Something that bothers me with the following question: $f: [0, \infty] \to \mathbb{R}$,$f \geq 0$, $\lim_{x \to \infty} f(x)$ exists and finite, and $\int_{0}^{\infty}f(x)dx$ converges, I need to show that $\int_{0}^{\infty}f^2(x)dx$
I separated the integral in the following way: $\int_{0}^{1}f^2(x)dx$+$\int_{1}^{\infty}f^2(x)dx$, while for the second one we know that $\lim_{x \to \infty}\frac{f^2(x)}{f(x)}$ so they converges together, but what happens in the first range, does my $\lim_{x \to 0}f(x)$ has to be finite? Do you have an example for an integral between $0$ and $\infty$ which converges and the $\lim_{x \to 0}$ is not finite? What should I do in my case?
Thanks!