Excuse the typing please, as I am posting from my phone. Let f have domain [0,infty) and range in R. Suppose as x goes to infinity, f'(x) goes to a constant b. I wish to show that f(x)/x goes to b as x goes to infinity. I have tried numerous applications of the MVT to solve this and cannot come up with f(x)/x. What I can show is for every h>0, (f(x+h)-f(x))/h goes to b as x goes to infinity. Does anyone see or can give me a hint on how to apply the MVT here? Thanks.