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

Assume the limit exists and has value $L$  Then
$$\log{L} = \lim_{x\to\infty} x \log{\left [ \frac{1}{e} \left ( 1+\frac{1}{x}\right)^x\right]} =\lim_{x\to\infty} x \left [ -1 + x \log{\left ( 1+\frac{1}{x}\right)}\right]$$
Taylor expand the log term for large $x$:
$$\log{L} = \lim_{x\to\infty} x \left [-1 + x \left (\frac{1}{x} - \frac{1}{2 x^2} + \cdots\right)\right]= -\frac12$$
Therefore $L=e^{-1/2}$.
