Answer
$$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}}{\left(x^{2}+1\right)\left(y^{2}+1\right)} =0$$
Work Step by Step
Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}}{\left(x^{2}+1\right)\left(y^{2}+1\right)} $$
by substituting $x=0, y=0$
So, we get
$$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}}{\left(x^{2}+1\right)\left(y^{2}+1\right)}=\frac{0}{(1)(1)}=0$$
