Find a function $f(x)$ so that $\lim\_{x \to \infty} f(x)=0$ and $\lim\_{x \to \infty}x^{\frac{1}{f(x)}}=\infty$

I have plugged maybe a hundred functions but I can't seem to figure this out. I'm wondering whether such a function exists at all since we need to find some function that goes to zero extremely slow.

Could somebody help me out?