limiten

Limiten, in mathematics, is the plural form of Limite in German. The term refers to limits, i.e., the values that functions or sequences approach under certain conditions. In English-language mathematics, the corresponding term is “limits.”

In calculus and analysis, a limit describes the value that a function f(x) approaches as x approaches a, or that a sequence (a_n) approaches as n grows without bound. Formally, lim_{x→a} f(x) = L means that for every ε > 0 there exists δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε. For a sequence, lim_{n→∞} a_n = L means that for every ε > 0 there exists N such that n ≥ N implies |a_n − L| < ε.

Limits can be finite or infinite. A limit is infinite if |f(x)| grows without bound near a or near infinity. Limits may be approached from one side, denoted lim_{x→a^+} and lim_{x→a^-}.

Examples include lim_{x→0} (sin x)/x = 1, lim_{n→∞} (1 + 1/n)^n = e, and lim_{x→∞} 1/x = 0.

Limit laws describe how limits behave under arithmetic operations: the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and the quotient limit holds when the denominator’s limit is nonzero; and similar rules apply to compositions in suitable cases. A function can have a limit at a point while its value there is different or undefined; this underpins the notion of continuity.

Historically, the rigorous use of limits developed in the 19th century, with contributions from Cauchy and Weierstrass, transforming intuitive ideas into precise ε-δ formulations. Limiten thus anchor much of real analysis and its applications.