Originally Posted by

**gusztav** Hi,

I would really appreciate any help with finding the following limit:

$\displaystyle \lim _{x \to 0} \frac{\text{arcsin} \sqrt{\text{sin }x}}{\sqrt{2x-x^2}}$

As $\displaystyle x \to 0$, both numerator and denominator $\displaystyle \to 0$, so we can apply the L'Hopital's rule:

$\displaystyle =\lim _{x \to 0} \frac{\frac{cos x}{\sqrt{sin x-sin^2 x}}}{\frac{2-2x}{\sqrt{2x-x^2}}}$

But this is stil an indeterminate form ($\displaystyle \frac{\infty}{\infty}$), and another application of the L'H rule further complicates the matter...

Thanks for any help!