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Math Help - Prove continuity at a point using epsilon/delta

  1. #1
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    Prove continuity at a point using epsilon/delta

    Hi,
    I need to prove that f(x) = cuberoot[x+3] is continuous at x = -2 (using: u^3 - v^3 = (u-v)(u^2 + uv + v^2). I know that I need to manipulate mod(f(x) - f(-2)) to be less than epsilon for some delta but I'm struggling with the algebra.
    Any help would be greatly appreciated!
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  2. #2
    Junior Member rubix's Avatar
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    Quote Originally Posted by s0791264 View Post
    Hi,
    I need to prove that f(x) = cuberoot[x+3] is continuous at x = -2 (using: u^3 - v^3 = (u-v)(u^2 + uv + v^2). I know that I need to manipulate mod(f(x) - f(-2)) to be less than epsilon for some delta but I'm struggling with the algebra.
    Any help would be greatly appreciated!
    f(x) = \sqrt[3](x+3) at x = -2

    what you want to show is the following:

    for \epsilon > 0, there exists \delta > 0 such that:

    |f(x) - f(c)| < \epsilon for all x that belongs to dom f\{c} s.t. |x-c| < \delta

    i remember working backward in problem like these. Start out with something like this...

    let \epsilon > 0,

    for |x+2| < \delta

    |\sqrt[3](x+3) - \sqrt[3](-2+3)| = |\sqrt[3](x+3) - 1|

    ... (you'll prolly need to use some algebraic formula for cube/cluberoots and perhaps triangle inequality as well
    ...
    ...

    < \epsilon


    P.S. I'll try this later on.
    Last edited by rubix; November 15th 2009 at 05:38 PM.
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