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

1. [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 $\delta=-1+\sqrt{1+\frac{\epsilon}{3}}>0$