Failure of L’Hôpital’s Rule: $ \lim\limits_{x\rightarrow 0^+} \dfrac{e^{-1/x}}{x^p} $ Consider
$$
\lim_{x\rightarrow 0^+} \frac{e^{-1/x}}{x^p}
$$
where $p\in \mathbb{R}^+\;(p>0)\;.$
Despite satisfying all of the criteria for L’Hôpital’s Rule, repeated applications continue to yield the same indeterminate limit, whereas it can be shown heuristically, that the limit is indeed zero.
I don't see how the Squeeze Theorem would help me here. Is there an alternative? Thanks.
Many thanks for all the answers. I wish I'd thought of the clever $u=1/x$ substitution beforehand!

You can still use it, after a change of variable $u=1/x$ that
\begin{align*}
\lim_{u\rightarrow\infty}\dfrac{u^{p}}{e^{u}}.
\end{align*}
Since $p>0$, repeating the L'Hopital $m$ times such that $p-m<0$ will do.
