Are there a condition to write this limit formula

Question

Are there a condition to write this limit formula:

$$\lim_{(x,y)\to(x_0,y_0)}(g_1(x,y),g_2(x,y))=(\lim_{(x,y)\to(x_0,y_0)}g_1(x,y),\lim_{(x,y)\to(x_0,y_0)}g_2(x,y)).$$

Thank you.

Answer 1

As in the one-variable case, you need the functions $g_1$ and $g_2$ to be continuous at $(x_0, y_0)$.