How to find $\lim_{n\to\infty}\log(1+\frac{1}{n})\sqrt[n]{n!}$ [closed]

As a physicist I may not be extremely rigorous, but as $n\rightarrow\infty$, one has that $1/n\rightarrow 0$, so the $\log(1+1/n)$ can be Taylor expanded to $1/n$ which determines its asymptotical behaviour as $n\rightarrow\infty$.
As for the $n!$ you already found the correct asymptotical behaviour, which is given by Stirling's approximation, $(n/e)^n$ (the $\sqrt{2\pi n}$ can be neglected as it represents higher order correction which is not useful in this case).

So one has: $$\log\left(1+\frac{1}{n}\right)\sqrt[n]{n!} \sim \frac{1}{n}\left( \frac{n}{e}\right) =\frac{1}{e}$$ So the limit is $$\lim_{n \to \infty}\left[ \log\left(1+\frac{1}{n}\right)\sqrt[n]{n!}\right]=\frac{1}{e}$$ Hope it helped!