Limit with parameter, $(e^{-1/x^2})/x^a$ How do I explore for which values of the parameter $a$
$$ \lim_{x\to 0} \frac{1}{x^a} e^{-1/x^2} = 0?
 $$
For $a=0$ it is true, but I don't know what other values.

It's true for all $a's$ (note that for $a\leqslant 0$ it's trivial) and you can prove it easily using L'Hopital's rule or Taylor's series for $\exp(-x^2)$.
