<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] let $y = \dfrac{x(x+1)^{x+1}}{(x+2)^{x+2}},$  then 
$\begin{align} \ln y &= \ln x + (x+1)\ln(x+1)-(x+2)\ln(x+2)\\
 &=\ln x +(x+1)[\ln x + \ln(1 + 1/x)] -(x+2)[\ln x+ \ln(1 + 2/x)]\\
&=\left(1+x+1-x-2  \right)\ln x +(x+1)\left(1/x  + \cdots\right)-(x+2)\left(2/x +\cdots\right)\\
&=-1 + \cdots
\end{align}$
so $$\lim_{x \to \infty} \ln y= -1 \text{ and } \lim_{x \to \infty} y= \frac{1}{e}.$$
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