Mathematics - Limits Question with Solution | TestHub

First, analyze the inner limit: ${\lim }_{n\to \infty }{\left(1+\frac{1}{n}\right)}^n=e$ and ${\lim }_{n\to \infty }\left(1+\frac{1}{n}\right)=1$.

Thus, ${\lim }_{n\to \infty }\left[{\left(1+\frac{1}{n}\right)}^n-\left(1+\frac{1}{n}\right)\right]=e-1$.

Therefore, $L={\lim }_{n\to \infty }(e-1{)}^{-n}$. Since $e-1>1$, $(e-1{)}^{-n}=\frac{1}{(e-1{)}^n}$ approaches 0 as $n\to \infty$.

Hence, $L=0$.

Final Answer: The final answer is $\boxed{0}$