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Rigorous definition of a limit

  1. Sep 6, 2012 #1
    First off I want to apologize for bombarding this subforum with my gazillion questions. If my continuous barrage of questions poses a problem just let me know and I'll stop.

    1. The problem statement, all variables and given/known data
    For each value of ε, find a positive value of δ such that the graph of the function leaves the window (a − δ) < x < (a + δ), (b − ε) < y < (b + ε) by the sides and not through the top or bottom.
    g(x) = −x^3 + 2
    a = 0
    b = 2
    ε = 0.1, 0.01, 0.001
    For ε = 0.1, δ must be less than or equal to what value?


    2. Relevant equations
    0<abs(x-a)<δ then abs(f(x)-b)<ε


    3. The attempt at a solution
    abs(−x^3 + 2-2)<ε
    abs(−x^3)<.1
    1.9<−x^3<2.1

    0<abs(x-0)<δ
    -δ<x<δ

    1.9<−x^3
    -(1.9^1/3)>x
    −x^3<2.1
    x>-(2.1^1/3)

    (-2.1^1/3)<=δ<x<δ<=-(1.9^1/3)
    δ<=-(1.9^1/3) || δ<=-1.2386

    Is this right? I tried to get everything to match of properly, but I'm not sure if I did it correctly. I'm not exactly sure what I even did just now :/ Thanks in advance and sorry for all the questions.
     
  2. jcsd
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