Finding limit of function $\lim_{x\to 0}\left(\frac{(1+x)^\frac1x}{e}\right)^\frac1x$

Find the limit of function

$$\lim_{x\to 0}\left(\frac{(1+x)^\frac1x}{e}\right)^\frac1x$$

Solution 1:

We have

$$(1+x)^{1/x}=e^{\frac 1 x\log(1+x)}=e^{1-\frac{x}{2}+o(x)}$$ hence $$\left(\frac{(1+x)^\frac1x}{e}\right)^\frac1x=e^{\frac{1}{x}(-\frac{x}{2}+o(x))}\sim e^{-1/2}$$

Solution 2:

Take logs. The log of your expression is $(\frac1x(\frac1x \log(1+x) -1)).$ Expanding $\log (1+x)$ in a power series at $0,$ get the limit of the log to be $-1/2,$ so the limit in the question is $1/\sqrt{e}.$

Solution 3:

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 $e^{-\frac{1}{2}}$.