Not every sequence has a limit; if it does, it is called convergent, and if it does not, it is divergent.If positive constants displaystyle lambda and displaystyle alpha exist with limnpn1ppnpdisplaystyle lim _nrightarrow infty frac lambda then pndisplaystyle p_n as ndisplaystyle n goes from 0displaystyle 0 to displaystyle infty converges to pdisplaystyle p of order displaystyle alpha, with asymptotic error constant displaystyle lambda Given.

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The limit of a sequence and the limit of a function are closely related.On one hand, the limit as n goes to infinity of a sequence a ( n ) is simply the limit at infinity of a function defined on the natural numbers.Conversely, the standard part of a hyperreal aandisplaystyle aa_n represented in the ultrapower construction by a Cauchy sequence (an)displaystyle (a_n), is simply the limit of that sequence: st(a)limnandisplaystyle operatorname st (a)lim _nto infty a_n.

Limit of a sequence edit Main article: Limit of a sequence Consider the following sequence:.79,.799,.7999.Contents Limit of a function edit Main article: Limit of a function Suppose f is a real-valued function and c is a real number.Nets and filters edit This section needs expansion.