User:IssaRice/Tao's notation for limits

Tao's notation for a limit is $\lim _{x\to x_{0};\,x\in E}f(x)$.

Can we write this in a more standard way? basically, if we give one additional definition, we have the very appealing formula $\lim _{x\to x_{0};\,x\in E}f(x)=\lim _{x\to x_{0}}f|_{E}(x)$.

The additional definition is this: if $f:X\to \mathbf {R}$, then we define $\lim _{x\to x_{0}}f(x):=\lim _{x\to x_{0};\,x\in X}f(x)$. In other words, by default we assume that the limit is taken over the entire domain of the function.

Now, given $f:X\to \mathbf {R}$ and some $E\subseteq X$, we have $f|_{E}:E\to \mathbf {R}$. Thus, $\lim _{x\to x_{0}}f|_{E}(x)=\lim _{x\to x_{0};\,x\in E}f|_{E}(x)$.

By exercise 9.4.6, $\lim _{x\to x_{0};\,x\in E}f(x)=\lim _{x\to x_{0};\,x\in E}f|_{E}(x)$

Combining these two equalities, we have $\lim _{x\to x_{0};\,x\in E}f(x)=\lim _{x\to x_{0}}f|_{E}(x)$ as promised.

Rather than thinking of this as a result per se, I think it's better to think of this as alternative notations. Why might one prefer one notation over the other? I think the strength of Tao's notation is that it works for anonymous functions/expressions. To be able to use the function restriction notation $f|_{E}$, we must have named our function beforehand. To give an example, we can write something like $\lim _{x\to 0;\,x\in (0,\infty )}|x|/x=1$, but this is difficult to write in the other notation; we would have to say something like, "Let $f:\mathbf {R} \setminus \{0\}\to \mathbf {R}$ be defined by $f(x):=|x|/x$. Then we have $\lim _{x\to 0}f|_{(0,\infty )}(x)=1$."

I think usually one would write the above like $\lim _{x\to 0+}|x|/x=1$. So then one gets to keep anonymous functions, but at the expense of adding more complexity to the notation (one now needs to assign meaning to "$0+$").

One should compare this to the function notation ($f'$) vs Leibniz notation (${\frac {d}{dx}}f(x)$) for derivatives.

Notation	Symbols	Strengths	Weaknesses
Tao's notation	$\lim _{x\to x_{0};\,x\in E}f(x)$	Works with anonymous functions.	Somewhat verbose: the "$x\in E$" part must be written out each time.
Function restriction notation	$\lim _{x\to x_{0}}f|_{E}(x)$		Doesn't work with anonymous functions.
Standard notation	$\lim _{x\to x_{0}}f(x)$	Works with anonymous functions.