(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Application of MVT

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