- #1

michonamona

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## Homework Statement

Find the limit of[tex] lim_{(x,y) \rightarrow (0,0)} xy(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}) [/tex]## Homework Equations

## The Attempt at a Solution

We were supposed to switch to polar coordinates to solve this problem. Thus we get,

[tex] lim_{(r) \rightarrow (0)}

rcos\theta rsin\theta (\frac{r^{2}cos^{2}\theta - r^{2}sin^{2}\theta}{r^{2}cos^{2}\theta+r^{2} sin^{2}\theta})[/tex]

cancel out the r squares in the fraction and using double angle formulas we get[tex] lim_{(r) \rightarrow (0)} r^{2} cos\theta sin\theta cos 2\theta[/tex]

I skipped a bunch of steps, I can write them out if I lose you guys.

My questions is as follows:

1.) What is the proper limit index after switching to polar coordinates, is r--->0 correct?

2.) What do we with the last line? Do we just let r approach zero and, thus, get limit = 0? or do we switch back to cartesian coordinate?

Thank you in advanced.

M

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