Answer
$\dfrac{1}{e^5}$
Work Step by Step
We are given the sequence:
$\left\{\left(\dfrac{n}{n+5}\right)^n\right\}$
Rewrite the general term of the sequence:
$\left(\dfrac{n}{n+5}\right)^n=\left(\dfrac{1}{\dfrac{n+5}{n}}\right)^n=\left(\dfrac{1}{1+\dfrac{5}{n}}\right)^n$
$=\dfrac{1^n}{\left(1+\dfrac{5}{n}\right)^n}=\dfrac{1}{\left(1+\dfrac{5}{n}\right)^n}=\dfrac{1}{\left(1+\dfrac{1}{\frac{n}{5}}\right)^n}$
$=\dfrac{1}{\left(\left(1+\dfrac{1}{\frac{n}{5}}\right)^{n/5}\right)^5}$
Compute the limit:
$\lim\limits_{n \to \infty}\dfrac{1}{\left(\left(1+\dfrac{1}{\frac{n}{5}}\right)^{n/5}\right)^5}=\dfrac{1}{\lim\limits_{n \to \infty}\left(\left(1+\dfrac{1}{\frac{n}{5}}\right)^{n/5}\right)^5}=\dfrac{1}{\left(\lim\limits_{n \to \infty}\left(1+\dfrac{1}{\frac{n}{5}}\right)^{n/5}\right)^5}$
Because $\lim\limits_{n \to \infty}\left(1+\dfrac{1}{n}\right)^n=e$, we have:
$\dfrac{1}{\left(\lim\limits_{n \to \infty}\left(1+\dfrac{1}{\frac{n}{5}}\right)^{n/5}\right)^5}=\dfrac{1}{e^5}$
