Hi, I've been sitting on this more than an hour and still can't find the solution

$\displaystyle \displaystyle\lim_{x \to{\infty}}{\displaystyle{x(}(1+\frac{1}{x})^{x} - {e})}$

I put this in a function grapher and it looks like the result should be $\displaystyle {-}\frac{e}{2}$, but I don't understand why this is true or how do I get to this solution. I assume that L'hospital should be used and supposedly there's another trick?

Thanks in advance