1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Spivak Chapter 5 Problem 26) a

  1. Jun 9, 2017 #1
    1. The problem statement, all variables and given/known data
    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 .

    2. Relevant equations


    3. 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.
     
  2. jcsd
  3. Jun 9, 2017 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    It seems sufficient to me. You've taken a function that looks discontinuous and shown it fits the 'definition'.
     
  4. Jun 9, 2017 #3

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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##
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Spivak Chapter 5 Problem 26) a
Loading...