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Epsilon-delta proof

  1. Sep 15, 2010 #1

    GBR

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    1. The problem statement, all variables and given/known data

    Use an epsilon-delta proof to show that f(x) = sqrt(x) is continuous in x = 4.


    2. Relevant equations



    3. The attempt at a solution

    |f(x) - f(4)| = |sqrt(x) - sqrt(4)| < ε
    = |x-4| < ε²

    And now what? I'm not even sure I'm on the right path. My biggest problem during these proofs is choosing an ε and relating it to |x-a| < δ to complete the proof. Please bear with me; I recently went back to school as a physics undergrad, and realize my math-fu is destroyed after six years as a journalist :)
     
  2. jcsd
  3. Sep 15, 2010 #2
    This will work for sufficiently small ε. We have
    [tex]|\sqrt{x}-2|=\frac{|\sqrt{x}+2|}{|\sqrt{x}+2|}|\sqrt{x}-2|=|x-4|/|\sqrt{x}+2|<\frac{\epsilon^2}{|\sqrt{x}+2|}[/tex].
    If, for example, ε<1, this is less than ε.
    My strategy is to start with what you want to get (epsilon inequality) and work it backwards to get bound for |x-x0|.
     
  4. Sep 15, 2010 #3

    Mark44

    Staff: Mentor

    You can't go from this inequality |sqrt(x) - sqrt(4)| < ε to this one |x-4| < ε². The reason is that |sqrt(x) - sqrt(4)|² is not equal to |x - 4|.
     
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