Chapter 15 - Differentiation in Several Variables - 15.2 Limits and Continuity in Several Variables - Exercises - Page 772: 31

$$\frac{1}{5}$$

Given $$\lim _{(x, y) \rightarrow(3,4)} \frac{1}{\sqrt{x^{2}+y^{2}}}$$ Since $ \dfrac{1}{\sqrt{x^{2}+y^{2}}}$ is continuous at $( 3,4)$, then by substitution, we get \begin{align*} \lim _{(x, y) \rightarrow(3,4)} \frac{1}{\sqrt{x^{2}+y^{2}}}&=\lim _{(x, y) \rightarrow(3,4)} \frac{1}{\sqrt{9+16}}\\ &=\frac{1}{5} \end{align*}