Question 11.8: Evaluate limn→∞∫¹0 nx/1 + n²x² dx. (11.28)

For every n ∈ \mathbb{N}, define the sequence of functions

f_{n}(x) = \frac{nx}{1 + n^{2}x^{2}}, x ∈ [0, 1],

which clearly converges to 0. Since f_{n} attains its maximum value at x = 1/n, we have

\underset{x∈[0,1]}{\sup} |f_{n}(x)| = \frac{1}{2},

so the convergence f_{n} → 0 is not uniform, and we cannot justify taking the limit inside the integral (11.28) on the basis of the properties of the Riemann integral. But we can also consider (11.28) as a Lebesgue integral, in which case we can use the bounded convergence theorem to write

\underset{n→∞}{\lim} \int_{0}^{1}{f_{n}(x)}dx =\int_{0}^{1}{\underset{n→∞}{\lim} f_{n}(x)}dx = 0.