calculus - Computing $lim_{x to infty} x biggl[ frac{1}{e} - left( frac{x}{x+1} right)^x biggr]$

I'm having some trouble solving finding the limit of:

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

I can see that $\left( \frac{x}{x+1} \right)^x = \left( 1 + \frac{-1}{x+1} \right)^x \rightarrow \frac{1}{e}$. That's why I thought the limit is 0. However, according to WolframAlpha it's $-\frac{1}{2e}$.

I tried to write $\frac{1}{e} - \left( \frac{x}{x+1} \right)^x$ as a series but I didn't find a way to...

What's the right approach to find this limit?