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.