teorema limiti a zero e ad infinito in R

Teorema

Se $\underset{x\to x_0}{\lim }f(x)=+\infty$, allora $\underset{x\to x_0}{\lim }\frac{1}{f(x)}=0$

se $\underset{x\to x_0}{\lim }f(x)=0$ e f(x) > 0 per $x\ne x_0$, allora $\underset{x\to x_0}{\lim }\frac{1}{f(x)}=+\infty$

Dimostrazione

Sia $x_0\in \mathbb{R}$,

$\underset{x\to x_0}{\lim }f(x)=+\infty$
$\Updownarrow$
$\forall M>0,\exists \delta >0:\forall x\in E,0<|x-x_0|<\delta$
$\Downarrow$
$f(x)>M>0$
$\Updownarrow$
$0<\frac{1}{f(x)}<\frac{1}{M}=\epsilon$

$\forall \epsilon >0$, fisso $M=\frac{1}{\epsilon }$

ottengo $\frac{1}{f(x)}<\frac{1}{M}=\epsilon \Rightarrow 0<\frac{1}{f(x)}<\epsilon$

conclusione $\underset{x\to x_0}{\lim }\frac{1}{f(x)}=0$

Risorse