Math Formulas || Limits and Derivaties || Limits

$\displaystyle \lim_{x \to a} ~\big[ f(x) \pm g(x) \big] $ $=\displaystyle \lim_{x \to a}~ f(x)\pm \lim_{x \to a}~g(x) $
$ \displaystyle \lim_{x \to a} ~\big[ f(x) \cdot g(x) \big] $ $=\displaystyle \lim_{x \to a}~ f(x)\cdot \lim_{x \to a}~g(x) $
$ \displaystyle \lim_{x \to a} \dfrac{f(x)}{g(x)} = \dfrac{\displaystyle \lim_{x \to a} ~f(x)}{\displaystyle \lim_{x \to a} g(x)} $
$\displaystyle \lim_{x \to a} c\cdot f(x) = c \cdot \lim_{x \to a}f(x) $
$\displaystyle \lim_{x \to a} \dfrac{1}{f(x)} = \dfrac{1}{\displaystyle \lim_{x \to a}f(x)} $
$\displaystyle \lim_{x \to \infty}~\left(1+\dfrac{1}{n}\right)^n = e $
$\displaystyle \lim_{x \to \infty}~(1 + n)^{1/n} = e $
$\displaystyle \lim_{x \to 0}~\dfrac{\sin x}{x} = 1 $
$\displaystyle \lim_{x \to 0}~\dfrac{\tan x}{x} = 1$
$\displaystyle\lim_{x \to 0}~\dfrac{\cos x-1}{x} = 0 $
$\displaystyle\lim_{x \to a}~\dfrac{x^n - a^n}{x-a} = n\,a^{n-1} $
$\displaystyle \lim_{x \to 0}~\dfrac{\sin^{-1} x}{x} = 1 $
$\displaystyle \lim_{x \to 0}~\dfrac{\tan^{-1} x}{x} = 1 $
$\displaystyle \lim_{x \to 0} ~a^x=1 $
$\displaystyle \lim_{x \to 0} ~\dfrac{\ln(1+x)}{x}=1 $