limitsis

Limitsis is a hypothetical mathematical construct proposed to unify various notions of convergence by collecting the limits produced by a family of summability methods. The term emphasizes that, depending on the method chosen, a sequence may be associated with one or several limiting values rather than a single limit.

Formally, let x = (x_n) be a sequence in a topological space, and let S be a specified family of summability methods (such as Cesàro, Abel, or Banach limits). The limitsis of x with respect to S, denoted L_S(x), is the set of all possible limit values lim_s x under methods s in S for which the transformed sequence converges. This makes L_S(x) a set rather than a single point in general.

Standard convergence yields a singleton L_S(x) = {L}. Oscillatory or divergent sequences may produce larger sets; for example, a sequence like (-1)^n can have different limit values under different methods, and a smoothing method may yield intermediate values such as 0.

Limitsis relates to traditional limits, limsup, liminf, and various summability methods, and it serves as a framework to analyze convergence behavior when no single limit exists. It is not unique and depends on the chosen family S; it is primarily a conceptual tool rather than a universally adopted operator.

In speculative or fictional mathematical literature, limitsis is discussed as a way to compare convergence notions and to guide approaches in numerical methods that must accommodate nonunique limits. See also: limit, limsup, liminf, Cesàro sum, Banach limit, summability.