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Math Help - Precise Def of a Limit Problem

  1. #1
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    Precise Def of a Limit Problem

    Given the limit as x approaches 1 of (4 + x -3x^3) = 2 find the value of delta that correspond to sigma = 0.25.

    I never got to these with my self-study. Can someone explain this type of problem quickly? I understand the concept but I don't know how to apply it.
    Last edited by bobsanchez; July 19th 2010 at 02:15 PM.
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  2. #2
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    I know it gets started like 4 + x - 3x^3 < .25 or something. But I don't know what to do after that
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  3. #3
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    d=delta
    sigma=e


    |x-1|<d ==> |4+x-3x^3-2|<1/4

    |2+x-3x^3|=|(x-1)(3x^2+3x+2|<d|3x^2+3x+2|

    |x-1|<d ==>x<d+1 , |3x^2+3x+2|<3(d+1)^2+3(d+1)+2=3d^2+6d+3+3d+3+2=3d^ 2+9d+8

    d|3x^2+3x+2|<d(3d^2+9d+8)<1/4

    ==> 3d^3+9d^2+8d-1/4<0

    ==> d<-1+1/36 (17496-216 sqrt(6497))^(1/3)+1/6 (81+sqrt(6497))^(1/3)~2.03
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  4. #4
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    Note that there is no such thing as the delta that corresponds to a particular epsilon. If one value of delta works, then so does any smaller value.
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