Lebesgue Integral: calculation of limit

Question

I need to calculate the following limit, using basic Lebesgue integral properties and lemmas:

$$\underset{n\rightarrow\infty}{\lim}\underset{}{\int_{(a,\infty)}\frac{n}{1+n^{2}x^{2}}}.$$

Answer 1

Hint:

1) if $n \ge 1$ is an integer then $0 \le \dfrac{n}{1 + n^2 x^2} \le \dfrac{1}{1 + x^2}$.

2) $0 \le \displaystyle \int_{(a,\infty)} \frac{1}{1 + x^2}\, dx \le \int_{(-\infty,\infty)} \frac{1}{1 + x^2} \, dx < \infty$.