TITLE: Evaluating $ \lim_{n\rightarrow\infty} \int_a^\infty \frac{n}{1+n^2x^2}dx$.
QUESTION [1 upvotes]: Evaluate$$ \lim_{n\rightarrow\infty}  \int_a^\infty \frac{n}{1+n^2x^2}dx$$ where $a \in \mathbb R$.
I know for $a>0$, we can take the limit inside of the integral to get the answer 0 by checking the conditions for the standard theorem on "putting the limit inside".(integrable majorant $1/x^2$ etc...)
However for the cases $a=0$ and $a<0$, the conditions break down, namely the integrand blows up as $n \rightarrow \infty$ for $x=0$. 
How can I resolve this problem?
Any hints is appreciated.

REPLY [2 votes]: Letting $u = nx$ , we have
$$\lim_{n \to \infty} \int_{an}^\infty \frac{1}{1+u^2} = \lim_{n \to \infty} \int_{-\infty}^\infty \mathbb{1}_{[an,\infty]}\frac{1}{1+u^2}  $$  Where $\mathbb{1}_A$ is the characteristic function on $A$. The  last integral is bounded by $\pi$, so we may apply the dominated convergence theorem and check the cases $a < 0$ ,  $a > 0$, and $a = 0$
{"set_name": "stack_exchange", "score": 1, "question_id": 2718166}