Example 2. Given the function

$$
f(x, y)=x \sin \frac{1}{y}+y \sin \frac{1}{x} \quad\left(x^{2}+y^{2} \neq 0\right), \quad f(0, y)=0, \quad f(x, 0)=0
$$

find $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} f(x, y)$.
Solution. Let $\varepsilon>0$, then for $|x|<\frac{\varepsilon}{2},|y|<\frac{\varepsilon}{2}$ we get $\rho=$

$$
=\sqrt{\left(\frac{\varepsilon}{2}\right)^{2}+\left(\frac{\varepsilon}{2}\right)^{2}}=\frac{\sqrt{2}}{2} \varepsilon . \text { We form the difference } f(x, y)-0 \text { and estimate it: }
$$

$$
|f(x, y)-0| \leq|x| \cdot\left|\sin \frac{1}{y}\right|+|y| \cdot\left|\sin \frac{1}{x}\right| \leq|x|+|y|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon
$$

Thus,

$$
|f(x, y)-0|<\varepsilon
$$

Based on formula (6.10), we conclude that

$$
\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} f(x, y)=0
$$