How can we redefine limits to remove problems with the current definition?

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Discussion Overview

The discussion revolves around the redefinition of limits in calculus, specifically addressing perceived issues with the traditional definition. Participants explore the implications of the term "as close to" in the context of limits, the nature of variables involved, and the conditions under which limits can be assigned.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant proposes an alternate definition of limits aimed at removing ambiguity related to the phrase "as close to as possible," questioning its mathematical meaning.
  • Concerns are raised about the implications of undefined values, particularly in cases where limits approach indeterminate forms like 0/0.
  • The concept of "natural variables" is introduced, suggesting that certain variables should encompass all values within a specified range without skipping any values.
  • A mathematical formulation for the proposed limit definition is presented, which includes conditions for assigning values to limits based on the behavior of functions around a point.
  • Another participant challenges the initial claims, asserting that the traditional definition of limits does not include the term "as close as possible" and emphasizes the importance of precise definitions in textbooks.
  • Critiques are made regarding the proposed redefinition, questioning the clarity and validity of the new formulation and its implications for understanding limits.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity of the proposed redefinition of limits. Some argue that the traditional definition is sufficient and well-defined, while others support the need for a new approach to address perceived shortcomings.

Contextual Notes

Limitations in the discussion include potential misunderstandings of standard terminology and definitions, as well as the challenge of articulating new concepts clearly. The discussion reflects varying levels of familiarity with calculus concepts among participants.

  • #31
ato said:
why do you think L or a has to be mentaioned ?
OK, there are equivalences where a variable is mentioned in one of the statements but not the other, for example
$$x=1 \text{ and } x\neq 1 \Leftrightarrow 0\neq 0.$$ However, in the case of the two statements we're talking about, it couldn't possibly be more obvious that the two statements are not equivalent. Note for example that given a function f and a number a the truth value (true or false) of the first statement depends on the value of the variable L, but the truth value of the second statement does not.
 
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  • #32
Ato, you are creating definition in order to solve a problem that was solved a few centuries ago. Your statements may be correct in your head (or maybe not) but do not follow normal practices in logic.
Your statement
\lim_{x\rightarrow a}f(x)=L\iff f([x_1,x_2]) is a real interval for all x_1,x_2 on the function's domain is not the definition of a limit. Take the example of f(x)=\frac{\sin x}{x} which has a limit of 1 as x\rightarrow 0 but f([0,\pi]) is not an interval because f(0) is not defined. Perhaps the idea that you have in your mind is correct, but your notation is wrong.

You should stop trying to come up with a new way to say something before you understand the old way. Learn calculus before you attempt to rewrite it. Even if you are some genius who came up with a better way to define limits, you are not capable of speaking the language of mathematics to get your point across. Go back to your books (or get new ones) and learn how limits work. They do. The definition is good. It really doesn't make sense to discuss this with you until you understand the definition of a limit.

Also, http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/ would make my eyes hurt a little less.
 
  • #33
I wholeheartedly agree with everything that DrewD said. I just want to add that we have put together a guide for using LaTeX here in the forum. Link.
 
  • #34
This thread has gone on for long enough.
Ato, I would suggest you to read a good calculus book such as Spivak or Apostol. Limits are very well understood these days. And the definition we have right now works.

Please make some effort to understand our definition. If you do, then we might discuss things again.
 

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