Find $$\lim_{n\to\infty} n^2\int_0^1 \frac{dx}{(1+x^2)^n }. $$
Actually, note that $f_n(x)= 1/(1+x^2)^n$ is a pointwise convergent and uniformly bounded sequence of functions where the limit function is $f(x)= 0$ if $x\neq 0$ and $f(0) = 1$.
Hence, by Arzela's theorem of interchanging of limit and integration, we have , $$\lim_{n\to\infty} \int_0^1 \frac{dx}{(1+x^2)^n } = \int_0^1 \lim_{n\to\infty} f_n(x) dx = 0, $$ hence the limit is of $\infty \cdot 0$ form , and I tried some inequalities but can't reach some decent step .
Best Answer
$$\int_0^1{\mathrm{dx}\over(1+x^2)^n}\geq\int_0^{1/\sqrt{n}}{\mathrm{dx}\over(1+x^2)^n}\geq\int_0^{1/\sqrt{n}}{\mathrm{dx}\over\left(1+\frac1n\right)^n}\geq{1\over\sqrt{n}}{1\over2e}$$ for large $n$. Therefore, the limit you seek is $\infty.$