Chapter 15 - Differentiation in Several Variables - 15.2 Limits and Continuity in Several Variables - Exercises - Page 772: 38

$$0$$

Since $(x+y+2) e^{-1 /\left(x^{2}+y^{2}\right)}$ is continuous at $(0,0)$, then \begin{align*} \lim _{(x, y) \rightarrow(0,0)}(x+y+2) e^{-1 /\left(x^{2}+y^{2}\right)}&=\lim _{(x, y) \rightarrow(0,0)}(x+y+2)\lim _{(x, y) \rightarrow(0,0)} e^{-1 /\left(x^{2}+y^{2}\right)}\\ &=(2)(0)=0 \end{align*}