today I was thinking on a problem which states:
let $e(x)$ be a function defined by
$e(x)=e^{-\frac{1}{x}}$ if $x>0$ and $e(x)=0$ if $x\le 0$. the problem asks to prove the function is smooth. I tried to prove $\lim_{x\to 0^+}\frac{e^{-\frac{1}{x}}}{x}=0$ using the L'Hospitals rule, but applying that results in more $x$'s in the denominator. what should I do?