Chapter 15 - Differentiation in Several Variables - 15.2 Limits and Continuity in Several Variables - Exercises - Page 772: 30

$$0$$

Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{1+y^{2}}$$ Since $ \dfrac{x^{2}+y^{2}}{1+y^{2}}$ is continuous at $(0,0)$, then by substitution, we get \begin{align*} \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{1+y^{2}}&=\lim _{(x, y) \rightarrow(0,0)} \frac{0}{1+0}\\ &=0 \end{align*}