Consider the function 𝑓(𝑥, 𝑦) = 𝑥𝑦 ( 𝑥 2−𝑦 2 𝑥 2+𝑦2 ) , (𝑥, 𝑦) ≠ (0, 0). Show that lim (𝑥,𝑦)→(0,0) 𝑓(𝑥, 𝑦) = 0
Convert to to polar coordinates: "x=r\\cos(\\theta), y=r\\sin (\\theta)"
"=\\lim\\limits_{r\\to0}\\dfrac{r\\cos(\\theta) r\\sin(\\theta)(r^2\\cos^2(\\theta)-r^2\\sin^2(\\theta))}{r^2\\cos^2(\\theta)+r^2\\sin^2(\\theta)}"
"=\\lim\\limits_{r\\to0}\\dfrac{r^4\\cos(\\theta) \\sin(\\theta)(\\cos^2(\\theta)-\\sin^2(\\theta))}{r^2}"
"=\\dfrac{1}{2}(0)^2 \\sin(2\\theta)\\cos(2\\theta)=0"
Therefore
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