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Math Help - [SOLVED] epsilon/delta limit in R^n. supposedly easy, but I need to learn the ropes

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    [SOLVED] epsilon/delta limit in R^n. supposedly easy, but I need to learn the ropes

    Use the epsilon/delta definition to find and prove the limit:

    \lim_{\textbf{X}\to\textbf{X}_0}f(\textbf{X})=\fra  c{x^3-y^3}{x-y}, \textbf{X}_0=(1,1).
    My notes so far:

    We have f(\textbf{X})=\frac{x^3-y^3}{x-y}=x^2+xy+y^2,x\neq y. So

    |f(\textbf{X})-f(\textbf{X}_0)|=|x^2+xy+y^2-x_0^2-x_0y_0-y_0^2|

    =|(x-x_0)(x+x_0)+(y-y_0)(y+y_0)+xy-x_0y_0|

    \leq|(x-x_0)(x+x_0)|+|(y-y_0)(y+y_0)|+|xy-x_0y_0|

    \leq|xy-x_0y_0|+|\textbf{X}-\textbf{X}_0|(|x+x_0|+|y+y_0|).

    That last inequality is because

    |\textbf{X}-\textbf{X}_0|=|(x-x_0,y-y_0)|=\sqrt{(x-x_0)^2+(y-y_0)^2}

    \geq|x-x_0|,|y-y_0|.

    But I don't seem to be getting anywhere with that. I've also looked at the Cauchy-Schwarz inequality, but that doesn't offer me any obvious (to me) help.

    Any ideas? Tricks? A complete solution? This isn't supposed to be a difficult problem, but it's the first of its kind which I have attempted!

    Thanks in advance!

    EDIT: I solved it on my own. Let \delta=-1+\sqrt{1+\frac{\epsilon}{3}}>0
    Last edited by hatsoff; October 26th 2009 at 10:59 PM.
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