Calculation of limit of a function sin

Question

How can I calculate the limit of this function: $\lim_{n\to\infty} \int_{(0,\pi)}\sqrt[n]{\sin(x)}\,dλ_1 (x) $?

Answer 1

Since $\vert\sqrt[\leftroot{-2}\uproot{2}n]{\sin(x)}\vert \leq 1$ for all $x$, one can use the dominated convergence theorem to pull the limit inside the integral. Then you can use the fact $\lim_{n\rightarrow\infty}\sqrt[\leftroot{-2}\uproot{2}n]{x}=1$ to obtain $\pi$ as the final result.