Lebesgue Integral calculating problem

Question

$$ \lim _{ n->\infty }{ \int _{ 0 }^{ 1 }{ { (1+nx^{ 2 })(1+x^{ 2 })^{ -n }\quad }dx } } $$

Please help me calculating the limit. Integral is Lebesgue Integral and what I learnt is Bounded convergence theorem

Answer 1

Let $f_n(x) = \frac{1 + nx^2}{(1+x^2)^n}$, then $1 + nx^2 \leq n + nx^2= n(1+x^2) \to \infty$ as $n \to \infty$ a.e.. We also have that $(1+x^2)^n \to \infty$ as $n \to \infty$ a.e.. Hence, by L'Hopital rule, $\lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \frac{2nx}{2nx (1+x^2)^{n-1}} \to 0$ a.e.

Moreover, $|f_n(x)| \leq 1$ for all $x$ so we may apply Lebesgue's dominated convergence theorem to conclude that $ \int f_n(x) dx \to 0$.