Limit notation convention

Question

I've seen in different sources that there is a prevalent notation convention regarding to limits.

If $f: X \rightarrow \mathbb{R}$ is a function and $x_0$ is an adherent point of $X$. It's very common to see that $\lim_{x\rightarrow x_0}f(x)=L$ means $f(x)\rightarrow L$ whenever $x\rightarrow x_0$ and $x\not= x_0$. In other words $x$ is not allowed to be equal to $x_0$ (in almost all the books that I've seen so far).

But in a very few other sources I've seen a more general approach defining $\lim_{x\rightarrow x_0; x\in E}f(x)=L$, $f(x)\rightarrow L$ whenever $x\rightarrow x_0$ in $E$; in particular in this notation the first limit can be expressed as $\lim_{x\rightarrow x_0;x\not= x_0}f(x)=L$.

So my questions are regarding to this: what is the advantage of one over the other in general? what could be the principal reason for which one is more common than the other which is a more general notation? Or is just a matters of taste to interpret either $\lim_{x \to x_0} f(x) = L$ as “$f(x) \to L$ whenever $x \to x_0$" in $E$, or “whenever $x \to x_0$ and $x \neq x_0$"?

Edit: I know that it may happen that $f(x_0)$ is not even defined, but this doesn't really matter because the limit perfectly exists as $\lim_{x\to x_0;x\in\text{dom}f\backslash \{x_0\}} f(x)$. So if the limit exists at the point or not is not important because we can simple restrict the set in which is defined and we're done the limit exists in this set.

Edit: If were the case of the example of Martin Argerami, then the only point of advantage of the second notation over the first is that for the first, if $x_n \to x_0$ then this not necessarily imply $f(x_n)\to L$ yet $\lim_{x\to x_0} f(x)= L$ but for the other case, the second notation, clearly $\lim_{x\to x_0; x\in\text{Dom}f \setminus \{x_0\} } f(x)= L$ and for any sequence $(x_n) \in \text{Dom}f\backslash \{x_0\}$ clearly would have $(f(x_n)) \to L$. Then maps convergent sequences in $E$ to convergent sequences which i think is nice.

But the main point here is this: what does the reason for which there is a prevalence for the first definition over the second which is more general? Is it just because is customary or by simplicity at time to write and save ink?

Thanks in advance

Answer 1

If we allow $x$ to be $x_0$ in $\lim_{x\to x_0}f(x)$ and $f(x_0)$, then the only possible value of the limit is $f(x_0)$ itself. In such a situation it wouldn't make much sense to waste ink on saying $\lim_{x\to x_0}f(x)$ because we could just have written $f(x_0)$ and get the same meaning whenever the expression is meaningful at all. The only reason to say $\lim_{x\to x_0}$ when $f$ is defined at $x_0$ would be to speak about whether the limit exists at all -- but we already have a good terminology for that with the usual conventions, namely "$f$ is continuous at $x_0$".

(This argument depends slightly on the fact that it is usually pretty clear whether $f$ is defined at $x_0$ or not).

The usual notion of limit (where we ignore the value, if any, of $f(x_0)$) is from a pragmatic standpoint more useful. There's only a difference when $x_0$ is in fact in the domain of $f$, but when it is, the usual notion allows us to use limits to define continuity (namely, $f$ is continuous at $x_0$ exactly when $f(x_0)=\lim_{x\to x_0}f(x)$).

It's true that in some cases it can be useful to allow a notation such as $\lim_{x\to x_0;x\in E}$, but it still makes sense to make the default space-saving meaning of $\lim_{x\to x_0}$ be one that excludes the value at $x_0$.

You may choose to consider $\lim_{x\to x_0}f(x)$ to be an abbreviation of $\lim_{x\to x_0;x\in\operatorname{Dom}f\setminus\{x_0\}} f(x)$ if you want.

Answer 2

Allowing $x=x_0$ would make the limit fail to exist in situations where you want to exist. For instance, let $$ f(x)=\begin{cases}0,&\ x\ne0,\\ 10,&\ x=0\end{cases} $$ This is a prototypical example of a function with an avoidable discontinuity at zero. But if you allow $x=0$, then $\lim_{x\to0}f(x)$ does not exist: for any $\delta>0$, you can choose $x\in(-\delta,\delta)$ with $f(x)=0$, and $x\in(-\delta,\delta)$ with $f(x)=10$.

Answer 3

I strongly doubt that anyone using the more general form ever intended to allow $x_0\in E$. It is always implicit that you are dealing with deleted neighborhoods of $x_0$, so it isn't usually stated. The general case lets you deal with things such as radial limits, where $x\rightarrow x_0$ along a ray, or in a sector, etc. In every case, it is implicit that you exclude $x_0$ from consideration.

It doesn't matter what the value of $f$ at $x_0$ is. It has no effect whatsoever on the the limiting value of $f$ there. That's the very meaning of a limit.