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.
Hint: Use that $1 + x^2 &lt; e^{x^2}$.
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 !