Spivak Chapter 5 Problem 26) a

In summary: I appreciate your feedback and suggestions. However, as an expert summarizer, I am only responsible for providing a summary of the content and not for answering questions or providing additional examples. Thank you for understanding. In summary, the given definition of limx→aƒ(x) = L is incorrect because it fails to hold for certain functions, such as piece-wise functions where the value at the point of interest does not match the limit. It is important to use specific numbers when finding examples and to consider both uncommon and common functions to fully illustrate the inadequacy of the definition.
  • #1
Derek Hart
14
1

Homework Statement


Give an example to show that the given "definition" of limx→aƒ(x) = L is incorrect.

Definition: For each 0<δ there is an 0<ε such that if 0< l x-a I < δ , then I ƒ(x) - L I < ε .

Homework Equations

The Attempt at a Solution


I considered the piece-wise function: ƒ(x) = (0 if x<0) = ( λ if x>0). I then chose an ε such that ε > I λ/2 I , and chose L = λ/2.
It is obviously true that for each positive δ , if 0 < I x-0 I < δ , then I ƒ(x) - λ/2 I < ε . But, by our definition, this means that limx→0ƒ(x) = λ/2 , which is blatantly false.

Is this sufficient? I think that spivak is expecting that I use some sort of "common sense" in my argument such as in my final statement.
 
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  • #2
Derek Hart said:

Homework Statement


Give an example to show that the given "definition" of limx→aƒ(x) = L is incorrect.

Definition: For each 0<δ there is an 0<ε such that if 0< l x-a I < δ , then I ƒ(x) - L I < ε .

Homework Equations

The Attempt at a Solution


I considered the piece-wise function: ƒ(x) = (0 if x<0) = ( λ if x>0). I then chose an ε such that ε > I λ/2 I , and chose L = λ/2.
It is obviously true that for each positive δ , if 0 < I x-0 I < δ , then I ƒ(x) - λ/2 I < ε . But, by our definition, this means that limx→0ƒ(x) = λ/2 , which is blatantly false.

Is this sufficient? I think that spivak is expecting that I use some sort of "common sense" in my argument such as in my final statement.

It seems sufficient to me. You've taken a function that looks discontinuous and shown it fits the 'definition'.
 
  • #3
Derek Hart said:

Homework Statement


Give an example to show that the given "definition" of limx→aƒ(x) = L is incorrect.

Definition: For each 0<δ there is an 0<ε such that if 0< l x-a I < δ , then I ƒ(x) - L I < ε .

Homework Equations

The Attempt at a Solution


I considered the piece-wise function: ƒ(x) = (0 if x<0) = ( λ if x>0). I then chose an ε such that ε > I λ/2 I , and chose L = λ/2.
It is obviously true that for each positive δ , if 0 < I x-0 I < δ , then I ƒ(x) - λ/2 I < ε . But, by our definition, this means that limx→0ƒ(x) = λ/2 , which is blatantly false.

Is this sufficient? I think that spivak is expecting that I use some sort of "common sense" in my argument such as in my final statement.

Your example is valid, but you may be missing the point somewhat.

First, when finding an example, you can (and perhaps it's better to) use some definite numbers. E.g. why not just have ##\lambda = 1##?

Also, very strictly speaking, you didn't actually find an ##\epsilon##. This does not mean what you did was wrong. But, why not give a specific ##\epsilon##? With ##\lambda = 1##, you could have taken ##\epsilon = 1##. Or, in your general case ##\epsilon = |\lambda|##.

The reason I mention this is that a reluctance (or inability) to choose a specific ##\epsilon## or ##\delta## can lead to difficulties in finding counterexamples or proving a limit does not exist.

Also, to show that the definition is not just invalid for unusual functions, I suggest finding another example for the function:

##\forall x \ f(x) = 0##
 

FAQ: Spivak Chapter 5 Problem 26) a

1. What is the purpose of Spivak Chapter 5 Problem 26(a)?

The purpose of this problem is to test your understanding of the concept of differentiability and to practice finding the derivative of a function using the limit definition.

2. How do I approach solving Spivak Chapter 5 Problem 26(a)?

To solve this problem, you need to carefully read and understand the given function, identify the points at which it is not differentiable, and then apply the limit definition of the derivative to calculate the derivative at the given point.

3. What is the limit definition of a derivative?

The limit definition of a derivative is a mathematical formula that describes the rate of change of a function at a particular point. It involves taking the limit of the average rate of change of the function as the distance between two points on the graph approaches zero.

4. What do I do if I am stuck on Spivak Chapter 5 Problem 26(a)?

If you are stuck on this problem, try breaking it down into smaller steps and make sure you understand the concept of differentiability. You can also consult your textbook or ask your instructor for help.

5. Why is Spivak Chapter 5 Problem 26(a) important?

This problem is important because it helps you develop a deeper understanding of the concept of differentiability and the limit definition of a derivative. It also allows you to practice applying these concepts to solve a real-world problem.

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