Formulario sui limiti

FORMULARIO 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

 

\({{e}^{x}}=\sum\limits_{n=0}^{+\infty }{\frac{1}{n!}{{x}^{n}}=\,\,\,}1+x+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}+o\left( {{x}^{3}} \right)\)

\(\sin x=\sum\limits_{n=0}^{+\infty }{{{\left( -1 \right)}^{n}}\frac{1}{\left( 2n+1 \right)!}{{x}^{2n+1}}}=\,\,\,x-\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}+o\left( {{x}^{5}} \right)\)

\(\cos x=\sum\limits_{n=0}^{+\infty }{{{\left( -1 \right)}^{n}}\frac{1}{\left( 2n \right)!}{{x}^{2n}}}=\,\,\,1-\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}+o\left( {{x}^{4}} \right)\)

\(\tan x=x+\frac{{{x}^{3}}}{3}+\frac{2{{x}^{5}}}{15}+\frac{17{{x}^{7}}}{315}+\frac{62{{x}^{9}}}{2835}+o({{x}^{9}})\)

\(\ln \left( 1+x \right)=\sum\limits_{n=1}^{+\infty }{{{\left( -1 \right)}^{n+1}}\frac{1}{n}{{x}^{n}}=\,\,\,x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}+o\left( {{x}^{3}} \right)}\) \(\arctan \left( x \right)=x-\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+o\left( {{x}^{5}} \right)\)

\(\arcsin x=x+\frac{{{x}^{3}}}{6}+\frac{3{{x}^{5}}}{40}+\frac{5{{x}^{7}}}{122}+\frac{35{{x}^{9}}}{1152}+o\left( {{x}^{9}} \right)\)

\(\arccos x=\frac{\pi }{2}x-\frac{{{x}^{3}}}{6}-\frac{3{{x}^{5}}}{40}-\frac{5{{x}^{7}}}{122}-\frac{35{{x}^{9}}}{1152}+o\left( {{x}^{9}} \right)\)

\(\sinh \left( x \right)=x+\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}+o\left( {{x}^{5}} \right)\)

\(\cosh x=1+\frac{{{x}^{2}}}{2}+\frac{{{x}^{4}}}{4!}+o\left( {{x}^{4}} \right)\)

\({{\tanh }^{-1}}\,x=1+\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+o\left( {{x}^{5}} \right)\)

\({{\sinh }^{-1}}\,x=x-\frac{{{x}^{3}}}{6}+\frac{3{{x}^{5}}}{40}-\frac{5{{x}^{7}}}{122}+\frac{35{{x}^{9}}}{1152}+o\left( {{x}^{9}} \right)\)