\[\mathop {\lim }\limits_{n \to + \infty } \left( {\frac{{{2^{2n}}}}{{\left( {\begin{array}{*{20}{c}} {2n} \\ n \end{array}} \right) \cdot \sqrt n }}} \right) = \sqrt \pi \]

\[\begin{array}{l} {\text{Найдите предел:}} \hfill \\ \mathop {\lim }\limits_{n \to + \infty } \frac{{\int\limits_0^\pi {{{\left( {\sin x} \right)}^{a \cdot n}}dx} }}{{\int\limits_0^\pi {{{\left( {\sin x} \right)}^{b \cdot n}}dx} }},{\text{ }}a,b > 0. \hfill \\ \end{array}\]