Very easy question about infinitesimals

Question

how can I prove that: $$ \lim_{x\to 0} \frac{e^{-1/x^2}}{x} = 0 ? $$ I suppose that the exponential "goes" to $0$ faster than linear, but I'm not sure.

Answer 1

Make the change of variable $y = \frac{1}{x^2}$, then $y \to +\infty $, when $x \to 0$ $$ \frac{e^{\frac{-1}{x^2}}}{x} = \sqrt{y}e^{-y} $$

Answer 2

As it is, your limit goes to $\infty$, since $1/x^2$ is very big whenever $x$ approaches zero. A very different story would be if you had to compute $$ \lim_{x\to 0}\frac{e^{-1/x^2}}{x}. $$