Question

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

Answer 1

Convert to to polar coordinates: $x=r\cos(\theta), y=r\sin (\theta)$

$$\lim\limits_{(x,y)\to(0,0)}\dfrac{xy(x^2-y^2)}{x^2+y^2}$$

$$=\lim\limits_{r\to0}\dfrac{r\cos(\theta) r\sin(\theta)(r^2\cos^2(\theta)-r^2\sin^2(\theta))}{r^2\cos^2(\theta)+r^2\sin^2(\theta)}$$

$$=\lim\limits_{r\to0}\dfrac{r^4\cos(\theta) \sin(\theta)(\cos^2(\theta)-\sin^2(\theta))}{r^2}$$

$$=\lim\limits_{r\to0}\dfrac{1}{2}r^2 \sin(2\theta)\cos(2\theta)$$

$$=\dfrac{1}{2}(0)^2 \sin(2\theta)\cos(2\theta)=0$$

Therefore

$$\lim\limits_{(x,y)\to(0,0)}\dfrac{xy(x^2-y^2)}{x^2+y^2}=0$$