## Calculus (3rd Edition)

Along the $x$-axis $y=0$, we have $$\lim\limits_{(x,y) \to (0,0)}\frac{x}{x^2+ y^2 } = \lim\limits_{(x,y) \to (0,0)}\frac{x}{x^2 }= \lim\limits_{(x,y) \to (0,0)}\frac{1}{x }=\infty.$$ hence the limit does not exist.