piirmäärad

Piirmäärad are an Estonian mathematical term that translates to limits or bounds. In mathematics, piirmäärad describe the values a quantity approaches under certain conditions. They are central in calculus and analysis and help characterize the behavior of functions, sequences, or processes.

In calculus, the limit of a function f at a point a is the value L that f(x) approaches as x tends to a. If this value is the same from the left and from the right, the function is said to have a limit at a. One-sided limits, infinite limits, and limits at infinity are all examples of piirmäärad. For sequences, the limit lim n→∞ a_n is the value the sequence approaches if it exists. If a limit does not exist, the sequence may still be bounded or have accumulation points.

Piirmäärad also serve as bounds for sets. The supremum (least upper bound) and infimum (greatest lower bound) are standard examples of piirmäärad of a set, when they exist. In this sense, piirmäärad describe the restrictions on the range of possible values a quantity can take.

In applied contexts, piirmäärad appear as boundary values in boundary value problems, where the solution to a differential equation must take specified values on the boundary of the domain.

Examples illustrate the concept: the limit of sin x / x as x → 0 is 1; the sequence a_n = 1/n tends to 0; the function 1/x has an infinite limit as x → 0+.

See also: limit (mathematics), boundary value problem, supremum, infimum.