1. Dec 12, 2006

### 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?

2. Dec 12, 2006

### StatusX

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

3. Dec 12, 2006

### ccox

I dunno but it would be a finite number

Any help would be appreciated

4. Dec 13, 2006

### 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.

5. Dec 13, 2006

### 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?