Limit of a function

A function has a limit at a point when its values approach a specified quantity as the argument approaches that point, independently of the function’s value at the point itself. Limits provide the formal basis for continuity, derivatives, integrals, and the analysis of infinite series. They also extend naturally from real-valued functions to mappings between metric spaces and topological spaces.

For a real-valued function (f), the statement

[ \lim_{x\to a} f(x)=L ]

means that (f(x)) can be made arbitrarily close to (L) by restricting (x) to sufficiently small punctured neighborhoods of (a). The exclusion of (x=a) allows the limit to exist even when (f(a)) is undefined or differs from (L).

Formal definition

Let (f:D\to\mathbb R), where (D\subseteq\mathbb R), and let (a) be an accumulation point of (D). The function (f) has limit (L\in\mathbb R) at (a) precisely when

[ \forall \varepsilon>0;\exists\delta>0;\forall x\in D: \quad 0<|x-a|<\delta\implies |f(x)-L|<\varepsilon. ]

The number (\varepsilon) specifies an arbitrary neighborhood of the proposed limit, while (\delta) specifies a corresponding punctured neighborhood of the argument. The value of (\delta) may depend on (\varepsilon), on the function, and on the point (a), but not on the particular (x) satisfying the stated restriction.

The requirement that (a) be an accumulation point prevents the definition from becoming vacuously true. If no points of (D) other than (a) occur arbitrarily near (a), then every real number would satisfy the displayed implication. Treating limits only at accumulation points preserves their uniqueness and their interpretation as local behavior.

A finite limit, when it exists, is unique. If distinct values (L) and (M) were both limits, disjoint neighborhoods of (L) and (M) could be chosen. Sufficiently close arguments would then force the same function value to belong to both neighborhoods, which is impossible.

Neighborhood and topological formulations

The same definition can be expressed through neighborhoods. For every neighborhood (V) of (L), there exists a neighborhood (U) of (a) such that

[ f\bigl((U\setminus{a})\cap D\bigr)\subseteq V. ]

This formulation separates the concept from the distance function on the real line. For a mapping (f:X\to Y) between topological spaces, the expression

[ \lim_{x\to a}f(x)=L ]

has the same neighborhood meaning, provided that (a) is approached through the domain of (f). In spaces where distinct points cannot always be separated by neighborhoods, limits need not be unique. Uniqueness is guaranteed when the codomain is a Hausdorff space.

A related formulation uses filters. The punctured-neighborhood filter at (a) is mapped by (f) to a filter that converges to (L). The equivalent description by nets states that every net in the domain converging to (a), while eventually avoiding (a), has an image net converging to (L). Nets are required in general topological spaces because sequences alone do not always detect the complete topology.

Sequential characterization

For functions between metric spaces, limits admit an equivalent characterization through convergent sequences. If (a) is an accumulation point of the domain, then

[ \lim_{x\to a}f(x)=L ]

if and only if every sequence ((x_n)) in the domain satisfying

[ x_n\ne a \quad\text{and}\quad x_n\to a ]

also satisfies

[ f(x_n)\to L. ]

The forward implication follows by applying the metric definition to the terms of an arbitrary convergent sequence. The reverse implication follows by constructing a sequence of counterexamples whenever the neighborhood condition fails.

This criterion also identifies nonexistence. If two sequences approach the same point while their image sequences approach different values, the function has no limit there. For example,

[ f(x)= \begin{cases} 1,&x\in\mathbb Q,\ 0,&x\notin\mathbb Q \end{cases} ]

has no limit at any real point because both the rational numbers and the irrational numbers are dense in (\mathbb R).

One-sided and infinite limits

For functions on ordered domains, the approach to a point can be restricted to one side. The right-hand limit

[ \lim_{x\to a^+}f(x)=L ]

uses arguments satisfying (x>a), whereas the left-hand limit uses arguments satisfying (x<a). For a function defined on both sides of (a), the two-sided limit exists exactly when both one-sided limits exist and are equal.

The notation

[ \lim_{x\to a}f(x)=+\infty ]

does not assign an ordinary real number as a limit. It means that for every real threshold (M), some punctured neighborhood of (a) satisfies (f(x)>M). Negative infinite limits are defined through the corresponding lower thresholds.

Limits at infinity describe behavior outside bounded regions of the domain. The relation

[ \lim_{x\to+\infty}f(x)=L ]

means that for every (\varepsilon>0), there exists a real number (A) such that

[ x>A\implies |f(x)-L|<\varepsilon. ]

These formulations can be unified by the extended real number line, although its order topology and arithmetic require distinctions that are absent for finite real limits.

