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## Main Question or Discussion Point

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?

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?