Can A function grow quicker than another but never catch up to it

I know this is a bit of a strange question, but it is one that has been on my mind for a little bit now. If we have two real valued functions f(x) and g(x) and they are both everywhere continuous and differentiable, could we choose f(x) and g(x) such that three conditions are satisfied

1. f(x) > g(x) for all x in R
2. g'(x) > f'(x) for all x in R
3. f(x) and g(x) both diverge as x tends to infinity

Essentially, I'm asking if g(x) can grow infinitely but never catch up to f(x)? I know this problem would be simple if f(x) and g(x) converged, but I am curious about if these criterion could be met while maintaining divergence. My initial thought is that no such functions exist, but I am not sure how I could prove that. If such functions did exist, I suspect g(x) would need to be ln(x) or something of the like, but I am not convinced it is possible for these criterion to all be met. If anyone has insight either way, that would be extremely helpful.

Solution 1:

Hint: Look at the difference $h:= f-g$. Can you quickly think of a function $h$ which is always positive but whose derivative is always negative?

Now, to make the the third condition true, it seems one only has to add another function $g$ which goes to $\infty$ for $x \to \infty$.

Then with $f = g+h$ you're done.