Maybe it's elementary, but is it possible that a non-negative continuous function satisfies $\int_{0}^{\infty}f(x)dx$ exists (finite) and the limit $\lim_{x\to\infty}f(x)$ doesn't exist (neither finite nor infinite)?
Also, is it possible to find a function such that the integral $\int_{0}^{\infty}f(x)dx$ exists (finite) but $f(x)$ is not bounded?
If you could show me examples it would be appreciated. If these are not possible, please explain why.
Both are possible. Here's a hint for how to come up with the example functions: consider a function that is usually $0$, but every once in a while spikes up to a positive number and then quickly returns to $0$. (Note that it should return to $0$ more quickly on each successive "bump".)