Proof of limit in analysis

Question

I've read the following question in a book :

''Prove that limit as $n$ goes to infinity of $n/(n!)^{1/n}$ = $e$.''

I've tried to prove it using the definition of a limit, but it's really complicated.

Answer 1

As the comments thus far have hinted, this is a nontrivial problem to do from first principles, but turns out to be essentially equivalent to a well-known result called Stirling's formula or Stirling's approximation. Note that since $\log x$ is continuous at $x=e$, it is equivalent to show that $$ \lim_{n\to\infty} \bigg( \log\frac n{(n!)^{1/n}} \bigg) = \log e \qquad\text{or}\qquad \lim_{n\to\infty} \big( \tfrac1n\log(n!) - \log n \big) = -1. $$ There are several ways to derive this limit, including some given at the link above.