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When defining the limit of a function at a point, Terence Tao (Analysis I, 2016, 3e) also adds "in $E$".

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I think with the example below, most texts would simply say that $\lim_{x\rightarrow 0}f(x)=0$.

But Tao makes a distinction between $\lim_{x\rightarrow 0;x\in \mathbb{R}- \{0\}}f(x)$ (equals $0$) and $\lim_{x\rightarrow 0;x\in \mathbb{R}}f(x)$ (undefined).

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Why does he do this? What's the point/advantage gained?

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    $\begingroup$ Tao's notation is more precise. There are many examples of functions $f: X\to R$, when $\lim_{x\to x_0} f(x)$ does not exist, while $\lim_{x\to x_0: x\in E} f(x)$ does. $\endgroup$ Dec 7 '21 at 9:20
  • $\begingroup$ I don't know for sure, but maybe he's doing this to prepare the way for those students who will later encounter various path limits used for functions of several real variables, functions of a complex variable, in boundary value problems, etc. -- radial limits, tangential and nontangential limits, (continued) $\endgroup$ Dec 7 '21 at 11:21
  • $\begingroup$ curvilinear limits, angular limits, $\ldots$ Many other types of limits also belong to this general type, such as unilateral limits for functions of a real variable and the approximate limit of classical real analysis. $\endgroup$ Dec 7 '21 at 11:22
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I think with the example below, most texts would simply say that $\lim_{x\rightarrow 0}f(x)=0$.

Indeed, but so would Tao, since he says

In the textbook, $\lim_{x\to x_0} f(x)$ is used as shorthand for $\lim_{x\to x_0;x\in X-\{x_0\}} f(x)$.

So Tao is not really defining limits in a way that would be in any way in conflict with the standard calculus notation. His notation is merely an expansion of the original one. I don't know the exact reason for using this notation, but it might make some expressions more clear. In particular, students are often confused by the fact that the value of $f(a)$ is completely irrelevant when calculating $\lim_{x\to a} f(x)$. This fact is much more clear in Tao's notation.

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