I need to find the functions, f(x), for which the following is satisfied (or alternatively the functions for which it is not):
$$\text{lim}_{\:x\to \infty} \large{{f(x)\over e^{-x^m\over2}}} \to \pm\infty $$
For example if $f(x) = e^{-x^n\over2},\: n\geq m $ then it is not satisfied. Are there any others? I've thought about using l'hopital's rule also which would suggest that something of the form $f(x) = -{n\over2}x^{n-1} e^{-x^n\over2},\: n\geq m$ would also not be satisfied. Finally I guess anything of the form $f(x) = g(x)e^{-x^n\over2},\: n\geq m$, so long as $g(x) \to 0$ as $x \to 0$. Are there any other functions I've missed. I need to prove that no more solutions exists. Cheers, guys!
I don't see the big problem here. Take any function $g(x)$ such that $\lim_{x\to\infty}g(x)=\infty$, then $f(x)=\pm{}g(x)\exp(-x^m/2)$ will do the trick. Furthermore, those are all the functions, since for all $g$ with $\lim_{x\to\infty}g(x)=a$ and $|a|<\infty$, you will have $\lim_{x\to\infty}f(x)/\exp(-x^m/2)=a$.