Limits

Standard limits

$x \to 0 lim \frac{sin x}{x} = 1,$ $x \to 0 lim \frac{tan x}{x} = 1,$ $x \to 0 lim \frac{1 - cos x}{x ^{2}} = \frac{1}{2}$
$x \to 0 lim \frac{e ^{x} - 1}{x} = 1,$ $x \to 0 lim \frac{a ^{x} - 1}{x} = ln a,$ $x \to 0 lim \frac{ln ( 1 + x )}{x} = 1$
$x \to a lim \frac{x ^{n} - a ^{n}}{x - a} = n a^{n - 1},$ $x \to 0 lim \frac{( 1 + x ) ^{n} - 1}{x} = n$

$x \to a lim f (x) = 1, lim g (x) = \infty \Rightarrow lim f (x)^{g (x)} = e^{l i m g (x) (f (x) - 1)}$
$x \to 0 lim (1 + x)^{1/ x} = e,$ $x \to \infty lim (1 + \frac{a}{x})^{x} = e^{a}$

Series
$e^{x} = 1 + x + \frac{x ^{2}}{2 !} + \dots,$ $ln (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots$
$sin x = x - \frac{x ^{3}}{3 !} + \dots,$ $cos x = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots,$ $tan x = x + \frac{x ^{3}}{3} + \dots$
L'Hospital
$\frac{0}{0}$ or $\frac{\infty}{\infty} : lim \frac{f ( x )}{g ( x )} = lim \frac{f ^{'} ( x )}{g ^{'} ( x )}$
Rational function at ∞
$x \to \infty lim \frac{a _{0} x ^{m} + \dots}{b _{0} x ^{n} + \dots} = ⎩ ⎨ ⎧ 0, a_{0} / b_{0}, \pm \infty, m < n m = n m > n$