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When we talk about: $$\lim_{x\to{c}}f(x)=L.$$ Is there a formal name for the number "$c$"?

I know that the notation means "$L$ is the limit of $f(x)$ as $x$ approaches $c$". It just would be nice to be able to refer to "$c$" separately without saying "what $x$ approaches". I tried to look at various textbooks but didn't find anything.

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    $\begingroup$ I am not aware of a standard terminology. $\endgroup$
    – Steven Gubkin
    Feb 5, 2020 at 20:13
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    $\begingroup$ I generally refer to it as "the limiting value of $x$". It is a little clunky, and probably not any better than "the thing which $x$ approaches", but works. $\endgroup$
    – Xander Henderson ♦
    Feb 6, 2020 at 2:21
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    $\begingroup$ @JosephO'Rourke I believe the "correct" term based on Latin would be "appropiand" (that which is to be approached). $\endgroup$
    – Raciquel
    Feb 7, 2020 at 0:30
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    $\begingroup$ Maybe, we could refer to $c$ as the limit point because, in the definition of limit, $c$ has to be a limit point of the domain. This expression would be good because, according to the general definition, a limit point of the domain is a point that (i) can be approximated by elements of the domain and (ii) need not be in the domain (which are the two properties of $c$ that have to be highlighted for the students). $\endgroup$
    – Pedro
    Feb 26, 2020 at 14:54
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    $\begingroup$ I was about to suggest the same as @Pedro and wanted to add, some might get potentially confused and stumble wondering whether limit point refers to $c$ or $L$ but one thing I try to do to subconsciously reinforce a distinction is to refer to "points" in the domain and "values" in the range (as consistently as I remember, and when e.g. you compose one function into another the terminology plays not so well, but I want to think it helps). $\endgroup$
    – Vandermonde
    May 28, 2020 at 18:44

3 Answers 3

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I vote for Pedro. We should call the point at which the limit is taken the limit point. In contrast, the value obtained by the limit (if it exists) is the limit's value. In particular, $$ \lim_{x \rightarrow c} f(x) = L $$ has limit point $c$ and the value of the limit is $L$. This terminology keeps with the usual usage of the term value for outputs of the function. In addition, while the term limit point does have a more abstract topological definition, I don't think there is much danger of confusion.

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  • $\begingroup$ "The limit point of the limit"? $\endgroup$
    – Peter Saveliev
    May 29, 2020 at 12:36
  • $\begingroup$ Yep. The limit point of the limiting operation. $\endgroup$
    – James S. Cook
    May 29, 2020 at 15:33
  • $\begingroup$ If the co-domain is a topological space, then point could also refer to a point in the co-domain. I think this will normally not be an issue. But if we want to disambiguate, we might call c the limit point of the domain. $\endgroup$
    – Klaas van Aarsen
    Aug 6, 2021 at 7:53
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    $\begingroup$ So we get to call it a limit point AND eat chimichangas next year? Sounds good to me! $\endgroup$
    – Thierry
    Aug 15 at 19:45
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    $\begingroup$ The problem is that limit point already has a meaning in analysis and topology. (One of the requirements of the definition of limit is that $c$ be a limit point of the domain of the function.) Pedro does not see that as a problem but having slightly different meanings seems worse to me than having completely different meanings. $\endgroup$
    – Raciquel
    Aug 16 at 20:21
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The "c" in limit notation is called the index (or at least that's what I have always been taught).

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    $\begingroup$ It would help to link to some reference which uses this term. $\endgroup$
    – Sue VanHattum ♦
    Aug 15 at 20:27
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Not standard but how about: the limit of $x$ or the limit of the independent variable?

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    $\begingroup$ Using limit of $x$ for $c$ would easily be confused with the limit's value $L$. $\endgroup$
    – Isaiah
    Aug 17 at 12:31

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