## Thomas' Calculus 13th Edition

Consider two different approaches to (0,0): 1. along x=2y, $\lim\limits_{x,y \to 0,0}f(x,y)=\lim\limits_{x,y \to 0,0}\frac{4y^{2}-y^{2}}{4y^{2}+y^{2}}=\frac{3}{5}$ 2. along x=3y, $\lim\limits_{x,y \to 0,0}f(x,y)=\lim\limits_{x,y \to 0,0}\frac{9y^{2}-y^{2}}{9y^{2}+y^{2}}=\frac{4}{5}$ Hence, the limit of function $f(x,y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$ at (0,0) does not exist, and we can not define $f(0,0)$ to make it continuous at the origin.