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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

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Your question is unclear to me, especially the second paragraph. Could you give an exact quote (and with a precise reference to the book or paper, the page number, and line number) for where you think a definition of limit is being made in which $x = x_{0}$ is allowed? –  Dave L. Renfro Feb 27 at 19:09
    
@DaveL.Renfro here I found it: math.ucla.edu/~tao/resource/general/131ah.1.03w/week6.pdf page number 11 to be specific. Also I think the Godement's Analysis uses this notation. Let me check that. –  Jose Antonio Feb 27 at 21:04
    
O-K, the definition of your 2nd paragraph limit is at the top of Tao's p. 11. Relevant to your question is the bottom of p. 14, and it does appear that Tao is allowing the value of $f(x_{0})$ to affect the value of the limit, at least he is building into the notation this option. I suppose this is being done for a little extra generality, as some results don't depend on this issue, but I must confess that I'm not entirely comfortable with the notions because of the high density of terms whose meaning is specific to this particular set of notes. –  Dave L. Renfro Feb 27 at 21:23
    
@DaveL.Renfro : I've checked and also the Godement's Analysis uses exactly the same notation. I admit that this is not the book that I'm reading but I have a copy here and it uses this notation. –  Jose Antonio Feb 27 at 21:38
    
@DaveL.Renfro : Also one advantage is that if $\lim_{x\to x_0; x\in E } f(x)= L$ then for any sequence $(x_n) \in E$ such that $x_n \to x_0$ clearly this would imply $(f(x_n)) \to L$ but this cannot be the case for the other definition, just consider the example given by Martin below. –  Jose Antonio Feb 27 at 21:46

3 Answers 3

up vote 1 down vote accepted

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.

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Thanks very helpful. Clearly both are essentially the same, so is more for abbreviation doesn't? Because any one could be transform in the other only restricting the set, if were the case in which $f(x_0)$ is undefined well that doesn't really matter as you says only setting $\text{dom} f \backslash \{x_0\}$ and the limit exists. –  Jose Antonio Feb 27 at 20:46

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 $f(x)=10$.

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But the limit perfectly exists as $\lim_ {x\to 0; x\not= 0} f(x)$ and indeed is zero. So there is no problem at all, I think. –  Jose Antonio Feb 27 at 20:30
    
Indeed in this case there is a lack of transitivity assuming this form of limit. For example if $x_n \to 0$ then not necessary would imply that $f(x_n) \to 0$ even though $\lim _{x\rightarrow 0} f(x)=0$. But this is not tha case if we define $\lim _{x\rightarrow 0; x\in \mathbb{R}\backslash \{0\}} f(x)=0$ any sequence $(x_n) \in \mathbb{R}\backslash \{0\}$ would imply $(f(x_n)) \to 0$. –  Jose Antonio Feb 27 at 20:36
    
For example setting $x_n=0$ clearly $x_n \rightarrow 0$ but not imply that $(f(x_n)) \to 0$. –  Jose Antonio Feb 27 at 20:50

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.

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The book of Terence Tao of Analysis uses this form of limit (the second one). I know that both are equivalent form. –  Jose Antonio Feb 27 at 20:28
    
Here I found it: math.ucla.edu/~tao/resource/general/131ah.1.03w/week6.pdf page number 11 to be specific and both are equivalent just restricting the set. –  Jose Antonio Feb 27 at 20:48
    
Also I think the Godement's Analysis uses this notation. Let me check that. –  Jose Antonio Feb 27 at 21:06

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