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 in general) 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

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 (→), as in

Suppose f is a real-valued function and c is a real number. Intuitively speaking, the expression

means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, the above equation can be read as "the limit of f of x, as x approaches c, is L".

Augustin-Louis Cauchy in 1821,^[2] followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit. The definition uses ε (the lowercase Greek letter epsilon) to represent any small positive number, so that "f(x) becomes arbitrarily close to L" means that f(x) eventually lies in the interval (L − ε, L + ε), which can also be written using the absolute value sign as |f(x) − L| < ε.^[2] The phrase "as x approaches c" then indicates that we refer to values of x whose distance from c is less than some positive number δ (the lower case Greek letter delta)—that is, values of x within either (c − δ, c) or (c, c + δ), which can be expressed with 0 < |x − c| < δ. The first inequality means that the distance between x and c is greater than 0 and that x ≠ c, while the second indicates that x is within distance δ of c.^[2]

The above definition of a limit is true even if f(c) ≠ L. Indeed, the function f need not even be defined at c.

For example, if

then f(1) is not defined (see indeterminate forms), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2:

Thus, f(x) can be made arbitrarily close to the limit of 2 just by making x sufficiently close to 1.

In addition to limits at finite values, functions can also have limits at infinity. For example, consider

As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish just by picking x sufficiently large. In this case, the limit of f(x) as x approaches infinity is 2. In mathematical notation,

Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are "approaching" 1.8, the limit of the sequence.

Formally, suppose a1, a2, ... is a sequence of real numbers. It can be stated that the real number L is the limit of this sequence, namely:

which is read as

to mean

Intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the absolute value |a**n − L| is the distance between a**n and L. Not every sequence has a limit; if it does, it is called convergent, and if it does not, it is divergent. One can show that a convergent sequence has only one limit.

The limit of a sequence and the limit of a function are closely related. On the one hand, the limit as n approaches infinity of a sequence {a**n} is simply the limit at infinity of a function a(n) defined on the natural numbers {n}. On the other hand, if X is the domain of a function f(x) and if the limit as n approaches infinity of f(x**n) is L for every arbitrary sequence of points {x**n} in {X – {x0} } which converges to x0, then the limit of the function f(x) as x approaches x0 is L.^[3] One such sequence would be {x0 + 1/n}.

In this sense, taking the limit and taking the standard part are equivalent procedures.

^[4]

Limits can be difficult to compute. There exists limit expressions whose modulus of convergence is undecidable. In recursion theory, the Limit lemma proves that it is possible to encode undecidable problems using limits.^[5]

Sequences in topological spaces can be generalized to the concepts of nets, along with a corresponding notion of a limit. A related notion, that of filters on topological spaces, also can have limits defined for them.