Finding limit of $\lim_{x\to 0^+}⁡\{[(1+x)^{1/x}]/e\}^{1/x}$ This is a question given in our weekly test.
$$f = \lim_{x\to 0^+}⁡\{[(1+x)^{1/x}]/e\}^{1/x}.$$
Find the value of $f$. I tried to use 1^ infinity form but I didn't get it. So anybody please help me.

Your limit $f$ exists if and only if its logarithm exists:
\begin{align}
\log f
&=\lim_{x\to0^+}\log\bigl(\bigl((1+x)^{1/x})/e\bigr)^{1/x}\bigr)
\\[6px]
&=\lim_{x\to0^+}\frac{\dfrac{1}{x}\log(1+x)-1}{x}
\\[6px]
&=\lim_{x\to0^+}\frac{\log(1+x)-x}{x^2}
\\[6px]
&=\lim_{x\to0^+}\frac{x-x^2/2+o(x^2)-x}{x^2}=-\frac{1}{2}
\end{align}
If you don't trust Taylor expansions (or cannot use them),
$$
\lim_{x\to0^+}\frac{\log(1+x)-x}{x^2}
\overset{*}{=}
\lim_{x\to0^+}\frac{\dfrac{1}{1+x}-1}{2x}
=
\lim_{x\to0^+}\frac{-x}{2x(1+x)}
$$
(where $\overset{*}{=}$ denotes an application of l'Hôpital).
