Limit :- $n!$ , $n^n$ and $\frac {1}{n}$

$\begin{aligned} L &= \lim_{n \to \infty} \left(\frac {n!}{n^n}\right)^\frac 1n \\ &= \lim_{n \to \infty} \left(\prod_{k=1}^n \frac kn \right)^\frac 1n \\ &= \lim_{n \to \infty} \exp \left(\ln \left(\prod_{k=1}^n \frac kn \right)^\frac 1n \right) \\ &= \exp \left( \lim_{n \to \infty} \ln \left(\prod_{k=1}^n \frac kn \right)^\frac 1n \right) \\ &= \exp \left( \lim_{n \to \infty} \frac 1n \sum_{k=1}^n \ln \frac kn \right) & \small \color{#3D99F6} \text{By Riemann sums } \\ &= \exp \left( \int_0^1 \ln x \ dx \right) \\ &= \exp \left( x \ln x -x \right) \bigg|_0^1 \\ &= \boxed{\dfrac 1e} \end{aligned}$