Finding the Limit (1)

We can write few terms of $\large \lim_{n\to\infty} \frac{n!}{n^n} = \Bigg(\frac{1}{n}\Bigg)\Bigg(\frac{2}{n}\Bigg)\Bigg(\frac{3}{n}\Bigg)\dots\Bigg(\frac{n}{n}\Bigg)$ Note as $n \rightarrow \infty$ , the denominator goes to $\infty$ ; thus the limit is $\large \lim_{n\to\infty} \frac{n!}{n^n} = 0$

It suffices to use Stirling's approximation : for large $n$ , $n! =\sqrt n\times {(\dfrac{n}{e})^n}$ , so that $\dfrac{n!}{n^n}=\dfrac{\sqrt n}{e^n}$ , which is $\boxed 0$ under the given limit.