User:Foxjwill/Limit Notes

Limit laws

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

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

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

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

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

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

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

Special limits

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

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

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

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

Limits by direct substitution

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

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

Derivitive

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

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

if this limit exists.