox{If } a < 1: \,$

$\lim_{x \to -\infty} a^x = \infty$

Trigonometric functions

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

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

If $x$ is expressed in radians:

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

$\lim_{x \to 0} \frac{\sin ax}{ax} = 1$ for a is a constant

$\lim_{x \to 0} \frac{1-\cos x}{x} = 0$

$\lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2}$

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

$\lim_{x \to 0} \frac{\sin ax}{x} = a$

$\lim_{x \to 0} \frac{\sin ax}{bx} = \frac{a}{b}$

Near infinities

$\lim_{x\to\infty}N/x=0 \text{ for any real }N$

$\lim_{x\to\infty}x/N=\begin{cases} \infty, & N > 0 \\ \text{does not exist}, & N = 0 \\ -\infty, & N < 0 \end{cases}$

$\lim_{x\to\infty}x^N=\begin{cases} \infty, & N > 0 \\ 1, & N = 0 \\ 0, & N < 0 \end{cases}$

$\lim_{x\to\infty}N^x=\begin{cases} \infty, & N > 1 \\ 1, & N = 1 \\ 0, & 0 < N < 1 \end{cases}$

$\lim_{x\to\infty}N^{-x}=\lim_{x\to\infty}1/N^{x}=0 \text{ for any } N > 1$

$\lim_{x\to\infty}\sqrt[x]{N}=\begin{cases} 1, & N > 0 \\ 0, & N = 0 \\ \text{does not exist}, & N < 0 \end{cases}$

$\lim_{x\to\infty}\sqrt[N]{x}= \infty \text{ for any } N > 0$

$\lim_{x\to\infty}\log x=\infty$

$\lim_{x\to0^+}\log x=-\infty$