limit of $ \lim\limits_{x \to ∞} \frac {x(x+1)^{x+1}}{(x+2)^{x+2}} $ Hello I am trying to find the limit of 
$ \lim\limits_{x \to ∞} \frac {x(x+1)^{x+1}}{(x+2)^{x+2}} $
I've tried applying L'H rule but it ends up getting really messy.
The answer is $ \frac {1}{e} $ so I assume it must simplify into something which I can apply the standard limit laws. 

$$\dfrac {x(x+1)^{x+1}}{(x+2)^{x+2}}=\dfrac{(x+1)^{x+2}-(x+1)^{x+1}}{(x+2)^{x+2}}$$
$$\dfrac {x(x+1)^{x+1}}{(x+2)^{x+2}}=\Big(1-\dfrac{1}{x+2}\Big)^{x+2}\Big(\dfrac{x+2}{x+1}\Big)-\Big(1-\dfrac{1}{x+2}\Big)^{x+1}\dfrac{1}{x+2}\to e^{-1}+0$$
