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

Revision as of 03:16, 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.

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 $f : R ∖ {0} \to R$ 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 ( $\frac{d}{d x} f (x)$ ) 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.

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 I think usually one would write the above like <math>\lim_{x\to 0+} |x|/x = 1</math>. 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 "<math>0+</math>").
+One should compare this to the function notation (<math>f'</math>) vs Leibniz notation (<math>\frac{d}{dx} f(x)</math>) for derivatives.
+{| class="sortable wikitable"
+|-
+! Notation !! Symbols !! Strengths !! Weaknesses
+|-
+| Tao's notation || <math>\lim_{x\to x_0;\, x\in E} f(x)</math> || Works with anonymous functions. || Somewhat verbose: the "<math>x\in E</math>" part must be written out each time.
+|-
+| Function restriction notation || <math>\lim_{x\to x_0} f|_E(x)</math> || || Doesn't work with anonymous functions.
+|-
+| Standard notation || <math>\lim_{x\to x_0} f(x)</math> || Works with anonymous functions. ||
+|}