calculus - Calculate the limit $lim_{xrightarrow 0}dfrac{e^{-frac{1}{x}}}{x^n}$

Let $n>0$ be an integer. Calculate the limit

$$\lim_{x\rightarrow 0}\dfrac{e^{-\frac{1}{x}}}{x^n}$$

The limit is of the form $\dfrac{0}{0}$. Using L'Hospital, the derivative of the denominator is $nx^{n-1}$, while the derivative of the numerator is $\dfrac{e^{-\frac1x}}{x^2}$, so that the new fraction is $\dfrac{e^{-\frac{1}{x}}}{nx^{n+1}}$, which is $\dfrac00$ again. It doesn't help much.

Answer

Indeed, L'Hôpital fails here. You want $x\to 0^+$. What you can argue is that is is equivalent to showing

$$\lim\limits_{t\to +\infty}t^n e^{-t}=0$$ for any $n$. And L'Hôpital works nicely in this case.