Engineering Handbook/Calculus/Limit

Definitions

To say that

$\lim _{x\to p}f(x)=L,\,$

means that ƒ(x) can be made as close as desired to L by making x close enough, but not equal, to p.

The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts.

List of Limit

Limits for general functions

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

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

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

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

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

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

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

Limits of general functions

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

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

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

Notable special limits

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

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

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

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

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

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

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

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

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

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

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

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

Trigonometric functions

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

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

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

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

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

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

Near infinities

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

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

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

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

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

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

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

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

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