These words have been pulled directly from Wikipedia, although I find the exact logical construction in my textbooks:(adsbygoogle = window.adsbygoogle || []).push({});

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The (ε, δ)-definition of the limit of a function is as follows:

Let ƒ be a function defined on an open interval containing c (except possibly at c) and let L be a real number. Then the formula:

[tex]\lim_{x \to c} f(x)=L[/tex]

means for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |ƒ(x) − L| < ε.

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My problem is that this seems to be an insufficient definition. A corresponding δ can be found for every positive value of ε even if δ is increasing as ε is decreasing. The inequalities can be satisfied for any limit L.

Say f(x) = 5x, and we claim the limit as x->3 is 20.

For every positive ε in |5x − 20| < ε, I can find a corresponding 0 < |x − 3| < δ, although in this case x will go to 4 in order to satisfy small ε.

My question is: should an added clause exist such that as ε tends to 0 so must δ, or that the product of their derivatives with respect to x must be greater than or equal to zero (so they both increase or decrease simultaneously), or is it implicitly assumed that one is choosing x closer to c. If it is the last case, it seems it would be more precise to explicitly mention this fact, as then there can only be one L.

I understand this question might be borderline, or full-line, pedantic, but I think we all understand the necessity for precision in mathematics and logic, and I'm concerned as to whether I am in fact missing a subtle nuance--say a subtle nuance that would indeed make the limit L unique despite what originally seems insufficient constraints.

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# Rigorous definition of a limit

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