The Limit of $frac {2^sqrt { log(log n)}}{log n}$

Wolfram tells me that the the limit is $0$ when $n$ goes to infinity.
Unfortunately, I have no idea how to prove it...

$$\lim_{n\to\infty}\frac {2^\sqrt { \log(\log n)}}{\log n}.$$

Any help would be appreciated,
thanks in advance.

Answer

Hints:

1. The logarithm of this quantity is $\log 2\cdot\sqrt{\log(\log n)}-\log(\log n)$.




2. When $n\to+\infty$, $\log(\log n)\longrightarrow$ $_________$.




3. When $x\to+\infty$, $\log2\cdot\sqrt{x}-x\longrightarrow$ $_________$.




4. Hence $\log2\cdot\sqrt{\log(\log n)}-\log(\log n)\longrightarrow$ $________$ when $n\to+\infty$.





5. And finally $2^{\sqrt{\log(\log n)}}/\log n\longrightarrow$ $_________$ when $n\to+\infty$.