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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Limit superior and limit inferior ： ウィキペディア英語版 Limit superior and limit inferior In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit. ==Definition for sequences== The limit inferior of a sequence (''x''''n'') is defined by :$\backslash liminf\_x\_n\; :=\; \backslash lim\_\backslash Big(\backslash inf\_x\_m\backslash Big)$ or :$\backslash liminf\_x\_n\; :=\; \backslash sup\_\backslash ,\backslash inf\_x\_m=\backslash sup\backslash .$ Similarly, the limit superior of (''x''''n'') is defined by :$\backslash limsup\_x\_n\; :=\; \backslash lim\_\backslash Big(\backslash sup\_x\_m\backslash Big)$ or :$\backslash limsup\_x\_n\; :=\; \backslash inf\_\backslash ,\backslash sup\_x\_m=\backslash inf\backslash .$ Alternatively, the notations $\backslash varliminf\_x\_n:=\backslash liminf\_x\_n$ and $\backslash varlimsup\_x\_n:=\backslash limsup\_x\_n$ are sometimes used. If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as real numbers or ±∞ (i.e., on the extended real number line). More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice. Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it; therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does ''not'' exist. Whenever lim inf ''x''''n'' and lim sup ''x''''n'' both exist, we have :$\backslash liminf\_x\_n\backslash leq\backslash limsup\_x\_n.$ Limits inferior/superior are related to big-O notation in that they bound a sequence only "in the limit"; the sequence may exceed the bound. However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e−''n'' may actually be less than all elements of the sequence. The only promise made is that some tail of the sequence can be bounded by the limit superior (inferior) plus (minus) an arbitrarily small positive constant. The limit superior and limit inferior of a sequence are a special case of those of a function (see below). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Limit superior and limit inferior」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.