I need some help with the following problem:
Suppose $f\colon(0,\infty) \to \mathbb{R}$ is differentiable. If $f$ is bounded and $f'(x) \to 0$ as $x \to +\infty$, does this imply that $\lim_{x \to +\infty}$ exists? Prove or give a counterexample.
It looks like it is false, but I am struggling to find the bounded counterexample.
You can consider a function like $\sin(\sqrt{x+1})$; the derivative of this tends to $0$ but the function clearly oscillates.
It may be helpful for me to explain how I came up with this example. We want a bounded function that doesn't have a limit at infinity; the trig functions are my go to example. The issue is that the derivatives also don't have a limit. How do you make a derivative smaller? Stretch out the graph horizontally. How do you make the derivatives tend to zero? Make the stretch get worse and worse as you head towards infinity.