[tex]\lim\limits_{x \searrow 0}\dfrac{\arcsin\sqrt{x}}{\sqrt{1-\ln(e-x)}} = \lim\limits_{x \searrow 0}\left[\dfrac{\arcsin\sqrt{x}}{\sqrt x}\cdot \dfrac{\sqrt{x}}{\sqrt{1-\ln(e-x)}}\right] = \\ \\ = \lim\limits_{x \searrow 0}\left[1\cdot \dfrac{\sqrt x}{\sqrt{1-\ln(e-x)}}\right] = \lim\limits_{x \searrow 0} \sqrt{\dfrac{x}{1-\ln(e-x)}} =[/tex]
[tex]= \sqrt{\lim\limits_{x \searrow 0}\dfrac{x}{1-\ln(e-x)}} \overset{\frac{0}{0}}{=} \sqrt{\lim\limits_{x \searrow 0}\dfrac{1}{-\dfrac{-1}{e-x}}} = \sqrt{\dfrac{1}{\dfrac{1}{e}}} = \boxed{\sqrt e}[/tex]
