Solve this limit $\lim_{x\to \frac{1}{2}^-}\frac{\arcsin{2x}-\frac{\pi}{2}}{\sqrt{x-2x^2}}$

I am trying to figure out how to make this limit, even with the hopital. I've tried using hopital two times, but the situation 0/0 is still there. I've tried to solve it using wolfram, but I don't the solution. Even with rationalization + hopital nothing comes out. $$\lim_{x\to \frac{1}{2}^-}\frac{\arcsin{2x}-\frac{\pi}{2}}{\sqrt{x-2x^2}}$$ I wonder if there is some way to solve it, and would really appreciate any suggestion.

Put $y = \sqrt{x-2x^2} \implies -2x^2+x -y^2 = 0\implies x = \dfrac{1+ \sqrt{1-8y^2}}{4}$. Can you substitute this into the arcsin and proceed to L'hospitale from here...with notice that this means $y \to 0^{+}$