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Math Help - Precise definition of a limit question

  1. #1
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    Precise definition of a limit question

    For the limit
    lim
    x → 2
    (x^3 − 4x + 7) = 7

    illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1. (Round your answers to four decimal places.)...note:let d=delta, E=epsilon

    My work:
    0</x-a/<d, then /f(x)-L/<E
    0</x-2/<d, then /x^3-4x/<E...

    so taking the right part of the statement further:

    /x(x^2-4)/<E
    /x(x-2)(x+2)/<E
    /x-2/< E/(x^2+2x)

    d<E/(x^2+2x)

    I'm not sure what to do here. Even if I plug in E=0.2, I'm still left with the x's...please help me out, any advice/tips are appreciated.
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Precise definition of a limit question

    What you have done is good, but \epsilon or \delta can't depending of the variable x.

    If we define the limit with the \epsilon-\delta definition:
    0<|x-2|<\delta \Rightarrow |f(x)-7|<\epsilon

    We calculate:
    |f(x)-7|=|x^3-4x+7-7|=|x^3-4x| and |x^3-4x|<\epsilon \Leftrightarrow |x||x+2||x-2|<\epsilon

    Now define \delta = 1 therefore |x-2|<1 (we want x close to 2).
    Determine |x| and |x+2| with this condition.

    But have you ever done examples like this before?
    Also I recommand you to read the sticky thread of Krizalid in forum calculus about epsilon-delta proofs.
    Last edited by Siron; September 18th 2011 at 11:06 PM.
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  3. #3
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    Re: Precise definition of a limit question

    How do you define 'd' as =1? I think I missed something...would you mind explaining this?

    so /x-2/<1
    so 2<x<3
    and /x/<1
    -1<x<1

    is this part right?? You caught me, this is my first time doing a problem like this...Strangely my book has no examples of this type of prob. Just" Prove" problems...If I found an example, I'd be all set. I'll go try to find that thread and read through it! But thanks for your help!
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  4. #4
    MHF Contributor Siron's Avatar
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    Re: Precise definition of a limit question

    Like I said before, it can be useful to take a look at the sticky thread of Krizalid in forum calculus about epsilon-delta proofs. In this document there're a lot of examples:

    http://www.mathhelpforum.com/math-he...ofs-47767.html
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  5. #5
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    Re: Precise definition of a limit question

    I read through the link.

    So interested only in numbers close to 2. And I can choose any x close to 2 for the restriction. So like you said, condition /x-2/<1. so assume d<1.

    Looking at #3 from the link,
    I don't understand
    /x-5/<1
    -1<x-5<1
    9<x+5<11 ****where did this come from???

    Could anyone explain that part to me?
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