1. The problem statement, all variables and given/known data Let f be diff. on (0,infinity) If the limit of f'(x) as x->infinity and limit of f(n) as n->infinity both exist and are finite, prove limit of f'(x) as x->infinity is 0. 2. Relevant equations Mean Value Theorem (applied below) 3. The attempt at a solution Suppose a>0 and b>0. Then by mvt there exists c in (a,b) such that f'(c)=(f(b)-f(a))/(b-a). Now taking the limit of both sides with respect to b as b->infinity, f'(c)=0 since the limit of f(n) as n->infinity is finite. Now, take the limit of both sides with respect to c as c->infinity and we have what we want? Not sure if this does it or is clear because the presence of f(a) might turn limit into indeterminate form? But f(a), and a is finite so taking the limit of both sides still yields what we want. This seemed a little too "convenient".... Thank you for looking.