Question

I need to show the following limit exists. lim(x,y) -> (0,0) of xy(x-y)/(x^2 + y^2). I can probably plug in for y=0 and x=0, but how can I show the limit exists for all lines?

Answer 1

Try using polar coordinates:

\[x = r\cos(\theta)\]\[y = r\sin(\theta)\]
Now your limit becomes: \[\lim_{(x,y) \rightarrow (0,0)} [x \times y \times (x-y)] \div [x^{2} + y^{2}] = \]
\[\lim_{r \rightarrow 0} [r \cos(\theta)\times r \sin(\theta)\times (r \cos(\theta) - r \sin(\theta))]\div[r^2 \cos^2(\theta) + r^2 \sin^2(\theta)] \] \[\lim_{r \rightarrow 0} [r^2\cos(\theta)\sin(\theta)\times r(\cos(\theta) - \sin(\theta))]\div r^2\]Now, observe that it's a limit of "r" going to 0, so r can never be 0, therefore we can divide by r² the top and the bottom our ratio:\[\lim_{r \rightarrow 0} r \times \cos(\theta)\sin(\theta)\times(\cos(\theta) -\sin(\theta)) = \]\[\cos(\theta)\sin(\theta)\times(\cos(\theta) -\sin(\theta))\lim_{r \rightarrow 0}r = 0\]Meaning that, independently of the angle (the theta) you are approaching the origin, it will always go to zero. Hope it helped!