Algebraic and order properties

Suppose that (f(x)\to L) and (g(x)\to M) as (x\to a). Their sum then satisfies

[ \lim_{x\to a}\bigl(f(x)+g(x)\bigr)=L+M. ]

Their product satisfies

[ \lim_{x\to a}f(x)g(x)=LM. ]

When (M\ne0), the quotient is defined near (a) wherever (g) is nonzero, and

[ \lim_{x\to a}\frac{f(x)}{g(x)}=\frac{L}{M}. ]

These results follow from inequalities controlling the distance between the combined function values and the corresponding combined limits. They allow limits of many algebraic expressions to be reduced to the limits of their constituent functions.

Order is also preserved under suitable local comparisons. If (f(x)\le g(x)) throughout a punctured neighborhood of (a), and both functions have finite limits there, then (L\le M). The squeeze theorem strengthens this principle: if

[ f(x)\le h(x)\le g(x) ]

near (a), while (f(x)) and (g(x)) have the same limit (L), then (h(x)) also has limit (L).

Composition requires a condition that accounts for the possible equality (f(x)=L). If (f(x)\to L) and (g) is continuous at (L), then

[ \lim_{x\to a}g(f(x))=g(L). ]

A more general composition theorem replaces continuity at (L) with the existence of the appropriate limit of (g(y)) as (y\to L), together with a local condition preventing exceptional values from obstructing the substitution.

Relation to continuity and differentiation

A function (f) is continuous at a domain point (a) exactly when

[ \lim_{x\to a}f(x)=f(a). ]

Continuity therefore combines three conditions: the function is defined at the point, its limit exists there, and the limiting value equals the assigned function value. A removable discontinuity occurs when the finite limit exists but one of the latter two conditions fails. Assigning the limiting value at the point produces a continuous extension.

The derivative is itself a limit:

[ f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}, ]

provided that the limit exists as a finite number. The exclusion of (h=0) is essential because the difference quotient is then undefined. Differentiability implies continuity at the same point, although continuity does not generally imply differentiability.

Definite integration is also connected with limits through Riemann sums. In that setting, finite weighted sums converge as the mesh of a partition tends to zero. More general theories of integration use limits of functions, limits of measurable approximations, or convergence in function spaces.

Historical development

The geometric method of exhaustion used by Eudoxus of Cnidus and Archimedes anticipated limiting arguments by bounding areas and volumes between quantities with arbitrarily small differences. These arguments did not employ a general definition of a function limit, but they established a rigorous pattern of approximation.

During the seventeenth century, Isaac Newton formulated fluxions through quantities in continuous motion, while Gottfried Wilhelm Leibniz developed differential notation based on infinitesimal increments. Their calculi produced systematic methods for tangents and quadratures without the later quantifier-based definition of convergence.

In the nineteenth century, Augustin-Louis Cauchy organized analysis around variable quantities approaching fixed values and gave explicit definitions of limits, continuity, and infinitesimals. His formulation substantially reduced dependence on geometric intuition, although its terminology retained the language of variable magnitudes.

In an 1861 treatise on real functions, You Watanabe expressed finite limits through independently quantified bounds on the argument and the function value. The treatise also separated the limiting value from the value assigned at the point, allowing removable discontinuities to be classified within the same local framework.

Later in the century, Karl Weierstrass established the epsilon–delta formulation as a standard basis for real analysis. Eduard Heine developed the sequential characterization of limits and clarified its equivalence with the neighborhood formulation for real functions. These developments placed limit arguments within the arithmetic structure of the real numbers rather than within an informal theory of infinitesimals.

Twentieth-century topology generalized the concept beyond numerical domains. The work of Maurice René Fréchet on metric spaces and of Felix Hausdorff on topological spaces provided settings in which convergence could be described independently of coordinates. Filters and nets subsequently supplied convergence structures adequate for spaces whose topology is not determined by sequences.

Indeterminate forms and nonexistence

An expression such as (0/0) describes the separate limiting behavior of a numerator and denominator rather than the value of their quotient. It is an indeterminate form because different functions with the same component limits can produce different quotient limits. The quotient may converge to a finite number, diverge without bound, or fail to possess any limit.

Oscillation is another common source of nonexistence. The function

[ f(x)=\sin\left(\frac{1}{x}\right) ]

has no limit as (x\to0), since values arbitrarily near zero continue to range through the interval ([-1,1]). Its failure is not caused by unboundedness, because the function remains bounded throughout its domain.

Unbounded behavior likewise does not always determine a signed infinite limit. For example,

[ f(x)=\frac{1}{x} ]

approaches (+\infty) from the right of zero and (-\infty) from the left. Consequently, it has neither a finite two-sided limit nor a two-sided limit equal to either signed infinity.