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.