8 Nov 2008

Dynamic and Static Definitions for the Limit

by Corry Shores
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A limit may be considered dynamically. We might consider something whose magnitude is in process of diminishing towards a zero-point. After reaching its destination it will equal zero, that is, it will have the being of zero. But immediately before that it was becoming-zero, which is to say, it was in the process of becoming zero without yet attaining the being of zero. We can consider it as becoming-zero, because we are treating the diminishment dynamically, so we can think of something becoming something else, without yet being that something else. This moment of becoming-zero is the limit. There is no longer an extensive magnitude between the becoming-zero-value and the zero-point, yet they are not equal to each other either. The distance between them is infinitesimally small. The derivative at the limit thus is not obtained by extensive magnitudes but rather only by intensive ones.

[Formal definition: “In considering the successive values of the difference quotient Δs/Δt, mathematics may continue indefinitely to make the intervals as small as it pleases. In this way an infinite sequence of values, r1, r2, r3, . . . rn, (the successive values of the ratio Δs/Δt) is obtained. This sequence may be such that the smaller the intervals, the nearer the ratio rn will approach to some fixed value L, and such that by taking the value of n to be sufficiently large, the difference |Lrn| can be made arbitrarily small. If this be the case, this value L is said to be the limit of the infinite sequence, or the derivative f’(t) of the distance function f(t), or the instantaneous velocity of the body. It is to be borne in mind, however, that this is not a velocity in the ordinary sense and has no counterpart in the world of nature, in which there can be no motion without a change of position. The instantaneous velocity as thus defined is not the division of a time interval into a distance interval, howsoever much the conventional notation ds/dt = f’(t) may suggest a ratio” (Boyer 7c.d).]

The limit may also be considered statically. Consider a series of values, each one smaller than its predecessor, continuing toward zero. Between zero and the sequence's smallest possible finite value there is a magnitude of difference that is smaller than any finite value, and the limit lies in this infinitesimally small magnitude. The limit, then, is the non-extensive 'region' of magnitude between zero and the smallest possible finite value. Thus, the derivative at the limit would be made of intensive and not extensive magnitudes.

[Formal definition: “L is said to be the limit of the above sequence if, given any positive number ε (however small), a positive integer N can be found such that for n > N the inequality |Lrn| < ε is satisfied” (8b).]

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.


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