Find $$\lim\_{x\rightarrow +\infty}\left(\frac{x^{1+x}}{(1+x)^x}-\frac{x}{e}\right).$$  

It first occurred to me that maybe taking the natural log, namely $\ln$, would do some help. Then the two terms immediately prevents me from doing so, because I can't separate these two terms, $\frac{x^{1+x}}{(1+x)^x}$ and $\frac{x}{e}$, under the operation of $\ln$.  

I have no idea now. Anyone could help?