- #36
ahrkron
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Originally posted by NeutronStar
We did many series convergence problems.
I disagreed with those conclusions. I mean from a static or absolute point of view.
Mathematicians use the same reasoning capabilities as you and I. Theorems are proven using such logic. "Points of view" (static, absolute or otherwise) need not enter in the discussion.
Well, if you learn that definition well, you can clearly see that mathematicians use it incorrectly all the time.
Either that, or you misunderstood the use they make of it.
Do you seriously sustain that all mathematicians since the formalization of calculus have used a definition incorrectly? I find that extremely hard to believe.
But it is not a matter of faith. I have gone over the resoning myself, as do quite a few high schools students (and college students and grad students and profesional mathematicians) every year, and find the to be quite clean.
The definition for the limit of f(x) at c clearly states: ...
Yet mathematicians never fail to claim that calculus can prove something about when epsilon equals zero
There's no such "claim". The definition of limit allows you to consider situations in which:
1. the case epsilon=0 is not relevant for the discussion at hand,
2. the value at epsilon=0 is not defined
3. the value at epsilon=0 is different from the limit.
4. such value is defined and is equal to the limit.
When studying the behavior of a series, it is possible to use these tools. No dogmas or errors are involved; rather, further concepts are founded upon these studies.
calculus cannot make any statements whatsoever about what might happen should epsilon actually become zero. Such statements are outside of the scope of calculus. [/B]
No, they're not. Look up the definition of continuity. It is one of the basic concepts in calculus and it is concerned with the case epsilon=0. It is not an invalid extrapolation, but a case that has to be considered due to its enormously frequent occurrence on the formalism.