calculus - Limit of $nleft(e-left(1+frac{1}{n}right)^nright)$

I want to find the value of $$\lim\limits_{n \to \infty}n\left(e-\left(1+\frac{1}{n}\right)^n\right)$$

I have already tried using L'Hôpital's rule, only to find a seemingly more daunting limit.

Answer

Let $P_n = (1+1/n)^n$. Then

$$\log{P_n} = n \log{\left ( 1+\frac1{n} \right )} = n \left (\frac1{n} - \frac1{2 n^2} + \cdots \right) = 1-\frac1{2 n} + \cdots$$

$$P_n = e^{1-1/(2 n)+\cdots} = e \left (1-\frac1{2 n} + \cdots \right )$$

Thus

$$\lim_{n \to \infty} n (e-P_n) = \frac{e}{2}$$