\[\mathop {\lim }\limits_{x \to \frac{1}{2}} \frac{{{{\arcsin }^{\frac{1}{2}}}x - {{\left( {\frac{\pi }{6}} \right)}^{\frac{1}{2}}}}}{{{x^{\frac{1}{2}}} - {{\left( {\frac{1}{2}} \right)}^{\frac{1}{2}}}}}\]