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 $\underset{x\to c}{\lim }f(x)=L_1$ and $\underset{x\to c}{\lim }g(x)=L_2$ then:

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

$\underset{x\to c}{\lim }[f(x)\cdot g(x)]=L_1\cdot L_2$

$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\frac{L_1}{L_2}\text{ if }{L}_2\ne 0$

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

$\underset{x\to c}{\lim }\sqrt[n]{f(x)}=\sqrt[n]{L_1}\text{ if }n\text{ is a positive integer, and if }n\text{ is even, then }{L}_1>0$

$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\underset{x\to c}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}\text{ if }\underset{x\to c}{\lim }f(x)=\underset{x\to c}{\lim }g(x)=0\text{ or }\underset{x\to c}{\lim }|g(x)|=\mathrm{\infty }$ (L'Hôpital's rule)

Limits of general functions

$\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}=f^{\prime }(x)$

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

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

Notable special limits

$\underset{x\to \mathrm{\infty }}{\lim }{(1+\frac{k}{x})}^m=e^{mk}$

$\underset{x\to \mathrm{\infty }}{\lim }{(1-\frac{1}{x})}^x=\frac{1}{e}$

$\underset{x\to \mathrm{\infty }}{\lim }{(1+\frac{k}{x})}^x=e^k$

$\underset{n\to \mathrm{\infty }}{\lim }\frac{n}{\sqrt[n]{n!}}=e$

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

Simple functions

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

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

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

$\underset{x\to c}{\lim }{x}^r=c^r\text{ if }r\text{ is a positive integer}$

$\underset{x\to 0^{+}}{\lim }\frac{1}{x^r}=\mathrm{\infty }$

$\underset{x\to 0^{-}}{\lim }\frac{1}{x^r}=\{\begin{array}{cc}-\mathrm{\infty },& \text{if }r\text{ is odd}\\ \mathrm{\infty },& \text{if }r\text{ is even}\end{array}$

Logarithmic and exponential functions

For $a>1$ :

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

$\underset{x\to \mathrm{\infty }}{\lim }{\log }_a(x)=\mathrm{\infty }$

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

$\underset{x\to \mathrm{\infty }}{\lim }{a}^x=\mathrm{\infty }$

Trigonometric functions

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

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

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

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

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

$\underset{x\to n^{\pm }}{\lim }\tan (\pi x+\frac{\pi }{2})=\mp \mathrm{\infty }\text{for any integer }n$

Near infinities

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

$\underset{x\to \mathrm{\infty }}{\lim }\frac{x}{N}=\{\begin{array}{cc}\mathrm{\infty },& N>0\\ \text{does not exist},& N=0\\ -\mathrm{\infty },& N<0\end{array}$

$\underset{x\to \mathrm{\infty }}{\lim }{x}^N=\{\begin{array}{cc}\mathrm{\infty },& N>0\\ 1,& N=0\\ 0,& N<0\end{array}$

$\underset{x\to \mathrm{\infty }}{\lim }{N}^x=\{\begin{array}{cc}\mathrm{\infty },& N>1\\ 1,& N=1\\ 0,& -1<N<1\end{array}$

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

$\underset{x\to \mathrm{\infty }}{\lim }\sqrt[x]{N}=\{\begin{array}{cc}1,& N>0\\ 0,& N=0\\ \text{does not exist},& N<0\end{array}$

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

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

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

Reference

1. List_of_limits