# 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.

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.

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

(although a few authors may use "Lt" instead of "lim"[2])
and is read as "the limit of *f* of x as x approaches c equals *L*". The fact that a function *f* approaches the limit *L* as x approaches c is sometimes denoted by a right arrow (→ or ), as in

which reads " of tends to as tends to ".