Let $X$ a metric space, $E \subseteq X$ and $p$ be a limit point of $E$. Let $f,g: E \subseteq X \to \mathbb{C}$ be functions. In Rudin and Apostol's books, I read the following:
Let $$\frac{f}{g}: Z:=\{x \in E: g(x) \neq 0\} \to \mathbb{C}: x \mapsto \frac{f(x)}{g(x)}$$
Let $A = \lim_{x \to p} f(x), B = \lim_{x \to p} g(x)$
Then $$\lim_{x \to p} \left(\frac{f}{g}\right)(x) = \frac{A}{B}$$
How are we sure that the limit makes sense? I.e., how do we know that $p$ is a limit point of $Z$ as well?
We have to require that $B = \lim_{x \to p} g(x) \ne 0$.
If this is the case, then $p$ is a limit point of $Z$.