This is an exercise from the Real Analysis by Folland.

Compute $$ \lim_{n\to\infty}\int_0^1(1+nx^2)(1+x^2)^{-n}dx. $$

Some thoughts:

Let $f_n(x)=(1+nx^2)(1+x^2)^{-n}$. One can see that $f_n(0)=1\to 1$ and $f_n(x)\to 0$ when $x\in(0,1]$. The convergence is not uniform and one cannot use uniform convergence to justify the exchange of $\lim$ and $\int$, which gives $0$ to the limit above. To use DCT, it suffices to show that $f_n$ are uniformly bounded, but I don't see how.


Hint: You can prove by induction that for any $a \geq 0$, $(1+a)^n \geq 1+na$. Therefore, $(1+nx^2) \leq (1+x^2)^n$, from which you can deduce that $0 \leq f_n \leq 1$ for every $n$. Now DCT.


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