Answer
$e^{-4}$
Work Step by Step
We are given the sequence: $\left\{\left(1-\dfrac{4}{n}\right)^{n}\right\}$
Rewrite the general term of the sequence as:
$\left(1-\dfrac{4}{n}\right)^{n}=\left(\left(\left(1+(\dfrac{1}{-\frac{n}{4}})\right)^{-n/4}\right)^{-4/n}\right)^n$
Now, we will use the fact that $\lim\limits_{n \to \infty} \left(1+\dfrac{1}{n}\right)^n=e$ to compute the limit of the given sequence to obtain:
$\lim\limits_{n \to \infty} \left(1-\dfrac{4}{n}\right)^{n}=\lim\limits_{n \to \infty}
\left(\left(\left(1+(\dfrac{1}{-\frac{n}{4}})\right)^{-n/4}\right)^{-4/n}\right)^n$
or, $\lim\limits_{n \to \infty} \left(1-\dfrac{4}{n}\right)^{n}=e^{\frac{-4}{n} (n)}=e^{-4}$
