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Delta-Epsilon Proof

  1. May 18, 2012 #1
    1. The problem statement, all variables and given/known data

    Determine the limit l for the given a, and prove that it is the limit by showing how to find a ∂ such that |f(x)-l| < ε for all x satisfying o < |x-a| < ∂.

    f(x) = x4 , arbitrary a

    (Spivak's Calculus 5-3iv)

    2. Relevant equations



    3. The attempt at a solution

    l = a4

    The best I can do is show that |x2 - a2| < ε for |x-a|<∂1 = min(ε/(2|a|+1),1). After that, I get lost.

    Since I found that |x2 - a2| < ε for |x-a|<∂1 = min(ε/(2|a|+1),1), can I rearrange |x4 - a4| to |(x2)2 - (a2)2| to find that ∂2=min(1,ε/(2|a|2+1)? This seems to be what Spivak's answer book is suggesting, but I'm not sure.

    Can someone walk me through this? I'm pretty new to delta-epsilon proofs.
     
  2. jcsd
  3. May 19, 2012 #2

    Bacle2

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    Science Advisor

    Have you considered the factoring:

    a2-b2=(a+b)(a-b)?

    It may help.

    And ,

    I heard it goes great with butter.....
     
  4. May 22, 2012 #3

    Bacle2

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    Sorry if the last comment above seemed weird: I was just making reference to

    your name 'Mantequilla' .
     
  5. May 22, 2012 #4
    Ahh, yes. It made me laugh when I read it. I would have responded earlier, but I've been terribly busy these past few days.

    I'm still having trouble with this proof. Even though I have Spivak's answer, I just don't see how he arrives at it.

    Using the (a2)2-(b2)2 factoring, we end up with (a2-b2)(a2+b2). The (a2+b2) part is giving me the most trouble, since I can't do difference of squares.
     
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