Application of Dominated Convergence Theorem help finding a Dominating function

Question

$$\lim_{n\to\infty}\int_0^\infty \frac{n\sin(x/n)}{x(1+x^2)}$$ I wish to use the Lebesgue Dominated Convergence theorem to solve this, but I'm having trouble finding a dominating function, $g(x)$. Taking the convention $ 0 * \infty = 0$, I believe that the sequence of $f_n$ converges to $0$ for all x. And so one would expect that the integral would go to $0$ for sufficiently large n, However that doesn't appear to be the case. I suspect my thoughts on the function that $fn$ converges to isn't correct.

Answer 1

Hint: We have $\lim_{z \to 0} \sin z /z = 1$ and for $z > 0$

$$\left|\frac{\sin z }{z}\right|\leqslant 1.$$

Now look at $n\sin(x/n)/x = \sin(x/n)/(x/n).$