Calculating a Lebesgue integral

Question

Calculate the Lebesgue integral of,

$$\lim_{n\to\infty}\int_{[0,1]}\frac{n\sqrt{x}}{1+n^2x^2}$$

I know I should use the Lebesgue dominated convergence theorem but what should be the dominating function? Can anyone give me a hint?

Answer 1

By AM-GM inequality: $\dfrac{n\sqrt{x}}{1 + n^2x^2} \leq \dfrac{n\sqrt{x}}{2nx} = \dfrac{1}{2\sqrt{x}}$. The dominating function is: $g(x) = \dfrac{1}{2\sqrt{x}}$