TITLE: How to calculate $\lim\limits_{n\to \infty} n^{\frac {\log n}{n^2}}$?
QUESTION [6 upvotes]: I need help to calculate this limit $$\large\lim_{n\to \infty} n^{\frac {\log n}{n^2}}.$$
I know  the limit of the exponent goes to zero, but the limit of the base goes to infinity, I think the limit goes to 1, am I right?
Thanks a lot

REPLY [7 votes]: Hint: Expanding on Andre's hint:
$$\log\left(n^{\frac{\log n}{n^2}}\right)=\frac{\log n\cdot\log n}{n^2}=\left(\frac{\log n}{n}\right)^2$$
Recall that by the continuity of $\log$ we have that $\lim_{n\to\infty}\log(f(n)) = \log(\lim_{n\to\infty} f(n))$.