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

In summary, the problem states that if a function is differentiable on the interval (0,infinity) and the limit of its derivative and itself as x approaches infinity exist and are finite, then the limit of the derivative must be equal to zero. The suggested approach is to use the Mean Value Theorem and show that if the limit of the derivative is not zero, then the function would not converge to a finite number as x approaches infinity. This is because the MVT states that there exists a point c in the interval (a,x) such that the derivative is equal to the gradient of a secant line between a and x. As x approaches infinity, this secant line becomes a tangent line, and if the gradient is anything other
  • #1
ccox
4
0
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?
 
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  • #2
If the limit of f'(x) was not zero, then what would f(x) look like for large x?
 
  • #3
I don't know but it would be a finite number

Any help would be appreciated
 
  • #4
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
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?
 

1. What is Advanced Calculus Proof?

Advanced Calculus Proof is a branch of mathematics that deals with the rigorous and logical proving of theorems and propositions in advanced calculus. It involves using mathematical concepts and principles to construct logical arguments and proofs.

2. What are the key components of an Advanced Calculus Proof?

The key components of an Advanced Calculus Proof include defining key terms and concepts, stating assumptions and axioms, using logical reasoning and deduction, and providing mathematical justifications and evidence for each step of the proof.

3. How is Advanced Calculus Proof different from regular calculus?

Regular calculus involves the study of limits, derivatives, and integrals, while Advanced Calculus Proof focuses on the logical and rigorous proving of theorems and propositions using these concepts. It requires a deeper understanding and application of mathematical principles and techniques.

4. What are some common strategies for approaching an Advanced Calculus Proof?

Some common strategies for approaching an Advanced Calculus Proof include breaking down the problem into smaller parts, using counterexamples to disprove a statement, applying mathematical induction to prove a statement for all cases, and using different proof techniques such as direct proof, proof by contradiction, or proof by contrapositive.

5. How can I improve my skills in Advanced Calculus Proof?

Practicing regularly, familiarizing yourself with common proof techniques and strategies, and seeking guidance from a teacher or mentor can help improve your skills in Advanced Calculus Proof. It is also important to have a strong understanding of basic calculus concepts and mathematical principles.

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