Finding the limit of $\lim_{n\to\infty} \frac{n^{log(n)}}{(\log n)^n}$

Question

I try to calculate the following limit:

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

I tried this:

$$ \lim_{n\to\infty} e^{(\log(n))^2 - n \log(\log(n))} $$

Is this useful? & what do now?

Answer 1

It is useful. Now, you could try to look at $\log^2 n - n\log\log n$ and see if it converges or diverges to $\pm \infty$ -- any of the two will give you the limit you're looking for.

As a start, rewrite $$\log^2 n - n\log\log n = n\left(\frac{\log^2n }{n} - \log\log n\right)$$ In the parenthesis, the second term goes to $-\infty$. What about the first?