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:
“Berowne they call him, but a merrier man,
Within the limit of becoming mirth,
I never spent an hour’s talk withal.”
—William Shakespeare (1564–1616)
“A dwarf who brings a standard along with him to measure his own size—take my word, is a dwarf in more articles than one.”
—Laurence Sterne (1713–1768)
“If you love music, hear it; go to operas, concerts and pay fiddlers to play to you; but I insist on your neither piping nor fiddling yourself. It puts a gentleman in a very frivolous, contemptible light.... Few things would mortify me more than to see you bearing a part in a concert, with a fiddle under your chin, or a pipe in your mouth.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)