Limit As Standard Part
In the context of a hyperreal enlargement of the number system, the limit of a sequence can be expressed as the standard part of the value of the natural extension of the sequence at an infinite hypernatural index . Thus,
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Here the standard part function "st" associates to each finite hyperreal, the unique finite real infinitely close to it (i.e., the difference between them is infinitesimal). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal represented in the ultrapower construction by a Cauchy sequence, is simply the limit of that sequence:
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In this sense, taking the limit and taking the standard part are equivalent procedures.
Read more about this topic: Limit (mathematics)
Famous quotes containing the words limit, standard and/or part:
“No one who cannot limit himself has ever been able to write.”
—Nicolas Boileau-Despréaux (1636–1711)
“An indirect quotation we can usually expect to rate only as better or worse, more or less faithful, and we cannot even hope for a strict standard of more and less; what is involved is evaluation, relative to special purposes, of an essentially dramatic act.”
—Willard Van Orman Quine (b. 1908)
“I could be worse employed
Than as watcher of the void,
Whose part should be to tell
What star if any fell.”
—Robert Frost (1874–1963)