How to use the Bounded convergence Lemma or the Monotone convergence Theorem to calcuate those Lebesgue Integrals?

Question

$\lim\limits_{n\rightarrow\infty}\int\limits_0^\infty n\sin(\frac{x}{n})(x(1+x^2))^{-1}dx$

$\lim\limits_{n\rightarrow\infty}\int\limits_0^1 \frac{1+nx^2}{(1+x^2)^n}dx$

I have tried to show that the first is an incresing fuction series, but I cant show that $n\sin(\frac{x}{n})(x(1+x^2))^{-1}\leq (n+1)\sin(\frac{x}{n+1})(x(1+x^2))^{-1}$

and the second one I think is bounded.

Thanks.

Answer 1

For the first one, use the fact that $ \left| \sin \! \left( \dfrac{x}{n} \right) \right| \leq \dfrac{x}{n} $ for all $ x \in [0,\infty) $. Then show that $ x \mapsto \dfrac{1}{1 + x^{2}} $ is integrable on $ [0,\infty) $.

For the second one, just observe that for all $ x \in [0,1] $, we have \begin{align} 0 & < \frac{1 + n x^{2}}{(1 + x^{2})^{n}} \\ & \leq \frac{1 + n x^{2}}{1 + n x^{2}} \qquad (\text{By Bernoulli’s Inequality.}) \\ & = 1. \end{align}