TITLE: Evaluating $\lim\limits_{x\rightarrow0}\frac{e^{-1/x^2}}{x}$
QUESTION [7 upvotes]: Today in my analysis class, we were preparing for the final and this question came up:

Evaluate $$\lim_{x\to0}\frac{e^{-1/x^2}}{x}$$

We tried taking the $\log$, using L'Hopitals and some other tricks but couldn't figure it out. 
I thought maybe viewing this limits as a sequence in the following way might help;
$$\lim_{x\rightarrow0}\frac{e^{-1/x^2}}{x} = \lim_{n\rightarrow \infty}\frac{e^{-n^2}}{1/n} = \lim_{n\rightarrow \infty}\frac{n}{e^{n^2}}$$
But from them I'm not sure. 
Thank you.

REPLY [3 votes]: As you noted, it sufficed to compute $\displaystyle \lim_{n\to\infty} \frac{n}{e^{n^{2}}}$. Well, using L'Hopital and the squeeze theorem and whatnot,
$$
0 \leq \lim_{n\to\infty} \frac{n}{e^{n^{2}}} \leq \lim_{n\to\infty} \frac{n}{e^{n}} = \lim \frac{1}{e^{n}} = 0.
$$