Calculate $\lim _{x \rightarrow 0} \frac{\int _{0}^{\sin x} \sqrt{\tan x} dx}{\int _{0}^{\tan x} \sqrt{\sin x} dx}$

Hints:

First, it should be

$$\;\lim_{x\to0}\frac{\int\limits_0^{\sin x}\sqrt{\tan t}\,dt}{\int\limits_0^{\tan x}\sqrt{\sin t}\,dt}$$

Observe that both integrands are continuous functions, and thus their integrals are differentiable functions of the upper limit...

Use now L'Hospital (why can you?), and for example in the numerator get $\;\cos x\sqrt{\tan \sin x}\;$ ...and continue.