Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value.^[1] Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

In formulas, a limit of a function is usually written as

${\displaystyle \lim _{x\to c}f(x)=L,}$

and is read as "the limit of f of x as x approaches c equals L". This means that the value of the function f can be made arbitrarily close to L, by choosing x sufficiently close to c. Alternatively, the fact that a function f approaches the limit L as x approaches c is sometimes denoted by a right arrow (→ or ${\displaystyle \rightarrow }$), as in

${\displaystyle f(x)\to L{\text{ as }}x\to c,}$

which reads "${\displaystyle f}$ of ${\displaystyle x}$ tends to ${\displaystyle L}$ as ${\displaystyle x}$ tends to ${\displaystyle c}$".

The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.

The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist.