calculus - Find the limit $lim_{x to infty} frac{e^{x}}{(1+frac{1}{x})^{x^{2}}}$

Find the limit

$$\lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}}.$$

I know that the limit is of indeterminate form type $\frac{\infty}{\infty}$, but it seems using L'Hopital's rule directly does not help here. Do I somehow use the fact that $\lim_{x \to \infty} (1+\frac{1}{x})^{x}=e$?

Answer

For any $x>1$,

$$x^2\log\left(1+\frac{1}{x}\right) = x -\frac{1}{2}+O\left(\frac{1}{x}\right),$$
so:
$$x-x^2\log\left(1+\frac{1}{x}\right) = \frac{1}{2}+O\left(\frac{1}{x}\right).$$
By exponentiating such identity, you get that the limit is $e^{1/2}=\color{red}{\sqrt{e}}$.