Estimate a common formula

Question

I know this should be easy, but I just can't find the proper search result. Thanks.

$\left(1-\frac1n\right)^n$, what is the estimation value when $n$ is very large?

Some follow-up,

If $n = 100$, what is the formula to calculate this?

Answer 1

The estimation value when n is very large is $1/e$, where $e$ is the well known mathematical constant.

\begin{gather} \lim\limits_{n\to\infty}\left(1-\dfrac{1}{n}\right)^{-n}=\lim\limits_{n\to\infty}\left(\dfrac{n-1}{n}\right)^{-n}= \\ =\lim\limits_{n\to\infty}\left(\dfrac{n}{n-1}\right)^{n}=\lim\limits_{n\to\infty}\left[\left(1+\dfrac{1}{n-1}\right)^{n-1}\left(1+\dfrac{1}{n-1}\right)\right]=e \end{gather}

Therefore, what you are seeking is $e^{-1}$, or $1/e$.