Find the limit $\lim_{n\to \infty}\{{n!}^{1/n}\}/n$

Question

Find the limit $$\lim_{n\to \infty}\{{(n!)}^{1/n}\}/n$$

I took exp log but getting answer as 1 but it should be 1/e. Required a nice approach.

Answer 1

For large $n$, we have $n! \simeq \exp (n \log n - n)$, Taking the limit then gives $\lim_{n\to \infty}\{{(n!)}^{1/n}/n\} = \frac1 e$

Answer 2

Hint : Evaluate an assymptotical equivalent to $$n!^{1/n} = e^{\sum_{k=1}^n \log(k)/n}$$ by first finding one of $$\sum_{k=1}^n \log(k)\over n$$