Nearlimit

Nearlimit is a term used informally in mathematics and related disciplines to denote a value that is approached by a sequence or function values but is not necessarily the limit of the entire sequence. In formal terms, a real number L is a near-limit (often called a limit point or accumulation point in standard terminology) of a sequence (a_n) if there exists a subsequence (a_{n_k}) that converges to L. The full sequence may fail to converge or may converge to a different value.

Properties: Near-limits depend only on accumulation behavior, not on the whole-sequence limit. The set of all near-limits of a sequence is exactly the set of its accumulation points. If the sequence converges, the unique near-limit equals the limit, and there are no other near-limits.

Examples: 1) a_n = (-1)^n has near-limits -1 and 1. 2) a_n = sin(n) has a full interval of near-limits equal to [-1, 1], since the set of accumulation points of sin(n) is the entire interval.

Origins and usage: The term near-limit is used in some teaching contexts to help distinguish convergence from accumulation behavior. It is not a standard term in most mathematical references; scholars usually say "accumulation point" or "limit point" and "lim sup/lim inf" for bounds.

See also: Limit, Accumulation point, Cluster point, Limit point.