Using convergence theorems, I am trying to compute the value of
$$ \lim_{n\to\infty}\int_a^\infty \frac n{1+n^2x^2}\,\mathbb{d}x $$
for $a \in \mathbb{R}$, and with respect to the Lebesgue measure. Firstly I tried using the dominated convergence theorem but it turns out that I can't find a dominating function (since $f_n(0) = n$).
The other theorem I have at hand is the monotone convergence theorem, but it is not clear that we have $f_1(x) \leq f_2(x) \leq \mathbb{ ...}$ , so I don't see how I can apply that one either. Can anyone point me in the right direction?
The substitution $y = nx$ yields
$$\int_a^\infty \frac{n}{1+n^2x^2}\,dx = \int_{na}^\infty \frac{dy}{1+y^2} = \int_\mathbb{R} \chi_{[na,\infty)}(y)\cdot\frac{1}{1+y^2}\,dy.$$
In that form, both, the dominated convergence theorem and the monotone convergence theorem, can be easily applied to find the limit.