Limitability

Limitability is the property of a mathematical object or process that it admits a well-defined limit under a given limiting process. The term is used informally in analysis and related fields to describe whether a limit exists and is unique, given a particular notion of approaching a value or state.

In mathematics, limitability appears in several contexts. For sequences, a_n is limitable with limit L if lim_{n→∞} a_n = L. For functions, a(x) is limitable at a if lim_{x→a} a(x) exists (via the epsilon–delta definition in real analysis). For series, the partial sums S_N converge to a limit; the series is limitable with limit S. In topology, one speaks of limits of nets or filters; limitability depends on the space and the convergence structure chosen.

Common tests and phenomena relate to limitability. Convergence criteria include the Cauchy criterion and monotone convergence. Examples include the geometric series with ratio |r| < 1 having a finite limit, or the limit lim_{n→∞} 1/n = 0. Interchanging limits and other operations is governed by results such as uniform convergence or the dominated convergence theorem, which affect whether a limit can be passed through sums or integrals.

Outside pure mathematics, limitability can describe termination of algorithms or the long-run behavior of dynamical systems. It is also used informally to discuss whether a process yields a computable, bounded, or otherwise well-defined limit. The term is not universally standardized and may vary in emphasis across disciplines.