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Talk:Limit (mathematics) (view source)
Revision as of 03:24, August 20, 2008
, 03:24, August 20, 2008→Function vs. Sequence: Order put back, but I still like it the other way.
I'd like to object. The idea of the limit of a sequence is the more basic one - and it's taught first, I suppose. Generally, you introduce the limit of a function in a point via the limit of sequences: What does it mean (in your example) that <math>x \rightarrow 3</math>? It means, you look at all sequences <math>(x_1, x_2,...)</math> with <math>\lim x_n = x</math>. And if for all these sequences, the corresponding sequence <math>(f(x_1),f(x_2), ...)</math> converges to the same value <math>z</math>, you'd say that the function has limit in x.
I'd like to object. The idea of the limit of a sequence is the more basic one - and it's taught first, I suppose. Generally, you introduce the limit of a function in a point via the limit of sequences: What does it mean (in your example) that <math>x \rightarrow 3</math>? It means, you look at all sequences <math>(x_1, x_2,...)</math> with <math>\lim x_n = x</math>. And if for all these sequences, the corresponding sequence <math>(f(x_1),f(x_2), ...)</math> converges to the same value <math>z</math>, you'd say that the function has limit in x.
But without this underlying concept, the expression <math> x \rightarrow 3</math> is unmotivated and baseless. --[[User:DiEb|DiEb]] 18:11, 17 August 2008 (EDT)
But without this underlying concept, the expression <math> x \rightarrow 3</math> is unmotivated and baseless. --[[User:DiEb|DiEb]] 18:11, 17 August 2008 (EDT)
:I have put the order back the way you want it, but I hope to convince you that putting sequences later is the right thing. That is, do ordinary functions first, functions going to infinity at finite X second, and functions going to finite Y for infinite X third. This establishes the notion of "for every epsilon there is an M such that x > M get f(x) within epsilon". The sequence version of a limit is just a simple variation of that notion.
:By the way, greetings. You and I seem to be the only math/science people left. Well, I notice that Lemonpeel has just come back, but things don't seem to be going well for him, and I suspect he will be out soon. My attempts to get the natural log page and the radioactivity page restored have met with no success at all, and I can't keep asking, since I'm on extremely thin ice. [[User:SamHB|SamHB]] 23:24, 19 August 2008 (EDT)