How can I correctly justify, demonstrate, and properly determine that this limit exists?

lim (x,y) --> (0,0) $\displaystyle \sqrt{x^2+y^2}$

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- Apr 25th 2011, 12:44 PMHypatiaDaVinciHow to prove that this limit does exists...
How can I correctly justify, demonstrate, and properly determine that this limit exists?

lim (x,y) --> (0,0) $\displaystyle \sqrt{x^2+y^2}$ - Apr 25th 2011, 01:05 PMPlato
- Apr 25th 2011, 06:17 PMHypatiaDaVinci
Thank you very much for the answer. How do you get to the conclusion that $\displaystyle \varepsilon=\delta$ I mean... how do you directly relate them?

- Apr 25th 2011, 06:21 PMTheChaz
To prove this by the definition, you should state that

"For all x,y in R, for all epsilon > 0, there exists a delta > 0 such that [d(x, y) < delta] implies [sqrt(x^2 + y^2) < epsilon]

since "d(x, y)" and "sqrt(x^2 + y^2)" are the same thing, you can choose delta to be the epsilon that is given. - Apr 26th 2011, 07:51 AMHypatiaDaVinci
Thank you very much, I think I got it now :)