Difference between revisions of "Talk:Limit"

Revision as of 07:05, 7 May 2008

AoPSWiki Article of the Day
Limit was the AoPSWiki Article of the Day for January 6th, 2008

Hello. I hastily created this article because I noticed we didn't have one on limits. I plan to add things such as proofs for uniqueness, multiplication, and addition, as well as stuff about continuity, left- and right-hand limits, infinite limits, continuous functions, etc., etc., etc. Please contribute if you'd like.

The article looks great. By the way, when you add something to a "Talk" page, add a "signature with time stamp" so everyone knows who said that. The code is two dashes (-) followed by four tildens (~).

--Xantos C. Guin 21:15, 29 June 2006 (EDT)

Oh, sorry; I thought everybody else could see it. That was me, by the way. Also, sorry about that stupid mixing-up of $\delta$ s and $c$ s. --~~

Whoops; I thought you had written "two tildes." All of the anonymous comments above were written by me. --Boy Soprano II 17:44, 30 June 2006 (EDT)

The "Existence of Limits" section needs work. (In particular, I think $f(x) = \frac{1}{x}$ is a bad example to start with -- $f(x) = \lfloor x \rfloor$ would be better.) Also, there is no mention in this article of infinite limits or limits to infinity, both of which are important. --JBL 14:41, 7 January 2008 (EST)

Regarding the limit of the function $f(x)=0$ for $x \ne 0$ and $f(x)=1$ for $x=0$ .

This function doesn't have a limit at x=0. Proof: Pick $\epsilon = 1/2$ . There is no $\delta>0$ such that $|x-0|<\delta \implies |f(x)-0| < 1/2$ . For any $\delta$ that you pick I can pick $x=0$ and $|f(0)-0|>1/2$ . Jep 00:15, 7 May 2008 (UTC)

Your last two sentences are correct, but they have little to do with the first one. For any $\epsilon, \delta >0$ the statement $0 < |x-0| < \delta \implies |f(x)-0| < \epsilon$ is true. Hence $\lim_{x\to 0} f(x) = 0$ . Do you see the difference between this (correct) criterion and yours?

On a different note, I think that this article should not be so rooted in real analysis. There is no discussion of the general, broad idea of "limit", just one narrow application and some details surrounding it. This is unfortunately, particularly as many of the details and adaptations of the reals obscure the main points about limits. Somebody (I?) should probably re-write most of the article and give it a different structure. —Boy Soprano II 02:53, 7 May 2008 (UTC)

I agree that the concept of limit is sufficiently important that it deserves more discussion.

Regarding the example, note that the definition of limit in the main text is different than the one you used above. Definition from main text.

We say that $\lim_{x\rightarrow c}f(x)=L$ iff

$\forall\epsilon>0\;\;\;\exists\delta>0$ such that

$|x-c|<\delta\implies|f(x)-L|<\epsilon$

The difference is that in the main text you have $|x-0|<\delta\implies|f(x)-0|<\epsilon$ (I replaced c and L with 0's) and above you used $0 < |x-0| < \delta \implies |f(x)-0| < \epsilon$ . Unless I really don't understand limit, the zero on the left hand side of the inequality is crucial, no?

@@ Line 25: / Line 25: @@
 On a different note, I think that this article should not be so rooted in real analysis.  There is no discussion of the general, broad idea of "limit", just one narrow application and some details surrounding it.  This is unfortunately, particularly as many of the details and adaptations of the reals obscure the main points about limits.  Somebody (I?) should probably re-write most of the article and give it a different structure.  &mdash;[[User:Boy Soprano II|Boy Soprano II]] 02:53, 7 May 2008 (UTC)
+I agree that the concept of limit is sufficiently important that it deserves more discussion.
+Regarding the example, note that the definition of limit in the main text is different than the one you used above. Definition from main text.
+We say that <math>\lim_{x\rightarrow c}f(x)=L</math> iff
+<math>\forall\epsilon>0\;\;\;\exists\delta>0</math> such that
+<math>|x-c|<\delta\implies|f(x)-L|<\epsilon</math>
+The difference is that in the main text you have <cmath>|x-0|<\delta\implies|f(x)-0|<\epsilon</cmath> (I replaced c and L with 0's) and above you used <cmath> 0 < |x-0| < \delta \implies |f(x)-0| < \epsilon </cmath>. Unless I really don't understand limit, the zero on the left hand side of the inequality is crucial, no?

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Difference between revisions of "Talk:Limit"

Revision as of 07:05, 7 May 2008