How would this problem (Lebesgue Integral using Dominant Convergence Theorem) change if it changed to
(1) lim$_{n\rightarrow\infty} \displaystyle\int_{0}^{1} \displaystyle\frac{nx}{1+n^2x^2} dx$
and
(2) lim$_{n\rightarrow\infty} \displaystyle\int_{0}^{1} \displaystyle\frac{n}{1+n^2x^2} dx$
The first one needs Arithmetic-Geometric inequality: \begin{align*} 1+n^{2}x^{2}\geq 2nx, \end{align*} and for $x\ne 0$, we have \begin{align*} \dfrac{nx}{1+n^{2}x^{2}}\leq\dfrac{nx}{2nx}=\dfrac{1}{2}, \end{align*} so Lebesgue Dominated Convergence Theorem goes through.
The second one need no big machine: \begin{align*} \int_{0}^{1}\dfrac{n}{1+n^{2}x^{2}}dx&=\int_{0}^{n}\dfrac{1}{1+x^{2}}dx=\tan^{-1}n\rightarrow\dfrac{\pi}{2}. \end{align*}