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

$lim_{(𝑥,𝑦)→(0,0)} 𝑓(𝑥, 𝑦)\\ =lim_{(𝑥,𝑦)→(0,0)} 𝑥𝑦 ( 𝑥^2−𝑦^2 𝑥^2+𝑦^2 )\\ \text{ put y=mx},\\ =lim_{x→0} mx^2 ( 𝑥^2−m^2 𝑥^4+m^2x^2 )\\ =0$