Approaching (0,0) along the line $y = mx+c$
$\Rightarrow$$\lim_{(x,y)\to(0,0)} \frac{x^2y}{x^4y + y^2}$
$\Rightarrow$$\lim_{(x,y)\to(0,0)} \frac{x^2.mx}{x^4.mx + m^2x^2}$
$\Rightarrow$$\lim_{(x,y)\to(0,0)} \frac{mx^3}{mx^5 + m^2x^2}$
$\Rightarrow$ $\lim_{(x,y)\to(0,0)} \frac{x}{x^2+ 1.m}$ $ = 0$
Approaching (0,0) along the parabola
$y = x^2$
$\Rightarrow$$\lim_{(x,y)\to(0,0)} \frac{x^2y}{x^4 + y^2}$ $ = $ $\lim_{x\to0} \frac{x^4}{x^4 + x^4}$
$= $$\lim_{x\to0} \frac{x^4}{2x^4 }$
$\Rightarrow \frac{1}{2}$
THe Limit does not exist.
answer is (D)