Example 1. Find $\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\sin x y}{y}$.
Solution. At the point $M(2,0)$, the function $z=\frac{\sin x y}{y}$ is undefined.

By multiplying and dividing the given function by $x \neq 0$, we get

$$
\frac{\sin x y}{y}=\frac{x \sin x y}{x y}=x \frac{\sin x y}{x y}
$$

Taking the limit in the last equality, we obtain

$$
\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\sin x y}{y}=\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} x \cdot \frac{\sin x y}{x y}=\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} x \cdot \lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\sin x y}{x y}=2 \cdot 1=2
$$

since $\lim _{\alpha \rightarrow 0} \frac{\sin \alpha}{\alpha}=1$.