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Math Help - epsilon delta continuity applied to polynomial

  1. #1
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    epsilon delta continuity applied to polynomial

    Hi, I've been asked to prove that f(x) = x^3 - 9x +2 is continous at x=3 by using epsilon delta continuity. I assume I'm not allowed to use the theorem that the sum of continuous functions is continious.

    So let e>0

    Then |f(x)-f(3)| < e
    |x^3 - 9x| < e
    | x (x+3) (x-3) | < e

    How do I find the requirements for delta so that |(x-3)| < delta?
    I don't think I'm allowed to divide the inequality by x(x+3) as I understand that delta has to be independent of x.

    It's clear that there is a solution for delta because cubic polynomials have solutioins, but it's very messy to solve.

    What other way can I find delta?

    Thanks!
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  2. #2
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    We know that \left| {x - 3} \right| < 1\, \Rightarrow \,\left| x \right| < 4\;\& \,\left| {x + 1} \right| < 5.
    So in picking \delta make sure it is less than one so the above will true.
    Given \varepsilon>0 pick \delta  = \min \left\{ {1,\frac{\varepsilon }<br />
{{20}}} \right\}
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  3. #3
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    Thanks, I didn't consider that
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