# if $\lim _{x\rightarrow \infty}{f'(x)}=0$ then does $\lim_{x \rightarrow \infty}{f(x)}$ exist in the broad sense

Let $$f$$ be a differentiable a function in $$\mathbb{R}$$, and let $$\lim _{x\rightarrow \infty}{f'(x)}=0$$

Does $$\lim_{x \rightarrow \infty}{f(x)}$$ exist in the broad sense?

I'm really lost here. This exercise is from a section on MVT, and intuitively it seems to be correct, but I can't seem to find a lead. If someone could just give me a hint that would be great.

So far my best shot has been using Heine's definition of the limit, but no dice.

• Do you want to know when this limit would exist ?or do you want to know when it won’t ? $f(x)=\log(x)$ and $f’(x)=\frac{1}{x}$ is a case where limit won’t exist for $f(x)$ Commented Jan 21, 2021 at 17:32
• I want to know if it always exists in the broad sense, so those examples wouldn't work Commented Jan 21, 2021 at 17:35
• Is there a formal definition of “broad sense”? Commented Jan 21, 2021 at 17:38
• I was sure that was the correct term but I can't find any evidence of that.now haha. Sorry, I just mean the limit can be + or - infinity aswell as finite Commented Jan 21, 2021 at 17:39
• At best you can use L'Hospital's Rule and say that $f(x) /x\to 0$ as $x\to\infty$. But limit of $f$ can be anything including $\pm\infty$ and oscillation. Commented Jan 22, 2021 at 7:27

You can think of the usual sine function as a spring, and imagine stretching it to the right. Try to understand why this make the function get straight as $$x$$ increases.
Consequently, $$\sin(\sqrt x)$$ would work.