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Calculus/Finding Continuity in 3D

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Expert: - 9/30/2008


Question
Having a bit of trouble with this question....

Define f(0,0) in a way that extends  f(x,y)= x^2 - y^2/x^2 + y^2 to be continuous at the origin.

I thought that I should take the limit and see if the limit can be calculated at the origin, but i can't figure out how to do that or if that is even the right approach.

Answer
It looks like the problem should be (x²-y²)/(x²+y²).
This factors into (x+y)(x-y)/(x²+y²).

If we stick to the line x=y, the top of the fraction is 0 and the bottom is not, so the limit is 0.

If we stick to the line y=0, the function reduces to x²/x², and the limit is 1.

If we stick to the line x=0, the function reduces to -y²/y², and the limit is -1.

I'm not sure how the function could be continuous at the origin.  

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