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,
- .
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:
- .
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)
“The urge for Chinese food is always unpredictable: famous for no occasion, standard fare for no holiday, and the constant as to demand is either whim, the needy plebiscite of instantly famished drunks, or pregnancy.”
—Alexander Theroux (b. 1940)
“Without our suffering, our work would just be social work, very good and helpful, but it would not be the work of Jesus Christ, not part of the Redemption.... All the desolation of the poor people, not only their material poverty, but their spiritual destitution, must be redeemed. And we must share it, for only by being one with them can we redeem them by bringing God into their lives and bringing them to God.”
—Mother Teresa (b. 1910)