Limit as $x to infty$ of $frac{log(x)^{log(log(x))}}{x}$

I want to compute $\lim\limits_{x \to \infty}\frac{\log(x)^{\log(\log(x))}}{x}$

By graphing it, clearly $x$ grows larger than $\log(x)^{\log(\log(x))}$, so the limit will go to $0$.

I tried iterating L'Hopital's rule, but after three derivations, the sequence of limits gets successively more complicated.

How can you prove that the limit is indeed $0$?

Answer

HINT:

Let $x=\exp(e^u)$. Then your limit is equal to

$$\lim_{u\to\infty}\frac{(e^u)^{\log(e^u)}}{\exp(e^u)}=\lim_{u\to\infty}\frac{e^{u^2}}{e^{e^u}}=\lim_{u\to\infty}e^{u^2-e^u}=\cdots$$