Suppose the limit L = _ n → ∞ n ∫ _ 0 ^ 1 1 (1 + x^ 2 )n dx exists...

We have,&

$L = \lim_{n \to \infty}\sqrt{n}{∫}_{0}^{1}\dfrac{dx}{\left(1 + x^{2}\right)n}$

We know that, \left(1 + x^{2}\right)n > 1 + nx^{2}$</p><p>$⇒ \dfrac{1}{\left(1 + x^{2}\right)n} < \dfrac{1}{1 + nx^{2}}$</p><p>$⇒ {∫}_{0}^{1}\dfrac{dx}{\left(1 + x^{2}\right)n} < {∫}_{0}^{1}\dfrac{1}{1 + nx^{2}}dx$</p><p>$⇒ {∫}_{0}^{1}\dfrac{1}{\left(1 + x^{2}\right)n} < \dfrac{1}{n}{\tan^{- 1}\sqrt{n}x}_{0}^{1}$</p><p>$= \dfrac{1}{n}\tan^{- 1}\sqrt{n}$</p><p>$L = \lim_{n \to \infty}\sqrt{n}{∫}_{0}^{1}\dfrac{1}{\left(1 + x^{2}\right)n}dx < \lim_{n \to \infty}\sqrt{n}\dfrac{1}{n}\tan^{- 1}\sqrt{n}$</p><p>$∴ L = \dfrac{π}{2}