In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. 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

lim

x

→

c

f

(

x

)

=

L

,

{\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

f

(

x

)

→

L

as

x

→

c

,

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

which reads "

f

{\displaystyle f}

of

x

{\displaystyle x}

tends to

L

{\displaystyle L}

as

x

{\displaystyle x}

tends to

c

{\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.

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