At th beginning of the third chapter of Baby Rudin, the author defines the limit of a function in the following manner:
Let $X$ and $Y$ be metric spaces; suppose $E \subseteq X$, $f$ maps $E$ into $Y$ and $p$ is a limit point of $E$. We write $f(x) \rightarrow q$ as $t \rightarrow p$, or $$\lim_{x \to p} f(x) = p$$
if for every $\epsilon > 0$ there exists some $\delta > 0$ such that
$$d_Y(f(x), q) < \epsilon$$
for all points $x \in E$ such that
$$d_X(x, p) < \delta$$.
At the end of the chapter he provides an alternative definition of a limit, one that is meant to be fit for the extended real number system. The definition reads: 
Does the latter definition require $x$ to be a limit point of $E$? If it does, how? And if it doesn't, how could this definition be equivalent to the first one then?