Definition 4.33 $\quad$ Let $f$ be a real function defined on $E.$ We say that $f(t)\to A$ as $t\to x$, where $A$ and $x$ are in the extended real number system, if for every neighborhood $U$ of $A$ there is a neighborhood $V$ of $x$ such that $V \cap E$ is not empty, and such that $f(t)\in U$ for all $t\in V\cap E, t\ne x.$
Definition 4.1 $\quad$ Let $X$ and $Y$ be metric space; suppose $E \subset X$, $f$ maps $E$ into $Y$, and $p$ is a limit point of $E$. We write $f(x)\to q$ as $x\to p$, or $\lim_{x\to p}f(x)=q$ if there is a point $q \in Y$ with the following property: For every $\epsilon > 0$ there exists a $\delta > 0$ such that $d_Y(f(x),q)<\epsilon$ for all points $x\in E$ for which $0<d_X(x,p)<\delta$.
Rudin writes "A moment's consideration will show that this coincides with Definition 4.1 when $A$ and $x$ are real."
It seems to me that $x$ need not be a limit point of $E$ when $A$ and $x$ are real in definition 4.33. How does it coincides with definition 4.1 when $A$ and $x$ are real?
In the case that $x =$ $+\infty$ (or $-\infty$), shouldn't we need $E$ to be unbounded above (below) in the (not extended) real number system? Then it would make sense to define the limit at infinity.
