Ques. -25

$\displaystyle{L=\lim _{ n\rightarrow \infty }{ { (\prod _{ r=1 }^{ n }{ \cfrac { r }{ n } } ) }^{ \cfrac { 1 }{ n } } } \\ \ln { L } =\lim _{ n\rightarrow \infty }{ { \cfrac { 1 }{ n } \sum _{ r=1 }^{ n }{ \ln { \cfrac { r }{ n } } } } } \equiv \int _{ 0 }^{ 1 }{ \ln { x } dx } =-1\\ \boxed { L={ e }^{ -1 } } }$

An alternative solution is to consider the sequence, $a_n = \frac {n!}{n^n}$ , we need to prove that it converge by root test.

Use Stirling's approximation!