TITLE: Finding limit of function $\lim_{x\to 0}\left(\frac{(1+x)^\frac1x}{e}\right)^\frac1x$
QUESTION [5 upvotes]: Find the limit of function
$$\lim_{x\to 0}\left(\frac{(1+x)^\frac1x}{e}\right)^\frac1x$$

REPLY [2 votes]: let $$L= \lim_{x\to 0}(\frac{(1+x)^\frac{1}{x}}{e})^\frac{1}{x}$$
taking $ln$ on both sides
$$ln L= \lim_{x\to 0} \frac{1}{x} ( \frac{1}{x} ln(1+x)-1)$$
now use when $x \to 0$
 $$ln(1+x)\approx x-\frac{x^2}{2}+....$$
the $RHS$ changes to$$\frac{-1}{2}+\frac{1}{x}-\frac{1}{x}+O$$
further removing $ln$.  limit becomes $e^{-\frac{1}{2}}$.