[Math] The Limit of $\frac {2^\sqrt { \log(\log n)}}{\log n}$

limits

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.

Best 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 $n\to+\infty$.

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