TITLE: Lebesgue Dominated Convergence example
QUESTION [9 upvotes]: How do I use Lebesgue Dominated Convergence Theorem to evaluate 
$$\lim_{n \to \infty}\int_{[0,1]}\frac{n\sin(x)}{1+n^2\sqrt x}dx$$
What dominating function to use here?

REPLY [2 votes]: $\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
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$$
0 < \int_{0}^{1}{n\sin\pars{x} \over 1 + n^{2}\,\sqrt{x\,}}\,\dd x
<
\int_{0}^{1}{n \over 1 + n^{2}\,\sqrt{x\,}\,}\,\dd x
=
2n\int_{0}^{1}{x \over 1 + n^{2}x}\,\dd x
=
{2 \over n}\,\ln\pars{1 + n^{2}}
$$


$$
\lim_{n \to \infty}{2 \over n}\,\ln\pars{1 + n^{2}}
=
2\lim_{n \to \infty}{2n/\pars{1 + n^{2}} \over 1} = 0
$$

$$
\lim_{n \to \infty}\int_{0}^{1}{n\sin\pars{x} \over 1 + n^{2}\,\sqrt{x\,}}\,\dd x
=
0
$$