Limit (mathematics)

Value that a function or sequence "approaches" as the input or index approaches some value

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.

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.[3] 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,[4] 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 (Lε, L + ε), which can also be written using the absolute value as |f(x) − L| < ε.[4] 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 lowercase Greek letter delta)—that is, values of x within either (cδ, c) or (c, c + δ), which can be expressed with 0 < |xc| < δ. The first inequality means that xc, while the second indicates that x is within distance δ of c.[4]

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

then f(1) is not defined (see Indeterminate form), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2:[5]

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

Now, since x + 1 is continuous in x at 1, we can now plug in 1 for x, leading to the equation

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

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—by making x sufficiently large. So in this case, the limit of f(x) as x approaches infinity is 2, or 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. One can state that the real number L is the limit of this sequence, namely:

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

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

Limits can be difficult to compute. There exist 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.[9]

This audio file was created from a revision of this article dated 6 June 2021 (), and does not reflect subsequent edits.