Limit of $n\left(e-\left(1+\frac{1}{n}\right)^n\right)$ 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.

Let $P_n = (1+1/n)^n$.  Then
$$\log{P_n} = n \log{\left ( 1+\frac1{n} \right )} = n \left (\frac1{n} - \frac1{2 n^2} + \frac1{3 n^3} - \cdots \right) = 1-\frac1{2 n} + \frac1{3 n^2} - \cdots$$
$$\therefore P_n = e^1e^{-1/(2 n)+ 1/(3n^2)-\cdots} = e \left (1-\frac1{2 n} + \frac{11}{24 n^2} \cdots \right ) $$
Thus
$$\lim_{n \to \infty} n (e-P_n) = \lim_{n \to \infty} e \left(\frac12-\frac{11}{24n} \cdots\right)=\frac{e}{2} $$
