When we talk about: $$\lim_{x\to{c}}f(x)=L$$ Is there a formal name for the number "$c$"?
I know of course that it 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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6$\begingroup$ I am not aware of a standard terminology. $\endgroup$– Steven GubkinFeb 5, 2020 at 20:13
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2$\begingroup$ Just for fun: The "approachand," pronounced ah-proach-and, analogous to "operand." $\endgroup$– Joseph O'RourkeFeb 6, 2020 at 0:36
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2$\begingroup$ 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 HendersonFeb 6, 2020 at 2:21
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2$\begingroup$ Maybe "approach value" for $c$ and "limit value" for $L$? $\endgroup$– Dave L RenfroFeb 6, 2020 at 19:16
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5$\begingroup$ 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). $\endgroup$– PedroFeb 26, 2020 at 14:54
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1 Answer
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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.
answered May 29, 2020 at 3:33
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$\begingroup$ Yep. The limit point of the limiting operation. $\endgroup$ May 29, 2020 at 15:33
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$\begingroup$ 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. $\endgroup$ Aug 6, 2021 at 7:53