Why don't we include $y$ approaches to $y_0$ as a limit?

Question

How comes we only use $x$ approaches to $x_0$ when $y$ approaches $y_0$ is equally important. Why don't we include $y$ approaches $y_0$?

Answer 1

Because $y$ depends on $x$... The points $(x_0,y_0)$ and $(x_1,y_1)$ belong to the graphic of a function $f$, so $y_1=f(x_1)$ and $y_0=f(x_0)$.