Advanced Calculus Proof

  • Thread starter ccox
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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

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If the limit of f'(x) was not zero, then what would f(x) look like for large x?
 
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I dunno but it would be a finite number

Any help would be appreciated
 

Gib Z

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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

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

[tex]\frac{f(x) - f(a)}{x - a} = f'(c)[/tex]

Now let [itex]x \rightarrow \infty[/itex]. What happens?
 

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