Mathematics - Definite Integration Question with Solution | TestHub

$L={\lim }_{n\to \infty }{\left(\frac{n!}{n^n}\right)}^{\frac{1}{n}}$

$$\begin{array}{rl}\ln L& =\underset{n\to \infty }{\lim }\frac{1}{n}\cdot (\ln n!-n\ln n)\\ \ln L& =\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{r=1}^n\ln \left(\frac{r}{n}\right)\\ \ln L& ={\int }_0^1\ln xdx\\ \ln L& =[\ln x\cdot x-x{]}_0^1\\ \ln L& =-1-\left(\underset{x\to 0}{\lim }\frac{\ln x}{1/x}\right)\\ \ln L& =-1-0\\ L& =\frac{1}{e}\end{array}$$