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

Revision as of 03:07, 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.

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 $lim_{x \to 0} f |_{(0, \infty)} (x) = 1$ ."

On the other hand, I think usually one would write the above like $lim_{x \to 0 +} | x | / x = 1$ .

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	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 <math>f\|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} \|x\|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := \|x\|/x</math>. Then we have <math>\lim_{x\to 0} f\|_{(0,\infty)}(x) = 1</math>."		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 <math>f\|_E</math>, we must have named our function beforehand. To give an example, we can write something like <math>\lim_{x\to 0;\, x\in (0,\infty)} \|x\|/x = 1</math>, but this is difficult to write in the other notation; we would have to say something like, "Let <math>f:\mathbf R\setminus \{0\} \to \mathbf R</math> be defined by <math>f(x) := \|x\|/x</math>. Then we have <math>\lim_{x\to 0} f\|_{(0,\infty)}(x) = 1</math>."

			On the other hand, I think usually one would write the above like <math>\lim_{x\to 0+} \|x\|/x = 1</math>.