Ótakmarkaður

Ótakmarkaður is Icelandic for unbounded or unlimited. The term is formed with the prefix ó- (negation) and takmarkaður (bounded). It is used primarily in mathematics and logic to describe objects that do not have finite bounds, but it also appears in everyday language to denote unlimited extent or capacity, such as ótakmarkaður aðgangur or ótakmarkaður fjöldi.

In mathematical usage, a set is ótakmarkaður if there is no real number M that bounds all its elements by absolute value; equivalently, for every M > 0 there exists an element x in the set with |x| > M. A function is ótakmarkaður on a domain if, for every M > 0, there exists a point in the domain where the function’s absolute value exceeds M. A sequence is ótakmarkaður if its terms can be made arbitrarily large in magnitude.

Examples illustrate the concept. The intervals (0, ∞) and (−∞, 0) are unbounded, each unbounded in one direction. The sequence a_n = n is unbounded above, and the function f(x) = x^2 is unbounded as |x| grows. By contrast, a set that is contained within a finite interval, such as [−1, 1], is takmörkuð (bounded).

Ótakmarkaður thus describes a fundamental property of magnitude or extent in mathematical contexts, while in ordinary language it conveys the idea of no practical limits or constraints.

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