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Math Help - Limit Discontinuity Question

  1. #1
    Senior Member slevvio's Avatar
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    Limit Discontinuity Question

    let  f(x) = x^3 + 1 for  -1 \le x < 0 ,
     \frac{1}{2} for  x=0 and
     x^2 for  0 < x \le 1 .

    Prove  \lim_{x \to 0^+} f(x) = 0, \lim_{x \to 0^-} f(x) = 1 and  \lim_{x \to 0} f(x) = \frac{1}{2} .

    The first two are easy but I get this problem with the third one:

     \forall \epsilon > 0, \exists \delta > 0 such that  \forall x \in (-\delta, \delta), |f(x) - \frac{1}{2}| < \epsilon

    How do i consider   |f(x) - \frac{1}{2}| < \epsilon if f(x) changes on either side of x = 0 ?? Thanks for any help.
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  2. #2
    MHF Contributor red_dog's Avatar
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    If \lim_{x\searrow 0}f(x)=0, \ \lim_{x\nearrow 0}=1 then \lim_{x\to 0}f(x) does not exist.
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