# Limit (mathematics)

In mathematics, a limit is the value that a function or sequence approaches as its input or index approaches some value: written \( \lim_{x \to c} f(x) = L \), it means that the outputs of \( f \) can be made arbitrarily close to \( L \) by taking inputs sufficiently close to \( c \). The limit is the foundational concept of mathematical analysis, underlying continuity, derivatives, integrals, and convergence.<sup>[1](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)</sup>

| Key fact | Detail |
|---|---|
| Formal definition | \( \lim_{x \to c} f(x) = L \) means: given any \( \epsilon > 0 \), there exists \( \delta > 0 \) such that for all \( x \neq c \), if \( |x - c| < \delta \) then \( |f(x) - L| < \epsilon \)<sup>[1](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)</sup> |
| Order of quantifiers | The \( y \)-tolerance \( \epsilon \) is given first; the limit exists if a working \( x \)-tolerance \( \delta \) can then be found<sup>[1](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)</sup> |
| Sequence form | \( x_n \to x \) if for any \( \epsilon > 0 \) there is a positive integer \( N(\epsilon) \) with \( |x_n - x| < \epsilon \) whenever \( n > N(\epsilon) \)<sup>[2](https://persson.berkeley.edu/math128a/slides/ch1math128a.html)</sup> |
| Role in calculus | Continuity at \( x_0 \) means \( \lim_{x \to x_0} f(x) = f(x_0) \); the derivative is the limit of the difference quotient \( \frac{f(x)-f(x_0)}{x-x_0} \)<sup>[2](https://persson.berkeley.edu/math128a/slides/ch1math128a.html)</sup> |
| Historical origin | Cauchy introduced the \( \delta \)-\( \epsilon \) notation in 1823 and defined the limit via inequalities in the 1820s; Weierstrass gave the full epsilon-delta definition in his 1861 lectures<sup>[3](https://arxiv.org/abs/1502.06942)</sup><sup> • </sup><sup>[4](https://www.jstage.jst.go.jp/article/jhsj/48/251/48_142/_article/-char/en)</sup> |
| Uniqueness | In a Hausdorff space a mapping can have only one limit at a point; in non-Hausdorff topologies a sequence may have several, even infinitely many, limits<sup>[5](https://encyclopediaofmath.org/wiki/Limit)</sup><sup> • </sup><sup>[6](https://mathworld.wolfram.com/Limit.html)</sup> |
| Practice | Limits are rarely evaluated directly from the definition; limit laws and techniques are used instead<sup>[1](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)</sup> |

## The epsilon-delta definition

Informal descriptions say that \( y \) "tends to" or "approaches" \( L \) as \( x \) "approaches" \( c \), or that \( y \) is "near" \( L \) whenever \( x \) is "near" \( c \). These words are not exact: they do not say how near \( x \) and \( y \) must be. The formal definition turns the informal statement into a precise claim about tolerances: if \( x \) is within \( \delta \) units of \( c \), then \( f(x) \) is within \( \epsilon \) units of \( L \).<sup>[1](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)</sup>

For a function \( f \) defined on an open interval containing \( c \), except possibly at \( c \) itself, the statement \( \lim_{x \to c} f(x) = L \) means that for every \( \epsilon > 0 \) there exists \( \delta > 0 \) such that \( 0 < |x - c| < \delta \) implies \( |f(x) - L| < \epsilon \). The exclusion \( x \neq c \) matters: the limit describes behavior near the point, not the value of the function at the point, so a function need not even be defined at \( c \) for its limit there to exist.<sup>[1](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)</sup>

The two tolerances play asymmetric roles. The \( \epsilon \) is a challenge issued first: an adversary may choose any positive output tolerance, however small. The proof succeeds only if one can then produce a \( \delta \), generally depending on \( \epsilon \), that meets the challenge for every input within \( \delta \) of \( c \). Reversing the order of the quantifiers gives a different, much weaker statement, which is why the definition cannot be read as "for some \( \delta \) there is an \( \epsilon \)".<sup>[1](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)</sup>

The same pattern defines limits of sequences. A sequence \( \{x_n\} \) of real or complex numbers has the limit \( x \) if, for any \( \epsilon > 0 \), there exists a positive integer \( N(\epsilon) \) such that \( |x_n - x| < \epsilon \) whenever \( n > N(\epsilon) \); one writes \( \lim_{n \to \infty} x_n = x \) and says the sequence converges to \( x \).<sup>[2](https://persson.berkeley.edu/math128a/slides/ch1math128a.html)</sup> The function definition can be viewed as a generalization of the sequence definition, since a sequence is just a function from the natural numbers into a space.<sup>[6](https://mathworld.wolfram.com/Limit.html)</sup>

