Differentiable function, limits, sequence

In summary, the conversation is discussing the existence of a sequence \{x_n\} that approaches \infty and has the property that the derivative of f at each point in the sequence approaches the limit A, as x approaches \infty. The conversation also explores the limit of [f(x) - f(a)]/[x-a] for any a and the use of the Mean Value Theorem to find a sequence that approaches A. However, the method of finding such a sequence is still unclear.
  • #1
alligatorman
113
0
f is differentiable on [tex](a,\infty)[/tex] and

[tex]\lim_{x\to\infty}\frac{f(x)}{x}=A[/tex]

I am trying to prove that there exists a sequence [tex]\{x_n\}, x_n\rightarrow \infty,[/tex] such that [tex]f'(x_n)\rightarrow A.[/tex]

Any help would be appreciated.
 
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  • #2
Can you give a heuristic explanation why it should be true?
 
  • #3
then what is the limit of [f(x) - f(a)]/[x-a], for any a?
 
  • #4
By MVT, it would equal f'(c), where c is in (a,x). But I don't see how I can get a sequence from this fact.

As x gets larger, there is always a c_n in (a,x) such that f'(c)=[f(x) - f(a)]/[x-a]. Perhaps it can be said that the sequence of x_n approaches A, but I don't know how to get there.
 

What is a differentiable function?

A differentiable function is a type of mathematical function that is smoothly changing and has a well-defined derivative at every point in its domain. This means that the function can be easily differentiated, or its rate of change can be calculated, at any given point.

What is a limit?

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as its input gets closer and closer to a specific value, but does not necessarily reach it.

What is a sequence?

A sequence is a list of numbers that follow a specific pattern or rule. Each number in a sequence is called a term, and the terms can be written in a specific order. Sequences are often used in mathematics to represent patterns and relationships, and they can be finite or infinite in length.

What is the difference between a convergent and divergent sequence?

A convergent sequence is one in which the terms get closer and closer to a specific limit as the sequence progresses. In other words, the terms approach a specific value. On the other hand, a divergent sequence is one in which the terms do not have a specific limit and instead, they either increase or decrease without bound.

How can I determine if a function is differentiable?

A function is differentiable if it is continuous and has a well-defined derivative at every point in its domain. This means that the function must be smooth, with no sharp turns or breaks, and the rate of change can be calculated at any point. Additionally, a function cannot be differentiable at a point where it is not defined or has a vertical tangent line.

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