prove limit of $ n^{(1/\log\log n)} /\bigl( ((\log n)^c\bigr)$

Question

I'm trying to prove that this limit is $\infty$ , c is a constant positive integer. I've tried using L'Hopital, with no success.. (not sure if it's the right way - the derivative looks too complicated) could anyone give me a hint?
$\displaystyle \lim_{n\to \infty} \left(\frac{n^{\left(\frac{1}{\log(\log(n))}\right)}}{\log^{c}(n)}\right)$

Answer 1

Hint:

Compute limit of the log: $$\frac{\log n}{\log\log n}-c\log\log n=\frac{\log n-c\log^2(\log n)}{\log\log n}.$$ and use equivalents.