User:IssaRice/Tao's notation for limits: Difference between revisions

Revision as of 03:00, 1 December 2018

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 $f : X \to R$ , 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 $f : X \to R$ and some $E \subseteq X$ , we have $f |_{E} : E \to R$ . 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.

@@ Line 5: / Line 5: @@
 The additional definition is this: if <math>f : X \to \mathbf R</math>, then we define <math>\lim_{x\to x_0} f(x) := \lim_{x\to x_0;\, x \in X} f(x)</math>. In other words, by default we assume that the limit is taken over the entire domain of the function.
-Now, given <math>f : X \to \mathbf R</math> and some <math>E \subseteq X</math>, we have <math>f|_E : E \to \mathbf R</math>. Thus, <math>\lim_{x\to x_0} f_E(x) = \lim_{x\to x_0;\, x \in E} f|_E(x)</math>.
+Now, given <math>f : X \to \mathbf R</math> and some <math>E \subseteq X</math>, we have <math>f|_E : E \to \mathbf R</math>. Thus, <math>\lim_{x\to x_0} f|_E(x) = \lim_{x\to x_0;\, x \in E} f|_E(x)</math>.
 By exercise 9.4.6, <math>\lim_{x \to x_0;\, x\in E} f(x) = \lim_{x\to x_0;\, x\in E} f|_E(x)</math>
-Combining these two equalities, we have <math>\lim_{x \to x_0;\, x\in E} f(x) = \lim_{x\to x_0} f_E(x)</math> as promised.
+Combining these two equalities, we have <math>\lim_{x \to x_0;\, x\in E} f(x) = \lim_{x\to x_0} f|_E(x)</math> as promised.