Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions 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.

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.

Notation

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

{\displaystyle f}

of

x

{\displaystyle x}

as

x

{\displaystyle x}

approaches

c

{\displaystyle c}

equals

L

{\displaystyle L}

". This means that the value of the function

f

{\displaystyle f}

can be made arbitrarily close to

L

{\displaystyle L}

, by choosing

x

{\displaystyle x}

sufficiently close to

c

{\displaystyle c}

. Alternatively, the fact that a function

f

{\displaystyle f}

approaches the limit

L

{\displaystyle L}

as

x

{\displaystyle x}

approaches

c

{\displaystyle 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,}

or in

f

(

x

)

→

x

→

c

L

,

{\displaystyle f(x){\xrightarrow[{x\to c}]{}}L,}

which reads "

f

{\displaystyle f}

of

x

{\displaystyle x}

tends to

L

{\displaystyle L}

as

x

{\displaystyle x}

tends to

c

{\displaystyle c}

".

History

According to Hankel (1871), the modern concept of limit originates from Proposition X.1 of Euclid's Elements, which forms the basis of the Method of exhaustion found in Euclid and Archimedes: "Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out."

Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."

In the Scholium to Principia in 1687, Isaac Newton had a clear definition of a limit, stating that "Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity".