TITLE: Evaluate a limit (probably involving L'Hôpital rule)
QUESTION [8 upvotes]: Evaluate the limit:
$$\mathop {\lim }\limits_{x \to \infty } x\left( {{{\left( {1 + {1 \over x}} \right)}^x} - e} \right)$$
My attempts didn't yield a result. I'd be glad for a guidance.
Thanks!

REPLY [7 votes]: There is a nice little trick to do this.
$$\lim\limits_{h\rightarrow 0+}{\frac{1}{h}\left((1+h)^{\frac{1}{h}}-e\right)}$$
$$(1+h)^{\frac{1}{h}}=\exp{\left(\frac{1}{h}\ln{(1+h)}\right)}=\exp\left({\frac{h-\frac{h^2}{2}+o(h^2)}{h}}\right)=e\left(1-\frac{h}{2}+o(h)\right)$$
Hence the limit is reduced to.
$$\frac{1}{h}\left((1+h)^{\frac{1}{h}}-e\right)=e\left(-\frac{1}{2}+o(1)\right)\longrightarrow -\frac{e}{2}$$ as $h\rightarrow 0^+$.