I've been tackling a few analysis problems today and have once again hit a bit of a block.

I need to either prove or find a counterexample to the following two propostions:

$\displaystyle \lim_{x \to \infty} f'(x) = 0 \Longrightarrow \lim_{x \to \infty} f(x) $ exists.

$\displaystyle \lim_{x \to \infty} f'(x) = 0 \Longrightarrow \lim_{x \to \infty} \frac{f(x)}{x} $ exists.

Noting here that I am saying a limit exists if it converges in $\displaystyle \mathbb{R}\cup\{\pm \infty \} $ In other words, if $\displaystyle \lim_{x \to \infty} \rightarrow \infty$ then itI know some take an infinite limit as not existing but I am taking it as existing for the purpose of this problem. A non-existent limit would be one like sin(x) where it just sort of oscillates or where the left limit and right limit are not equal.DOES exist.

So, any pointers here? I am pretty sure that both are true because I've had no luck finding a counter example. In a similar problem to this I had earlier (which was effectively the reverse) I was able to employ the Mean Value Theorem.

But basically I am completely stuck.

I thought about saying the following:

We know that if a function is differentiable then it must be continuous. We know thata function is continuous if and only if it has a limit at every point. Hence, if $\displaystyle f'(x) \rightarrow 0$ as $\displaystyle x \rightarrow \infty$ then f(x) is continuous as $\displaystyle x \rightarrow \infty$

Thus f(x) has a limit as x $\displaystyle \rightarrow \infty$

$\displaystyle \qedsymbol$

But I'm pretty sure I'm mis-remembering something with that claim in italics and thus not making any sense at all. Any pointers would be greatly appreciated.