@Sreshtha.
Even I think (C) should be the answer. Lets see why.
$\lim x\rightarrow0 y\rightarrow0$ $\frac{xy}{x^{2}+y^{2}}$
= $\lim x\rightarrow0 y\rightarrow0$ $\frac{xy}{x^{2}+y^{2}}$
= $\lim x\rightarrow0 y\rightarrow0$ $\frac{x}{x^{2}+y^{2}}$ + $\lim x\rightarrow0 y\rightarrow0$ $\frac{x}{x^{2}+y^{2}}$ .........Y times
Now, $\frac{x}{x^{2}+y^{2}}$ tends to 1. ........................(1)
Also, we are taking this sum Y times.
Let , Y = 1/Z and let, Z -> infinity
Thus, we are dividing (1) by infinity.
Thus, this tends to 0.
Dont know if right approach or wrong.