limN

limN is a shorthand used in some texts to denote the limit of a sequence as the index N tends to infinity. The standard notation is lim_{N→∞} a_N, indicating the limit L if the sequence a_N approaches L as N grows without bound.

Formal definition: A sequence {a_N} converges to L if for every ε > 0 there exists an N0 such that for all N ≥ N0, |a_N − L| < ε. If such an L exists, the limit is L; otherwise the sequence diverges.

Examples: a_N = 1/N has limit 0. a_N = (−1)^N has no limit because the sequence oscillates between −1 and 1. a_N = log N / N has limit 0.

Notes: In rigorous writing, lim_{N→∞} is preferred; limN without an arrow is informal or used in certain contexts. Limits are fundamental in defining convergence and in establishing asymptotic behavior in analysis, number theory, and probability. They underpin constants defined by limiting procedures and are central to many theorems about sums, integrals, and series.

Related concepts: limsup and liminf describe the limiting superior and inferior values when a sequence does not converge. Interchange of limits with sums or integrals can be subtle and is governed by convergence theorems such as dominated convergence or monotone convergence.

See also: Notation in calculus, convergence of sequences, asymptotic notation, and limit theorems.