Answer
The limit doesn't exist.
Work Step by Step
Approach (0,0) along x-axis, $\lim\limits_{x \to 0}f(x,0) = \frac{0}{x^{2}} = 0$.
Approach (0,0) along y-axis, $\lim\limits_{y \to 0}f(0,y) = \frac{0}{y^{8}} = 0$.
Now, approach (0,0) along $x = y^{4}$ , we get $\lim\limits_{y \to 0}f(y^{4},y) = \frac{y^{8}}{2y^{8}}$=1/2
Since there are different limits along different paths, the limit doesn't exist.
