Solve: $$ \lim_{n \rightarrow \infty} \int_{[0, \infty)} \frac{n \sin(\frac{x}{n})}{x(1+x^2)}\,\mathrm dm(x) $$
I am aware that the problem should be solved by using the Lebesgue's theorem of dominated convergence, and the dominated function could be $g(x) = \frac{n}{x(1+x^2)}$ but the point $0$ gives me headaches.
How could I solve problems like this one?
Your intuition is the right one, but you have to use a stronger approximation than only $|\sin(x)|\leq 1$. For instance, using $|\sin(x)|\leq |x|$, you get $$\left|\frac{n\sin(\frac{x}{n})}{x(1+x^2)}\right|\leq \frac{1}{1+x^2}\in L^1(0,\infty ).$$