<s>[INST] 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. 
 [/INST] $$\lim_{x\rightarrow \infty }\frac{x(x+1)^{x+1}}{(x+2)^{x+2}}=\\\lim_{x\rightarrow \infty }\frac{(x+1)^{x+1}}{(x+2)^{x+1}}*\frac{x}{x+2}=\\\lim_{x\rightarrow \infty }(\frac{x+1}{x+2})^{x+1}\frac{x}{x+2}=\\(1-\frac{1}{x+2})^{x+1}* \lim_{x\rightarrow \infty }\frac{x}{x+2}=\\e^{-1}*\lim_{x\rightarrow \infty }\frac{x}{x+2}\\e^{-1}*1\\=\frac{1}{e}$$
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