If $f_n (x)=\frac{n \sin x}{x (1+n^2 x^2)}$ then evaluate limit of integration $f_n(x)$ over $0 \to 1$ as $n \to \infty$ In this problem,
I tried to dominated convergence theorem but I couldn't get any dominated function. How to find limit of this integration?
Any hints or comments are welcomed.

I am assuming that you have
$$
f_n(x)=\frac{\sin(x)}{x}\frac{n}{1+n^2x^2}
$$
and want
$$
\lim_{n\to\infty}\int_0^1f_n(x)\,\mathrm{d}x
$$
First note that
$$
\begin{align}
\lim_{n\to\infty}\int_0^1\frac{n}{1+n^2x^2}\,\mathrm{d}x
&=\lim_{n\to\infty}\int_0^n\frac1{1+x^2}\,\mathrm{d}x\\
&=\frac\pi2
\end{align}
$$
For $n\gt\frac1x$, $\frac{n}{1+n^2x^2}$ tends monotonically to $0$. Thus, using Dominated convergence on intervals avoiding $0$, we can deduce that
$$
\lim_{n\to\infty}\int_0^1g(x)\frac{n}{1+n^2x^2}\,\mathrm{d}x=\frac\pi2\lim_{x\to0}g(x)
$$
for any $g$ continuous on $[0,1]$.
For this question, $g(x)=\frac{\sin(x)}{x}$.
