I have a problem regarding the function f (x,y) = {x*y*(x^2-y^2)/(x^2+y^2) if (x,y)!=(0,0) and f(x,y)=0 if (x,y)=(0,0).(adsbygoogle = window.adsbygoogle || []).push({});

I am asked if this function is differentiable. Running it through a graphing program it looks differentiable. I know the partial derivatives of it in terms of x and y are 0 and 0 respectively and I know that a function is differentiable at (0,0) if fx(0,0) and fy(0,0) exists which they do and if lim (x,y)->(0,0) [f(x,y)-h(x,y)/||(x,y)-(a,b)||]=0 where h(x,y) describes the tangent plane of the function at point (a,b) or in our case (0,0).

So after a lot of algebra which I will not repeat here using the above definition I boil it all down to lim (x,y)->(0,0) [x*y*(x^2-y^2)/sqrt(x^2+y^2)] and the question is does this = 0? I have tried to prove it is using delta epsilon proofs but I just get a bunch of gibberish unless I am doing something wrong, I might be I am not sure and I have also tried to prove this is wrong by trying to come into (0,0) at some odd angle like y=mx and showing it doesn't = 0 but all attempts have failed. Help.

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# Homework Help: Is f Differentiable? - Vector Calculus

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