User:Foxjwill/Limit Notes

${\displaystyle \lim _{x\rightarrow a}\left[f(x)+g(x)\right]=\lim _{x\rightarrow a}f(x)+\lim _{x\rightarrow a}g(x)}$

${\displaystyle \lim _{x\rightarrow a}\left[f(x)-g(x)\right]=\lim _{x\rightarrow a}f(x)-\lim _{x\rightarrow a}g(x)}$

${\displaystyle \lim _{x\rightarrow a}\left[cf(x)\right]=c\lim _{x\rightarrow a}f(x)}$

${\displaystyle \lim _{x\rightarrow a}\left[f(x)g(x)\right]=\lim _{x\rightarrow a}f(x)\cdot \lim _{x\rightarrow a}g(x)}$

${\displaystyle \lim _{x\rightarrow a}\left[{\frac {f(x)}{g(x)}}\right]={\frac {\lim _{x\rightarrow a}f(x)}{\lim _{x\rightarrow a}g(x)}}{\mbox{ if }}\lim _{x\rightarrow a}g(x)\neq 0}$

${\displaystyle \lim _{x\rightarrow a}\left[f(x)\right]^{n}=\left[\lim _{x\rightarrow a}f(x)\right]^{n}}$

${\displaystyle \lim _{x\rightarrow a}{\sqrt[{n}]{f(x)}}={\sqrt[{n}]{\lim _{x\rightarrow a}f(x)}}}$

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

${\displaystyle \lim _{x\rightarrow a}x=a}$

${\displaystyle \lim _{x\rightarrow a}x^{n}=a^{n}}$

${\displaystyle \lim _{x\rightarrow a}{\sqrt[{n}]{x}}={\sqrt[{n}]{a}}}$

If ${\displaystyle f\!}$ is a polynomial or a rational function and ${\displaystyle a\!}$ is in the domain of ${\displaystyle f\!}$, then

${\displaystyle \lim _{x\rightarrow a}f(x)=f(a)}$

The derivative of a function ${\displaystyle f\!}$ at a number ${\displaystyle a\!}$, denoted by ${\displaystyle f'(a)\!}$ is

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

if this limit exists.