Chapter 15 - Differentiation in Several Variables - 15.2 Limits and Continuity in Several Variables - Exercises - Page 772: 32

$$0$$

Given $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}$$ Consider the line $ y=mx$ that passes through $(0,0)$: \begin{align*} \lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}&=\lim _{x\to 0 } \frac{mx^2}{\sqrt{x^{2}+m^{2}x^2}}\\ &=\lim _{x\to 0} \frac{mx}{\sqrt{1+m^2}}\\ &=0 \end{align*}