Q:

Limit of Lebesgue Integrals $\lim_{n\to \infty} \int_{[0,\infty]} \frac{n\sin(\frac{x}{n})}{x(1+x^2)}dm$

Can you assist me in solving this limit:
$$\lim_{n\to \infty} \int_{[0,\infty]} \frac{n\sin(\frac{x}{n})}{x(1+x^2)}\,dm$$
where $m$ is the Lebesgue Measure on $\mathbb{R}$?
I thought I should try to use the dominated convergence theorem, but didn't succeed in bounding that integrand, through substitution either.

A:

One may use that
$$
|\sin x | \le |x|,\qquad x \in \mathbb{R},
$$ giving, as $n \to \infty$,
$$
\left|\sin\Big(\frac{x}{n}\Big)\right|\le \left|\frac{x}{n}\right| \implies n\left|\sin\Big(\frac{x}{n}\Big)\right|\le |x|,\qquad x \in \mathbb{R},
$$ then
$$
\left|\int_{[0,\infty)} \frac{n\sin(\frac{x}{n})}{x(1+x^2)}\:dm\right|\le \int_{[0,\infty)} \frac{ |x|}{x(1+x^2)}\:dm=\int_{[0,\infty)} \frac{ 1}{1+x^2}\:dm=\frac{\pi}2.
$$ Can you take it from here?

