Answer
See the proof below.
Work Step by Step
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.
