User:IssaRice/Tao's notation for limits

Tao's notation for a limit is $\underset{x\to x_0;x\in E}{lim}f(x)$.

Can we write this in a more standard way? basically, if we give one additional definition, we have the very appealing formula $\underset{x\to x_0;x\in E}{lim}f(x)=\underset{x\to x_0}{lim}f{|}_E(x)$.

The additional definition is this: if $f:X\to \mathbf{R}$, then we define $\underset{x\to x_0}{lim}f(x):=\underset{x\to x_0;x\in X}{lim}f(x)$. 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, $\underset{x\to x_0}{lim}f{|}_E(x)=\underset{x\to x_0;x\in E}{lim}f{|}_E(x)$.

By exercise 9.4.6, $\underset{x\to x_0;x\in E}{lim}f(x)=\underset{x\to x_0;x\in E}{lim}f{|}_E(x)$

Combining these two equalities, we have $\underset{x\to x_0;x\in E}{lim}f(x)=\underset{x\to x_0}{lim}f{|}_E(x)$ 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 $\underset{x\to 0;x\in (0,\mathrm{\infty })}{lim}\left|x\right|/x=1$, 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 $f(x):=\left|x\right|/x$. Then we have $\underset{x\to 0}{lim}f{|}_{(0,\mathrm{\infty })}(x)=1$."

An amusing fact/potential advantage of the function restriction notation is that one can write it in a point-free notation. Namely, we can write $\underset{x_0}{lim}f{|}_E$.

I think usually one would write the above like $\underset{x\to 0+}{lim}\left|x\right|/x=1$. 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 ($f^{\prime }$) vs Leibniz notation ($\frac{d}{dx}f(x)$) for derivatives.

Notation	Symbols	Strengths	Weaknesses
Tao's notation	$\underset{x\to x_0;x\in E}{lim}f(x)$	Works with anonymous functions.	Somewhat verbose: the "$x\in E$" part must be written out each time.
Function restriction notation	$\underset{x\to x_0}{lim}f{|}_E(x)$		Doesn't work with anonymous functions.
Standard notation	$\underset{x\to x_0}{lim}f(x)$	Works with anonymous functions.