List of Limit

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

Limits for general functions

If ${\displaystyle \lim _{x\to c}f(x)=L_{1}}$ and ${\displaystyle \lim _{x\to c}g(x)=L_{2}}$ then:

${\displaystyle \lim _{x\to c}{\Big [}f(x)\pm g(x){\Big ]}=L_{1}\pm L_{2}}$

${\displaystyle \lim _{x\to c}{\Big [}f(x)\cdot g(x){\Big ]}=L_{1}\cdot L_{2}}$

${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad {\text{ if }}L_{2}\neq 0}$

${\displaystyle \lim _{x\to c}f(x)^{n}=L_{1}^{n}\qquad {\text{ if }}n{\text{ is a positive integer}}}$

${\displaystyle \lim _{x\to c}{\sqrt[{n}]{f(x)}}={\sqrt[{n}]{L_{1}}}\qquad {\text{ if }}n{\text{ is a positive integer, and if }}n{\text{ is even, then }}L_{1}>0}$

${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}\qquad {\text{ if }}\lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\lim _{x\to c}{\bigl |}g(x){\bigr |}=\infty }$ (L'Hôpital's rule)

Limits of general functions

${\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=f'(x)}$

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

${\displaystyle \lim _{h\to 0}\left({\frac {f(x(1+h))}{f(x)}}\right)^{\frac {1}{h}}=\exp \left({\frac {xf'(x)}{f(x)}}\right)}$

Notable special limits

${\displaystyle \lim _{x\to \infty }\left(1+{\frac {k}{x}}\right)^{m}=e^{mk}}$

${\displaystyle \lim _{x\to \infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}}$

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

${\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e}$

${\displaystyle \lim _{n\to \infty }2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\cdot +{\sqrt {2}}}}}}}} _{n}=\pi }$

Simple functions

${\displaystyle \lim _{x\to c}a=a}$

${\displaystyle \lim _{x\to c}x=c}$

${\displaystyle \lim _{x\to c}ax+b=ac+b}$

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

${\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x^{r}}}=\infty }$

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

Logarithmic and exponential functions

For ${\displaystyle a>1}$ :

${\displaystyle \lim _{x\to 0^{+}}\log _{a}(x)=-\infty }$

${\displaystyle \lim _{x\to \infty }\log _{a}(x)=\infty }$

${\displaystyle \lim _{x\to -\infty }a^{x}=0}$

${\displaystyle \lim _{x\to \infty }a^{x}=\infty }$

Trigonometric functions

${\displaystyle \lim _{x\to a}\sin(x)=\sin(a)}$

${\displaystyle \lim _{x\to a}\cos(x)=\cos(a)}$

${\displaystyle \lim _{x\to 0}{\frac {\sin(x)}{x}}=1}$

${\displaystyle \lim _{x\to 0}{\frac {1-\cos(x)}{x}}=0}$

${\displaystyle \lim _{x\to 0}{\frac {1-\cos(x)}{x^{2}}}={\frac {1}{2}}}$

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

Near infinities

${\displaystyle \lim _{x\to \infty }{\frac {N}{x}}=0{\text{ for any real }}N}$

${\displaystyle \lim _{x\to \infty }{\frac {x}{N}}={\begin{cases}\infty ,&N>0\\{\text{does not exist}},&N=0\\-\infty ,&N<0\end{cases}}}$

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

${\displaystyle \lim _{x\to \infty }N^{x}={\begin{cases}\infty ,&N>1\\1,&N=1\\0,&-1<N<1\end{cases}}}$

${\displaystyle \lim _{x\to \infty }N^{-x}=\lim _{x\to \infty }{\frac {1}{N^{x}}}=0{\text{ for any }}N>1}$

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

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

${\displaystyle \lim _{x\to \infty }\log(x)=\infty }$

${\displaystyle \lim _{x\to 0^{+}}\log(x)=-\infty }$

Reference

1. List_of_limits