#### 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

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

#### ccox

I dunno but it would be a finite number

Any help would be appreciated

#### Gib Z

Homework Helper
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

Homework Helper
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?

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