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I Spivak Limit Theorem 1

  1. Sep 13, 2016 #1

    I am struggling with proving theorem 1, pages 98-99, in Spivak's Calculus book: "A function f cannot approach two different limits near a."

    I understand the fact that this theorem is correct. I can easily convince myself by drawing a function in a coordinate system and trying to find two different limits at a given x coordinate and it will not work.

    But when proving the theorem I fail to see the notion behind two choices that Spivak made in proving this theorem:

    (i) he chooses delta = min(d1, d2), and
    (ii) he chooses epsilon = |L - M| / 2

    I understand the structure of the proof, which is a proof by contradicting the assumption that L unequal M. But I am stuck at the above two mentioned choices of delta and epsilon.

    I apologize sincerely for not using Latex symbols and notation and for not posting pictures of the text, but atm I am on my smartphone and do not have access to a computer.

    Any reference/help is appreciated!
  2. jcsd
  3. Sep 13, 2016 #2
    I downloaded the PF app. And I noticed that one can use it to upload pictures.

    The problem and part 1 of the proof:


    Part 2 of the proof:

  4. Sep 13, 2016 #3


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    In terms of choosing epsilon to be half the difference between the limits:

    Eventually the function must get within epsilon of both limits. But it can't be less than half the distance from them both at the same time.

    if you and a friend stand 1m apart. No one can stand within 0.5m of you both at the same time.
  5. Sep 13, 2016 #4
    Now that you put it in terms of distance it makes sense. But how does one develop the intuition to "see" what value for a variable one should choose when proving theorems?

    I figured the delta part out:

    Since the definition of a limit states, that "for all epsilon > 0, there is some delta > 0,..." It means that if we have the two deltas above mentioned one can always choose a smaller one, thus by taking the min(d1, d2) makes also sense.

    Thank you for your reply!
  6. Sep 13, 2016 #5


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    I guess a lot of people see the epsilon- delta definition as mysterious, but it always seemed to me a fairly logical way of formalising the geometric behavior of a continuous function.

    Take a fresh look at it from that perspective perhaps.
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