Answer
See the proof below.
Work Step by Step
Along the $ x $-axis $ y=0$, the function $\frac{xy}{x^2+y^2}=\frac{0}{x^2+0}=0$ and along the $ y $-axis $ x=0$, the function $\frac{xy}{x^2+y^2}=\frac{0}{0+y^2}=0$.
Now, along the line $y=x $, we have
$$ \lim\limits_{(x,y) \to (0,0)}\frac{xy}{x^2+y^2}= \lim\limits_{(x,y) \to (0,0)}\frac{xy}{x^2+y^2}\\
=\lim\limits_{(x,y) \to (0,0)}\frac{x^2}{x^2+x^2}\\
=\lim\limits_{(x,y) \to (0,0)}\frac{1}{2}\\
=\dfrac{1}{2}$$
Thus the limits are not the same and hence the overall limit does not exist.
