<s>[INST] Limit of $x\left(e-\left(\frac{x+2}{x+1}\right)^x\right)$ when $x\to+\infty$ 
Find $$\lim_{x\to\infty}x\left(e-\left(\frac{x+2}{x+1}\right)^x\right)$$

I calculated $\lim\limits_{x\to\infty}\left(\frac{x+2}{x+1}\right)^x=e$ but then the limit in question becomes $0\times \infty $ form, and further solution becomes messy.

Please tell a solution without the use of series expansions because I have no knowledge of these.
 [/INST] \begin{align}
\lim_{x\to\infty}x\left(e-\left(\frac{x+2}{x+1}\right)^x\right)&=\lim_{x\to\infty}\frac{e-\left(\frac{x+2}{x+1}\right)^x}{\frac1x}\\
&=\lim_{x\to\infty}\frac{\frac{\left(\frac{1}{x+1}+1\right)^x \left(x-(x+1) (x+2) \log
   \left(\frac{1}{x+1}+1\right)\right)}{(x+1) (x+2)}}{-\frac1{x^2}}\\
&=-e\lim_{x\to\infty}\left(x-(x+1) (x+2) \log
   \left(\frac{1}{x+1}+1\right)\right)\\
&=-e\lim_{x\to0}\frac{x-(x+1) (2 x+1) \left(x-\frac{3 x^2}{2}+\frac{7 x^3}{3}-\frac{15 x^4}{4}+o(x^4)\right)}{x^2}\\
&=-e \lim_{x\to 0}(\frac{15 x^4}{2}+\frac{79
   x^3}{12}-\frac{x^2}{4}+\frac{x}{6}-\frac{3}{2}+o(x^2))\\
&=\frac32e
\end{align}
</s>