What is the Use of Mean Value Theorem for Infinity in Advanced Calculus Proof?

[ccox]

Here is the problem Let f be differentiable on (0,infinity) if the limit as x approaches infinity f'(x) f(x) both exist are finite prove that limit as x approaches infitity f'(x)=0.


I have trouble proving this problem I was told to use Mean Value Theorem to find a contridiction. However, I have not seen how to use the MVT when we are dealing with infinity any hints?

 

[StatusX]

If the limit of f'(x) was not zero, then what would f(x) look like for large x?

 

[ccox]

I don't know but it would be a finite number

Any help would be appreciated

 

[Gib Z]

We'll you see, if the gradient was anything other than an infinitesimal, then the limit as it approaches infinity will not converge and not be finite. Hopefully this helps.

 

[morphism]

Fix a \in (0, \infty), then by the MVT there exists a c \in (a, x) such that

\frac{f(x) - f(a)}{x - a} = f'(c)

Now let x \rightarrow \infty. What happens?