Solve the given integral. Find:
$$
\lim_{n \to \infty} n^{2} \int_{0}^{1} \frac{dx}{(1+x^{2})^{n}}
$$
This is indeterminate form $\infty \times 0$.
How can I initiate the problem.
Just need an hint.
 A: Hint: Use that $1 + x^2 < e^{x^2}$.
A: Let $u=\sqrt{n}x$ Then the integral becomes
$$\lim_{n\to\infty} n^{\frac{3}{2}} \int_0^{\sqrt{n}} \frac{du}{\left(1+\frac{u^2}{n}\right)^n}$$
Now let's focus only on the integral portion. If you only want a hint stop here, but I continue the solution below.

By dominated convergence we have that 
$$\int_0^{\sqrt{n}} \frac{du}{\left(1+\frac{u^2}{n}\right)^n} \longrightarrow \int_0^\infty e^{-u^2}\:du = \frac{\sqrt{\pi}}{2}$$
Since this limit is finite and multiplied by a limit that goes to infinity, the original limit is infinite.
A: The same limit can also be written as
$$
\lim_{n \to \infty} \frac{\int_{0}^{1} \frac{dx}{(1+x^{2})^{n}}}{\frac{1}{n^2}}
$$
And, then this becomes a $0/0$ form of limit so now we can apply L-Hospitals Rule, but just take care about the fact that differentiation has to be carried out with respect to $n$ because the limit is applied to $n$ and correspondingly to the entire expression. So, the differentiation with respect to $n$ can be taken inside the integral and then can be differentiated.
Hope this helps you ! 
