Calculus Course/Limit

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

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

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

$\lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad {\text{ if }}L_{2}\neq 0$$\lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad {\text{ if }}L_{2}\neq 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)^{n}=L_{1}^{n}\qquad {\text{ if }}n{\text{ is a positive integer}}$

$\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$$\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$

$\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$$\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

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

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

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

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

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

$\lim _{x\to \infty }\left(1+{\frac {k}{x}}\right)^{x}=e^{k}$$\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 }{\frac {n}{\sqrt[{n}]{n!}}}=e$

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

Simple functions

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

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

$\lim _{x\to c}ax+b=ac+b$$\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 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}}}=\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}}$$\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 $a>1$$a>1$ :

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

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

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

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

Trigonometric functions

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

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

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

$\lim _{x\to 0}{\frac {1-\cos(x)}{x}}=0$$\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 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 n^{\pm }}\tan \left(\pi x+{\frac {\pi }{2}}\right)=\mp \infty \qquad {\text{for any integer }}n$

Near infinities

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

$\lim _{x\to \infty }{\frac {x}{N}}={\begin{cases}\infty ,&N>0\\{\text{does not exist}},&N=0\\-\infty ,&N<0\end{cases}}$$\lim _{x\to \infty }{\frac {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 }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,&-1<N<1\end{cases}}$$\lim _{x\to \infty }N^{x}={\begin{cases}\infty ,&N>1\\1,&N=1\\0,&-1<N<1\end{cases}}$

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

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

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

Reference

1. List_of_limits