4.5 LIMIT OF A FUNCTION
In this section we consider two metric spaces $(S, d_S)$ and $(T, d_T)$, where $d_S$ and $d_T$ denote the respective metrics. Let $A$ be a subset of $S$ and let $f:A\to T$ be a function from $A$ to $T$.Definition 4.11. If $p$ is an accumulation point of $A$ and if $b\in T$, the notation $$\lim_{x\to p} f(x) = b,$$ is defined to mean the following:
For every $\epsilon>0$ there is a $\delta>0$ such that $$d_T(f(x),b)<\epsilon\,\,\,\,\,\text{whenever }x\in A, x\neq p,\text{ and }d_S(x,p)<\delta.$$
Let $f:\mathbb{R}\to(-\frac{\pi}{2},\frac{\pi}{2})$ be a function such that $f(x) =\arctan(x).$
Here we consider $+\infty$ is an accumulation point of $\mathbb{R}$.
Then, $\frac{\pi}{2}\notin (-\frac{\pi}{2},\frac{\pi}{2})$, so the author cannot use the notation $$\lim_{x\to +\infty} f(x) = \frac{\pi}{2}.$$ But we use the notation $$\lim_{x\to +\infty} f(x) = \frac{\pi}{2}.$$
Let $g:\mathbb{R}\to\mathbb{R}$ be a function such that $g(x) =\arctan(x).$
Then, the author can use the notation $$\lim_{x\to +\infty} g(x) = \frac{\pi}{2}$$ since $\frac{\pi}{2}\in \mathbb{R}$.
For example, when we use a real valued function $f$, is it a good practice to think the codomain of $f$ is $\mathbb{R}$?
Is the following definition bad?
4.5' LIMIT OF A FUNCTION
In this section we consider two metric spaces $(S, d_S)$ and $(T, d_T)$, where $d_S$ and $d_T$ denote the respective metrics. Let $A$ be a subset of $S$ and $U$ be a subset of $T$ and let $f:A\to U$ be a function from $A$ to $U$.Definition 4.11'. If $p$ is an accumulation point of $A$ and if $b\in T$, the notation $$\lim_{x\to p} f(x) = b,$$ is defined to mean the following:
For every $\epsilon>0$ there is a $\delta>0$ such that $$d_T(f(x),b)<\epsilon\,\,\,\,\,\text{whenever }x\in A, x\neq p,\text{ and }d_S(x,p)<\delta.$$
