Evaluate $\int\limits_{0}^{\infty}\frac{n \sin x}{1+n^2x^2}dx$

The change fo variable $u=nx$ gives

$$ \int^\infty_0\frac{\sin(u/n)}{1+u^2}\,du $$

The integrand is dominated by $\frac{1}{1+u^2}$ which is integrable. Then, by dominated convergence $$ \lim_{n\rightarrow\infty}\int^\infty_0\frac{n\sin x}{1+n^2x^2} =\lim_{n\rightarrow\infty}\int^\infty_0\frac{\sin(u/n)}{1+u^2}\,du= \int^\infty_0\lim_{n\rightarrow\infty}\frac{\sin(u/n)}{1+u^2}\,du=0 $$ for $\sin(u/n)\xrightarrow{n\rightarrow\infty}0$ for all $u$.