Solve $$\lim\limits_{x\to0}\left(1+\frac{1}{2x}\right)^x$$
i tried solve the limit by this way
$$\begin{align} \lim\limits_{x\to0}\left(1+\frac{1}{2x}\right)^x&\stackrel{?}{=}\left[\lim\limits_{x\to0}\left(1+\frac{1}{2x}\right)\right]^{\lim\limits_{x\to0}x}\\ &=(\pm\infty)^0\\ &\stackrel{?}{=}1 \end{align}$$
but im not sure if i made it correct, how i solve it?