## Variants and related notions

**One-sided and infinite limits.** The limit may be taken from below, \( \lim_{z \to a^-} \), or from above, \( \lim_{z \to a^+} \); if the two one-sided limits are equal, "the" limit is said to exist. The definition also adapts to infinite limits: \( \lim_{z \to a} f(z) = +\infty \) means that for every \( N > 0 \) there is a \( \delta \) such that \( f(z) > N \) whenever \( 0 < |z - a| < \delta \), with a corresponding form for \( -\infty \) and for \( z \to \pm\infty \).<sup>[6](https://mathworld.wolfram.com/Limit.html)</sup>

**Upper and lower limits.** For a real function, the upper and lower limits \( \limsup_{x \to x_0} f(x) \) and \( \liminf_{x \to x_0} f(x) \) capture the largest and smallest cluster of values near a point; the function has a finite limit \( L \) exactly when the lower and upper limits coincide and equal \( L \).<sup>[7](https://encyclopediaofmath.org/wiki/Upper_and_lower_limits)</sup>

**General spaces.** In a metric space, a sequence \( \{x_n\} \) has limit \( L \) if for all \( \epsilon > 0 \) there is \( n_0 \) with \( |x_n - L| < \epsilon \) for all \( n \geq n_0 \). In a general topological space the condition is stated with neighborhoods: a sequence converges to \( x \) if every neighborhood of \( x \) contains a tail of the sequence. In spaces that fail to be Hausdorff (\( T_2 \)), a sequence may have several, even infinitely many, different limits.<sup>[6](https://mathworld.wolfram.com/Limit.html)</sup> Sequences themselves are not always adequate: in a space that is not first countable, convergent sequences can fail to characterize closure, continuity, and compactness, and the notion of a net, a function from a directed set into the space, is used instead; a net converges to \( x \) if for every open set \( U \) containing \( x \) there is an index \( d \) such that \( w(e) \in U \) for all \( e \geq d \).<sup>[8](https://www.uvm.edu/~smillere/TProjects/MShi20s.pdf)</sup><sup> • </sup><sup>[6](https://mathworld.wolfram.com/Limit.html)</sup>

**Existence criteria.** A function with a finite non-zero limit at a point is bounded and keeps its sign in some neighborhood of that point. The Cauchy criterion gives an intrinsic test: when the domain space is first countable at \( x_0 \) and the target is a complete metric space, the limit exists if and only if for each \( \epsilon > 0 \) there is a neighborhood \( U \) of \( x_0 \) such that all points \( x' , x'' \) in \( E \cap U \setminus \{x_0\} \) satisfy \( \rho(f(x''), f(x')) < \epsilon \).<sup>[5](https://encyclopediaofmath.org/wiki/Limit)</sup>

## Role in calculus and analysis

The limit defines the basic objects of calculus. A function \( f \) is continuous at \( x_0 \) when \( \lim_{x \to x_0} f(x) = f(x_0) \), and differentiable at \( x_0 \) when the limit of the difference quotient \( \frac{f(x)-f(x_0)}{x-x_0} \) exists; differentiability implies continuity.<sup>[2](https://persson.berkeley.edu/math128a/slides/ch1math128a.html)</sup> Limits also enter numerical analysis, where the same definition underwrites error analysis and the convergence of iterative methods.<sup>[2](https://persson.berkeley.edu/math128a/slides/ch1math128a.html)</sup>

In practice, limits are rarely evaluated from the definition. Basic limits serve as building blocks, and limit laws, such as the facts that the limit of a sum is the sum of the limits and, when the denominator's limit is non-zero, the limit of a quotient is the quotient of the limits, reduce most computations to algebra.<sup>[1](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)</sup><sup> • </sup><sup>[5](https://encyclopediaofmath.org/wiki/Limit)</sup> Indeterminate forms of the types \( \infty/\infty \) and \( 0/0 \) can often be resolved with L'Hospital's rule, and forms like \( 0 \cdot \infty \) are converted to \( 0/0 \) by writing \( f(x)g(x) = f(x)/(1/g(x)) \).<sup>[6](https://mathworld.wolfram.com/Limit.html)</sup>

## History

Newton and Leibniz invented the calculus in the seventeenth century using "infinitesimal" quantities backed by geometric or mechanical considerations, without a rigorous definition of what these quantities were; the difficulty was noted from early on, and [George Berkeley](https://www.edgepedia.org/george-berkeley)'s 1734 *Analyst* attacked the logical basis of the calculus.<sup>[9](https://math.vanderbilt.edu/schectex/courses/past/history/)</sup><sup> • </sup><sup>[10](https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/)</sup> The intuitive, graphical understanding of limits sufficed for the first century or more of calculus, from Newton and Leibniz in the late 1600s to Cauchy in the early 1800s.<sup>[11](https://math.libretexts.org/Courses/Edmonds_College/Contemporary_Calculus/1_Limits_and_Continuity/1.4_Definition_of_Limit)</sup>

