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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

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 ), as in

which reads " of tends to as tends to ".

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.

**^**Stewart, James (2008).*Calculus: Early Transcendentals*(6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8.