Limit of logarithm function, $\lim\limits_{n\to\infty}\frac{n}{\log \left((n+1)!\right)}$ Determine $$\lim_{n \to \infty} \frac{n}{\log \left((n+1)!\right)}.$$
Now, I know that $\log x < \sqrt{x} < x$ and trying to apply comparison test, but it does not work. Please help.

By Stirling's approximation $\log n!=O(n\log n)$.
Also $O((n+1)\log(n+1))=O(n\log n)$.
Therefore, $$\lim_{n\to\infty}\frac{n}{\log((n+1)!)}=\lim_{n\to\infty}\frac{n}{n\log n}=\lim_{n\to\infty}\frac{1}{\log n}=0$$.
