Chapter 14: Partial Derivatives - Section 14.2 - Limits and Continuity in Higher Dimensions - Exercises 14.2 - Page 797: 60

$0$

$ \lim\limits_{(x,y) \to (0,0) } |f(x,y)|=\lim\limits_{(x,y) \to (0,0) }|xy \times \dfrac{x^2-y^2}{x^2+y^2}|$ or, $=\lim\limits_{(x,y) \to (0,0) } \dfrac{|xy|}{x^2+y^2}|x^2-y^2|$ or, $\lim\limits_{(x,y) \to (0,0) } \dfrac{|xy|}{x^2+y^2}|x^2-y^2| \leq \lim\limits_{(x,y) \to (0,0) } \dfrac{1}{2} \times |x^2-y^2|$ or, $=0$ So, by the Sandwich Theorem$ \lim\limits_{(x,y) \to (0,0) } |f(x,y)|=0$