Formulario sui limiti
I PRINCIPALI LIMITI NOTEVOLI
LIMITE DI NEPERO
$\underset{x\to 0}{\mathop{\lim }}\,{{\left( 1+x \right)}^{\frac{1}{x}}}=\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{x} \right)}^{x}}=e$
LIMITI CHE DERIVANO DA QUELLO DI NEPERO
$\underset{x\to 0}{\mathop{\lim }}\,\frac{\ln \left( 1+x \right)}{x}=1$ $\Rightarrow$ $\ln \left( 1+x \right)\underset{x\to 0}{\mathop{\sim }}\,\,\,x$
$\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{x}}-1}{x}=1$ $\Rightarrow$ ${{e}^{x}}\underset{x\to 0}{\mathop{\sim }}\,\,\,1+x$
$\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\left( 1+x \right)}^{\alpha }}-1}{x}=\alpha$ $\Rightarrow$ ${{\left( 1+x \right)}^{\alpha }}\underset{x\to 0}{\mathop{\sim }}\,\,\,1+\alpha x$
LIMITI GONIOMETRICI
$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x}{x}=1$ $\Rightarrow$ $\sin x\underset{x\to 0}{\mathop{\sim }}\,\,\,x$
$\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan x}{x}=1$ $\Rightarrow$ $\tan x\underset{x\to 0}{\mathop{\sim }}\,\,\,x$
$\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos x}{x}=0$
$\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos x}{{{x}^{2}}}=\frac{1}{2}$ $\Rightarrow$ $\cos x\underset{x\to 0}{\mathop{\sim }}\,\,\,1\,\,-\,\,\frac{{{x}^{2}}}{2}$
SVILUPPI DI MC LAURIN
${{e}^{x}}=\sum\limits_{n=0}^{+\infty }{\frac{1}{n!}{{x}^{n}}=\,\,\,}1+x+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}+o\left( {{x}^{3}} \right)$
$\sin x=\sum\limits_{n=0}^{+\infty }{{{\left( -1 \right)}^{n}}\frac{1}{\left( 2n+1 \right)!}{{x}^{2n+1}}}=\,\,\,x-\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}+o\left( {{x}^{5}} \right)$
$\cos x=\sum\limits_{n=0}^{+\infty }{{{\left( -1 \right)}^{n}}\frac{1}{\left( 2n \right)!}{{x}^{2n}}}=\,\,\,1-\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}+o\left( {{x}^{4}} \right)$
$\tan x=x+\frac{{{x}^{3}}}{3}+\frac{2{{x}^{5}}}{15}+\frac{17{{x}^{7}}}{315}+\frac{62{{x}^{9}}}{2835}+o({{x}^{9}})$
$\ln \left( 1+x \right)=\sum\limits_{n=1}^{+\infty }{{{\left( -1 \right)}^{n+1}}\frac{1}{n}{{x}^{n}}=\,\,\,x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}+o\left( {{x}^{3}} \right)}$ $\arctan \left( x \right)=x-\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+o\left( {{x}^{5}} \right)$
$\arcsin x=x+\frac{{{x}^{3}}}{6}+\frac{3{{x}^{5}}}{40}+\frac{5{{x}^{7}}}{122}+\frac{35{{x}^{9}}}{1152}+o\left( {{x}^{9}} \right)$
$\arccos x=\frac{\pi }{2}x-\frac{{{x}^{3}}}{6}-\frac{3{{x}^{5}}}{40}-\frac{5{{x}^{7}}}{122}-\frac{35{{x}^{9}}}{1152}+o\left( {{x}^{9}} \right)$
$\sinh \left( x \right)=x+\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}+o\left( {{x}^{5}} \right)$
$\cosh x=1+\frac{{{x}^{2}}}{2}+\frac{{{x}^{4}}}{4!}+o\left( {{x}^{4}} \right)$
${{\tanh }^{-1}}\,x=1+\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+o\left( {{x}^{5}} \right)$
${{\sinh }^{-1}}\,x=x-\frac{{{x}^{3}}}{6}+\frac{3{{x}^{5}}}{40}-\frac{5{{x}^{7}}}{122}+\frac{35{{x}^{9}}}{1152}+o\left( {{x}^{9}} \right)$