How do epsilon-delta proofs prove the limit?

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Discussion Overview

The discussion revolves around the understanding of epsilon-delta proofs in the context of limits in calculus. Participants explore how these proofs establish the validity of a limit and the relationship between epsilon (ε) and delta (δ) in this framework.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how the proof demonstrates that the limit is 35, expressing confusion about the role of epsilon in the argument.
  • Another participant clarifies that ε can be any positive number and that continuity is established if a corresponding δ can be found for any ε.
  • There is a discussion about the relationship between δ and ε, with some participants suggesting that δ can be defined as a constant multiple of ε.
  • Concerns are raised about whether δ being a multiple of ε is sufficient to prove the limit, with some participants seeking further clarification on this point.
  • One participant notes that δ can sometimes be defined in more complex ways, such as being proportional to ε squared.
  • A suggestion is made to study additional resources for a better understanding of epsilon-delta proofs, indicating the importance of grasping this concept for further mathematical analysis.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and agreement regarding the epsilon-delta argument, with some finding it clear and others still confused about its implications and proof structure. The discussion remains unresolved regarding the sufficiency of δ being a multiple of ε as a proof method.

Contextual Notes

Some participants highlight the need for a deeper understanding of the definitions and relationships between ε and δ, indicating that the discussion may depend on specific interpretations of these terms and their application in proofs.

Shaybay92
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I would really appreciate if someone could please explain to me how the final result of the proof of a limit actually PROVES the limit? http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preclimsoldirectory/PrecLimSol.html

I was particularly looking at Solution 2 on this website. I understand how |x-10| was factorized out etc. and therefore it can equal delta... but when it says that if delta = epsilon/3, |x-10| < epsilon/3 and therefore |f(x)-35| is less than epsilon... how is this true? I mean we don't even know what epsilon is?

how does this even prove that the limit is 35
 
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I understand how |x-10| was factorized out etc. and therefore it can equal delta

Does it?

Do you understand the epsilon-delta argument. It is a two part formal method much used in mathematical analysis. Would this be your difficulty?
 
Hi Shaybay92! :smile:

(have a delta: δ and an epsilon: ε :wink:)
Shaybay92 said:
… I mean we don't even know what epsilon is?

ε is anything (> 0).

Continuity is proved if, for any ε, you can find a δ (which depends on that ε) which works. :wink:
 
ok but how does this proof tell us that there is a delta for any epsilon? i don't really see how if delta = cepsilon (c is some constant) then that is a proof...
 
Hi Shaybay92! :smile:

(what happened to that δ and ε i gave you? :wink:)
Shaybay92 said:
ok but how does this proof tell us that there is a delta for any epsilon? i don't really see how if delta = cepsilon (c is some constant) then that is a proof...

because, if δ is defined as cε, then for any ε, there is a δ ! :smile:
 
so as long as x is less than a particular multiple of ε (cε) then |f(x)-L| < ε?
 
Shaybay92 said:
so as long as x is less than a particular multiple of ε (cε) then |f(x)-L| < ε?

That's right. :smile:

(Warning: although δ is usually a multiple of ε, somtimes it's something more awkward, like ε2 :wink:)
 

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