calculus - How to compute the limit $lim_{x to infty}left[xleft(1+frac{1}{x}right)^x-exright]$?

How to compute the limit. My first instinct was to convert the expression in a fraction and use l'hopitals rule, but the didnt seem like it was going anywhere. Are there any better approaches to evaluating this limit?

$$\lim_{x \to \infty}\left[x\left(1+\frac{1}{x}\right)^x-ex\right]$$

Answer

$$\ln\left[\left(1+\frac1x\right)^x\right] =x\ln\left(1+\frac1x\right)=1-\frac1{2x}+O(x^{-2})$$
so
$$\left(1+\frac1x\right)^x =e\exp\left(-\frac1{2x}+O(x^{-2})\right)=e\left(1-\frac1{2x}+O(x^{-2}) \right)$$
and so

$$x\left(1+\frac1x\right)^x-ex\to-\frac e2$$
as $x\to\infty$.