**Cauchy and Weierstrass.** In the 1820s [Augustin-Louis Cauchy](https://www.edgepedia.org/augustin-louis-cauchy) described the limit concept in terms of inequalities, freeing it of geometric and temporal intuition, and built his calculus on this arithmetical foundation; he introduced the \( \delta \)-\( \epsilon \) notation in his *Cours d'analyse* and provided definitions of function, limit, and derivative close to those used today.<sup>[4](https://www.jstage.jst.go.jp/article/jhsj/48/251/48_142/_article/-char/en)</sup><sup> • </sup><sup>[12](https://old.maa.org/node/418939)</sup> Cauchy, however, never specified a functional relationship giving \( \delta \) as a function of \( \epsilon \). That step came with Karl Weierstrass, who began his 1861 lectures on the differential calculus by defining the fundamental concepts in terms of epsilon-delta inequalities, with no geometrical image behind them; in his formulation the epsilon-delta method appeared in full.<sup>[3](https://arxiv.org/abs/1502.06942)</sup><sup> • </sup><sup>[4](https://www.jstage.jst.go.jp/article/jhsj/48/251/48_142/_article/-char/en)</sup>

This nineteenth-century program, the "arithmetization" of analysis pursued by Cauchy, Bolzano, Weierstrass, and others, found a way to avoid infinitesimals through the epsilon-delta characterization of limits, and the limit concept took its rigorous and final form in the latter half of the nineteenth century.<sup>[13](https://plato.stanford.edu/entries/dedekind-foundations/)</sup><sup> • </sup><sup>[14](https://plato.stanford.edu/archives/fall2013/entries/continuity/index.html)</sup> The rigorous limit also made possible the discovery and control of pathological examples, such as continuous functions that are nowhere differentiable, whose existence showed that informal intuition was not a reliable guide.<sup>[15](https://plato.stanford.edu/archives/sum2026/entries/mathematical-style/)</sup>

**Alternatives.** The epsilon-delta definition is not the only rigorous foundation. [Abraham Robinson](https://www.edgepedia.org/abraham-robinson)'s non-standard analysis gave a legitimate mathematical status to infinitely small and infinitely large quantities, and within it every statement of ordinary analysis involving limits has a succinct translation into the language of infinitesimals; the traditional and non-standard definitions of limits, continuity, and derivatives have been formally proved equivalent.<sup>[14](https://plato.stanford.edu/archives/fall2013/entries/continuity/index.html)</sup><sup> • </sup><sup>[16](https://arxiv.org/abs/1406.1560)</sup> In smooth infinitesimal analysis, a framework built on category theory, the use of limits in defining the basic notions of the calculus is replaced by nilpotent infinitesimals.<sup>[14](https://plato.stanford.edu/archives/fall2013/entries/continuity/index.html)</sup>

## References

1. [1.2 Epsilon-Delta Definition of a Limit, APEX Calculus LT](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)
2. [Chapter 1: Mathematical Preliminaries and Error Analysis, Math 128A, UC Berkeley](https://persson.berkeley.edu/math128a/slides/ch1math128a.html)
3. [On the history of epsilontics, arXiv:1502.06942](https://arxiv.org/abs/1502.06942)
4. [Cauchy's and Weierstrass's Contributions to the Formation of Analysis in Epsilonics, Journal of History of Science, Japan](https://www.jstage.jst.go.jp/article/jhsj/48/251/48_142/_article/-char/en)
5. [Limit, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Limit)
6. [Limit, Wolfram MathWorld](https://mathworld.wolfram.com/Limit.html)
7. [Upper and lower limits, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Upper_and_lower_limits)
8. [Nets and Filters, University of Vermont](https://www.uvm.edu/~smillere/TProjects/MShi20s.pdf)
9. [History of Math, Math 252, Vanderbilt University](https://math.vanderbilt.edu/schectex/courses/past/history/)
10. [The rise of calculus, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/)
11. [1.4: Definition of Limit, Contemporary Calculus, LibreTexts](https://math.libretexts.org/Courses/Edmonds_College/Contemporary_Calculus/1_Limits_and_Continuity/1.4_Definition_of_Limit)
12. [Mathematical Treasure: Cauchy's Cours d'Analyse, Mathematical Association of America](https://old.maa.org/node/418939)
13. [Dedekind's Contributions to the Foundations of Mathematics, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/dedekind-foundations/)
14. [Continuity and Infinitesimals, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/archives/fall2013/entries/continuity/index.html)
15. [Mathematical Style, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/archives/sum2026/entries/mathematical-style/)
16. [Equivalence of the Traditional and Non-Standard Definitions of Concepts from Real Analysis, arXiv:1406.1560](https://arxiv.org/abs/1406.1560)

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