In mathematics, the limit of a sequence is the value that the terms of an infinite sequence tend toward as the index of the sequence increases toward infinity. If a sequence possesses a limit, it is characterized as convergent; if it does not approach a specific value, it is described as divergent. The concept of a limit is a fundamental pillar of calculus and mathematical analysis.
Formally, a sequence of real numbers $(a_n)$ is said to converge to a limit $L$ if, for every positive real number $\epsilon$ (epsilon), there exists a corresponding natural number $N$ such that for all $n > N$, the absolute difference $|a_n - L|$ is less than $\epsilon$. This relationship is conventionally expressed using the notation $\lim_{n \to \infty} a_n = L$. This definition implies that for any arbitrarily small distance around the limit, there is a point in the sequence beyond which all subsequent terms remain within that distance.
The limit of a convergent sequence is unique; a sequence cannot converge to two distinct values. Mathematical operations such as addition, subtraction, multiplication, and division can be applied to limits of sequences, provided that the individual limits exist and, in the case of division, the limit of the divisor is non-zero. These properties are often referred to as the algebraic limit theorems.
While the concept is most commonly applied to sequences of real or complex numbers, it extends to more general mathematical structures. In a metric space, convergence is defined using the distance function (metric) of that space. In a topological space, a sequence $(x_n)$ converges to $x$ if every neighborhood of $x$ contains all but a finite number of terms of the sequence. The study of limits of sequences provides the necessary framework for defining other core concepts, such as the sum of an infinite series, the continuity of functions, and the derivative.