Finding limits without l'Hopital

Question

Trying to figure out $$\lim_{x\to0}\bigg(1+\dfrac x2\bigg)^{^\tfrac1{2x}}$$

So far the closest thing we've learned is $\lim_{x\to0}(1+x)^{^\tfrac1x}=e$, but I can't figure out how to get it into that format.

Answer 1

$$\lim_{x\to0}\left(1+\frac x2\right)^{\dfrac1{2x}}$$

$$=\left[\lim_{x\to0}\left(1+\frac x2\right)^{\dfrac2x}\right]^{\dfrac14}$$

Set $\dfrac2x=n$ to use $\lim_{n\to\infty}\left(1+\frac1n\right)^n=e$

or set $\dfrac x2=h$ to use $\lim_{h\to0}\left(1+h\right)^\frac1h=e$