finding the limit $\lim\limits_{x \to \infty }(\frac{1}{e}(1+\frac{1}{x})^x)^x$ Can someone show me how to calculate the limit:
$$\lim_{x \to \infty }\left(\frac{1}{e}\left(1+\frac{1}{x}\right)^x\right)^x $$
I tried to use taylor series but failed.
Thanks

Take the logarithm,
$$\begin{align}
\log \left(\frac{1}{e}\left(1+\frac{1}{x}\right)^x\right)^x &= x\left(\log \left(1+\frac1x\right)^x - 1\right)\\
&= x\left(x\log\left(1+\frac1x\right)-1\right)\\
&= x\left(-\frac{1}{2x} + O\left(\frac{1}{x^2}\right)\right).
\end{align}$$
