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?