Limittyvät

Limittyvät is a term used in Finnish mathematical literature to describe objects that converge to a finite limit under a given process. The plural form limittyvät refers to several such objects, while the related adjective is limittyvä. In practice, limittyvät can denote sequences, functions, or more general nets that have a well-defined finite limit.

Definitionally, for a sequence (a_n), the sequence is limittyvä if lim_{n→∞} a_n exists and is finite. For a function f defined near a point a, f is limittyvä at a if lim_{x→a} f(x) exists and is finite. The concept aligns with standard convergence, but the term is sometimes used to emphasize the existence of a finite limit rather than other aspects of the convergence process.

Examples include the sequence a_n = 1/n, which is limittyvä with limit 0, and the function f(x) = x^2, which is limittyvä at 0 with limit 0. A non-example is a_n = (-1)^n, which does not converge and thus is not limittyvä.

Basic properties follow standard limit rules: if (a_n) converges to L and c is a scalar, then (c a_n) converges to cL; if (a_n) converges to L and (b_n) converges to M, then (a_n + b_n) converges to L + M. The concept of limittyvät is a convenient emphasis in analysis for discussing convergence, stability, and asymptotic behavior, though terminology usage can vary among authors.