Suppose that $f$ is differentiable function such that $f(x)\ge 0$ and $\int_0^\infty f(x)dx < \infty$. Which of the followings are true?
A. $\lim_{x\to\infty} f(x) = 0$
B. $\lim_{x\to\infty} f'(x) = 0$
C. For every $\epsilon>0$, there is an $M$ such that $\int_{x>M} f(x) dx < \epsilon$
D. $f$ is bounded
I know that C is true but I couldn't come up with counter examples for the other three. The few examples in my head are $f(x)=\frac{1}{x}$ and Gaussians.
On
An interesting example is
$$f(x) = x\left (\frac{2+\cos x}{3}\right )^{x^5}.$$
Then $\int_0^\infty f < \infty,$ and $f$ fails A,B, D and satisfies C.
It's humorous to use a graphing calculator on this. You'll see a function that looks like it's $0$ except for ever-growing spikes at $0,2\pi,4\pi, 6\pi, \dots $ Zoom in and you'll see the graph is actually smooth (in fact the function is real analytic). I don't recommend this as a solution your first time around this topic; Barry's answer is much better for intuition.
Hint: You can have a series of (smooth) spikes that are extremely tall and even more extremely skinny.