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Limit of a sequence
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==Hyperreal numbers== The definition of the limit using the [[hyperreal numbers]] formalizes the intuition that for a "very large" value of the index, the corresponding term is "very close" to the limit. More precisely, a real sequence <math>(x_n)</math> tends to ''L'' if for every infinite [[hypernatural]] <math display="inline">H</math>, the term <math>x_H</math> is infinitely close to <math display="inline">L</math> (i.e., the difference <math>x_H - L</math> is [[infinitesimal]]). Equivalently, ''L'' is the [[Standard part function|standard part]] of <math>x_H</math>: :<math> L = {\rm st}(x_H)</math>. Thus, the limit can be defined by the formula :<math>\lim_{n \to \infty} x_n= {\rm st}(x_H)</math>. where the limit exists if and only if the righthand side is independent of the choice of an infinite ''<math display="inline">H</math>''.
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