Help with a limit using $\ln$ and l'Hôpital

Question

I need help with the following limit: $$\lim\limits_{n\to\infty} \left(\frac{\left(1+\frac{1}{n}\right)^{n}}{e}\right)^{n}$$

I tried using $\ln$, taking out the $n$, writing it as $\frac{1}{n}$ in the denominator and performing l'Hôpital (and then using the exponent to get the limit). But for some reason I'm not reaching the correct limit. I would greatly appreciate if someone could show me steps for solving this limit. Thank you.

Answer 1

Hint:

$$ \begin{align} \lim_{n \to \infty} n\big(n \ln(1+\frac{1}{n})-1\big) & = \lim_{x \to 0^+} \frac{1}{x}\big(\frac{1}{x} \ln(1+x)-1\big) = \\ &= \lim_{x \to 0^+} \frac{\ln(1+x)-x}{x^2} \\ &= \lim_{x \to 0^+} \frac{\cfrac{1}{1+x}-1}{2 x} = \cdots \\ \end{align} $$