Answer
$$0$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{x{y^2}}}{{{x^2} + {y^2}}} \cr
& {\text{Rewrite the limit using polar coordinates}} \cr
& x = r\cos \theta ,{\text{ }}y = r\sin \theta \cr
& \left( {x,y} \right) \to \left( {0,0} \right),{\text{ so }}r \to 0 \cr
& {\text{Substituting}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{x{y^2}}}{{{x^2} + {y^2}}} \Rightarrow \mathop {\lim }\limits_{r \to 0} \frac{{\left( {r\cos \theta } \right){{\left( {r\sin \theta } \right)}^2}}}{{{{\left( {r\cos \theta } \right)}^2} + {{\left( {r\sin \theta } \right)}^2}}} \cr
& = \mathop {\lim }\limits_{r \to 0} \frac{{{r^3}\cos \theta {{\sin }^2}\theta }}{{{r^2}{{\cos }^2}\theta + {r^2}{{\sin }^2}\theta }} \cr
& = \mathop {\lim }\limits_{r \to 0} \frac{{{r^3}\cos \theta {{\sin }^2}\theta }}{{{r^2}\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)}} \cr
& = \mathop {\lim }\limits_{r \to 0} \frac{{r\cos \theta {{\sin }^2}\theta }}{{\left( 1 \right)}} \cr
& = \mathop {\lim }\limits_{r \to 0} r\cos \theta {\sin ^2}\theta \cr
& {\text{Evaluate the limit when }}r \to 0 \cr
& = \left( 0 \right)\cos \theta {\sin ^2}\theta \cr
& = 0 \cr
& {\text{Then, we can conclude that}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{x{y^2}}}{{{x^2} + {y^2}}} = 0 \cr} $$
