I would like to understand why $ \lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}= \lim_{n\to\infty}(1+\frac{1}{n})^{n}$.
I was given a solution, but it gives no further details than $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}=\lim_{n\to\infty}\sqrt[n]{\frac{n^n}{n!}}=\lim_{n\to\infty}\frac{(n+1)^{(n+1)}/(n+1)!}{n^n/n!}=\lim_{n\to\infty}(1+\frac{1}{n})^n.$$
What happened between $\lim_{n\to\infty}\sqrt[n]{\frac{n^n}{n!}}$ and $\lim_{n\to\infty}\frac{(n+1)^{(n+1)}/(n+1)!}{n^n/n!}$?