limitsas

Limitsas is a term used in some mathematical circles to denote the operation of taking a limit of a function as its input approaches a specified value. The term is not standard, but appears in pedagogical discussions and online references as a label for the limiting process rather than a distinct operator.

Definition and notation: In standard real analysis, the limit of f(x) as x approaches a is written lim_{x->a} f(x) = L. When spoken or used informally as limitsas, the idea is the same: as x gets arbitrarily close to a, f(x) approaches L. In a formal formulation, lim_{x->a} f(x) = L means that for every open neighborhood U of L there exists a neighborhood V of a such that x in V and x ≠ a implies f(x) in U. This formalism can be extended to sequences, nets, and filters.

Existence and properties: The limit, when it exists, is unique. If f is continuous at a and a lies in the domain of f, then lim_{x->a} f(x) = f(a). If the limit does not exist, the limitsas is undefined.

Examples: The function f(x) = (x^2 - 1)/(x - 1) for x ≠ 1 has limit 2 as x approaches 1. The function sin x / x has limit 1 as x approaches 0.

Usage and reception: Limitsas is largely a pedagogical label and is not widely adopted in rigorous texts. It may appear in introductory discussions or in forums to emphasize the behavior of a function near a point rather than to introduce new notation.

See also: limit, limit superior, limit inferior, convergence, epsilon-delta definition, L'Hôpital's rule.