Investigate the limiting behavior at the origin on the three lines $x = 0 , y = 0$ and $y = x$. 
Investigate the limiting behavior at the origin on the three lines $x = 0 , y = 0$ and $y = x$. 

For the function $f :\mathbb R^2 \to \mathbb R$ given by
$$f(x,y)=\begin{Bmatrix} 0, \quad \{x,y\} = {0,0} \\ \frac{x^2}{x^2+y^2}  \quad\{x,y\}, \neq {0,0} \end{Bmatrix}$$
$$\lim_{x \to y } \frac{x^2}{x^2+y^2}  = \frac{x^2}{x^2+x^2} = \frac{x^2}{2x^2} =\frac{1}{2}   $$
$$ \lim_{y \to x } \frac{x^2}{x^2+y^2}= \frac{y^2}{y^2+y^2} = \frac{1}{2} $$
$$\lim_{x \to 0 } \frac{x^2}{x^2+y^2} = \frac{0}{0+y^2} = 0$$
$$\lim_{y \to 0 } \frac{x^2}{x^2+y^2} = \frac{x^2}{x^2} = 1$$
How's that?

You have the right idea but note that approaching along the straight line $x=0$, it is not always true that $\lim_{x,y \to 0} f(0,y) = \lim_{x\to 0} f(x,y)$. The notation is incorrect but you did arrive at the right conclusion (the limit does not exist due to differing values as we change paths).
