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Precise Definition of a Limit

  1. Nov 16, 2004 #1

    JasonRox

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    How do I find g?

    It's so confusing.

    I'm trying to learn this on my own, so bare with me.

    I'm going with an example I know the answer to, and maybe someone can work with me here. I'll ask questions through the solution.

    We will do x^3 since that is complicated enough, but I understand the steps, just not the logic to moving on to the next step.

    [tex]|x^3-a^3|<e[/tex]
    [tex]0<|x-a|<g[/tex]

    Find a value for g that satisfies the above.

    [tex]|x^3-a^3|=|x-a||x^2+ax+a^2|[/tex]
    [tex]|x^2+ax+a^2|<=|x|^2+|a||x|+|a|^2[/tex]

    Note: [tex]|x|-|a|<=|x-a|<1[/tex]
    If you don't understand why one is chosen maybe this isn't for you. In case you forgot, we take 1 because that guarantees that the difference won't be too big.

    [tex]|x|<1+|a|[/tex]

    Take the above and you get...

    [tex](1+|a|)^2+|a|(1+|a|)+|a|^2

    WARNING: This is in the works. I will be back to complete it.
     
  2. jcsd
  3. Nov 17, 2004 #2

    matt grime

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    You'd be better using that |a|< |x| +g

    or better yet, assuming that g is chosen such that |a|<2|x|, since if some g works, a smaller g has to work too, so there's no harm in placing a maximal size on g that helps eliminate a (assuming x is not zero. if x is zero it's quite easy)
     
    Last edited: Nov 17, 2004
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