multi variable calculus problem

Question

In this problem putiing it in limit 1 / n f(k/n) form it turns out to be π\4 which is not in the option . its answer is given as the first three options . how can it have three possible answers?

Answer 1

You can't always interchange integration and limit operations. Easiest way is to integrate before taking the limit \begin{align*} \lim_{n\to \infty}\int_a^\infty \frac{n}{1+n^2x^2}dx&=\lim_{n\to \infty}\int_a^\infty\frac{d(nx)}{1+(nx)^2}\\ &=\lim_{n\to \infty}\lim_{M\to \infty}\arctan(nx)\Big|_a^M\\ &=\lim_{n\to \infty}\left(\frac\pi2-\arctan(na)\right). \end{align*} We notice that if $a=0$, then $L=\frac\pi2$, $a>0$, then $L=0$ and $a<0$, then $L=1$. This comes from following properties: $$\arctan 0=0,$$ $$\lim_{x\to \infty}\arctan x=\frac\pi2,$$ $$\lim_{x\to -\infty}\arctan x=-\frac\pi2.$$