limitswhether

Limitswhether is a term used in discussions of calculus and formal analysis to denote a predicate or function that decides whether the limit of a given function exists as the variable approaches a specified point (or infinity). It is not part of the standard mathematical vocabulary, but it appears in computational contexts and in expository writing as a concise way to ask the question: does the limit exist?

Definition and variants: For a function f defined on a punctured neighborhood of a, the predicate limitswhether(f, a) is true if the limit lim_{x→a} f(x) exists. This notion can be extended to one-sided limits or to limits at infinity. In cases where a limit exists only from one side or does not converge, limitswhether is false or may be defined to reflect the relevant notion (two-sided, one-sided, or extended limits).

Computation and use: Conceptually, limitswhether can be established by applying the Cauchy criterion, showing that left and right limits agree, or using direct algebraic simplification or known theorems. In practice, algorithmic approaches in symbolic computation attempt to determine the existence of a limit and return true or false. Some questions about limit existence are not decidable in full generality, especially for arbitrary functions described by procedures rather than explicit formulas.

Examples: limitswhether((x^2-1)/(x-1), a=1) is true, with the limit equal to 2. limitswhether(sin(1/x), a=0) is false, since the limit does not exist due to oscillation.

See also: limit, one-sided limit, limit at infinity, oscillation, symbolic computation.