# Limit of a function

The **limit of a function** is the value that a function's output approaches as its input approaches a given point, written \( \lim_{x \to c} f(x) = L \). The concept makes precise the informal idea that "\( f(x) \) gets close to \( L \) as \( x \) gets close to \( c \)," and it underlies the definitions of continuity, the derivative, and the integral in calculus.<sup>[1](https://opentext.uleth.ca/apex-calculus/sec_limit_def.html)</sup><sup> • </sup><sup>[2](https://www.geeksforgeeks.org/maths/real-life-applications-of-limits/)</sup>

| Key fact | Detail |
|---|---|
| Formal statement | \( \lim_{x \to c} f(x) = L \) means: for every \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that \( 0 < \lvert x - c \rvert < \delta \) implies \( \lvert f(x) - L \rvert < \varepsilon \)<sup>[1](https://opentext.uleth.ca/apex-calculus/sec_limit_def.html)</sup> |
| Value at the point | The definition says nothing about \( f(c) \); the value may differ from \( L \) or may not be defined at all<sup>[3](https://web.mit.edu/wwmath/calculus/limits/formal.html)</sup> |
| Order of quantifiers | The \( y \)-tolerance \( \varepsilon \) is given first; the limit exists if an \( x \)-tolerance \( \delta \) can be found that works for it<sup>[1](https://opentext.uleth.ca/apex-calculus/sec_limit_def.html)</sup> |
| Variants | One-sided limits, limits at infinity, and infinite limits each have their own ε-δ-style definitions; one count enumerates fifteen distinct combinations of two-sided, one-sided, and infinite cases<sup>[4](https://www.math.uci.edu/~ndonalds/math140b/notes.pdf)</sup> |
| Sequential form | \( q \) is the limit of \( f \) at \( p \) exactly when every sequence \( x_m \to p \) with \( x_m \neq p \) has \( f(x_m) \to q \)<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_%28Zakon%29/04%3A_Function_Limits_and_Continuity/4.02%3A_Some_General_Theorems_on_Limits_and_Continuity)</sup> |
| History | Cauchy introduced the symbols ε and δ in 1823; the full ε-δ definition of a limit appeared in Weierstrass' work of 1861<sup>[6](https://arxiv.org/abs/1502.06942)</sup> |
| Place in teaching | In AP Calculus AB, Unit 1 (Limits and Continuity) carries a 10%–15% weighting in the multiple-choice section of the exam<sup>[7](https://apcentral.collegeboard.org/courses/ap-calculus-ab)</sup> |

## The informal idea and its problem

Before a formal definition is stated, a limit is usually described in words: "\( y \) tends to \( L \) as \( x \) tends to \( c \)," or "\( y \) is near \( L \) whenever \( x \) is near \( c \)." These phrasings fail as definitions because words like "tends," "approaches," and especially "near" are not exact; the definition does not say how near \( x \) and \( y \) must be.<sup>[1](https://opentext.uleth.ca/apex-calculus/sec_limit_def.html)</sup>

The repair is to treat "near" as a matter of tolerances. If \( x \) is within \( \delta \) units of \( c \), then \( f(x) \) should be within \( \varepsilon \) units of \( L \). In symbols, the requirement is \( \lvert x - c \rvert < \delta \implies \lvert f(x) - L \rvert < \varepsilon \), with \( \delta \) and \( \varepsilon \) positive numbers that can be chosen arbitrarily small.<sup>[1](https://opentext.uleth.ca/apex-calculus/sec_limit_def.html)</sup>

## The epsilon-delta definition

Let \( I \) be an open interval containing \( c \), and let \( f \) be a function defined on \( I \), except possibly at \( c \). The statement \( \lim_{x \to c} f(x) = L \) means that given any \( \varepsilon > 0 \), there exists \( \delta > 0 \) such that for all \( x \) in \( I \) with \( x \neq c \), if \( \lvert x - c \rvert < \delta \), then \( \lvert f(x) - L \rvert < \varepsilon \).<sup>[1](https://opentext.uleth.ca/apex-calculus/sec_limit_def.html)</sup> Written without words:

\[ \lim_{x \to c} f(x) = L \iff \forall\, \varepsilon > 0,\ \exists\, \delta > 0 \text{ such that } 0 < \lvert x - c \rvert < \delta \implies \lvert f(x) - L \rvert < \varepsilon. \]

Two features of this definition deserve attention. First, the condition \( x \neq c \) (equivalently \( 0 < \lvert x - c \rvert \)) means the limit is concerned only with behavior *around* the point, not at it. The value of \( f(c) \) does not affect the limit and may not even be defined.<sup>[3](https://web.mit.edu/wwmath/calculus/limits/formal.html)</sup>

In practice, limits are rarely evaluated directly from the definition. The definition's role is to make the notion precise so that theorems, such as the limit laws, can be proved from it; computational work then proceeds by those theorems.<sup>[8](https://openstax.org/books/calculus-volume-1/pages/2-5-the-precise-definition-of-a-limit)</sup><sup> • </sup><sup>[9](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)</sup>

## One-sided, infinite, and limits at infinity

The two-sided definition adapts to several other situations by changing the condition on \( x \).<sup>[8](https://openstax.org/books/calculus-volume-1/pages/2-5-the-precise-definition-of-a-limit)</sup>

**One-sided limits.** The right-hand limit \( \lim_{x \to x_0^+} f(x) \) is the limit at \( x_0 \) restricted to inputs greater than \( x_0 \), that is, within the set \( (x_0, \infty) \); the left-hand limit is defined analogously on \( (-\infty, x_0) \).<sup>[10](https://people.tamu.edu/~yvorobets/MATH409-2020A/Lect3-05web.pdf)</sup>

**Infinite limits.** The statement \( \lim_{x \to a} f(x) = \infty \) means that for every number \( M > 0 \) there is a \( \delta > 0 \) such that \( f(x) > M \) whenever \( 0 < \lvert x - a \rvert < \delta \); the corresponding definition for \( -\infty \) requires \( f(x) < N \) for every \( N < 0 \). No matter how large a threshold \( M \) is chosen, some interval around \( a \) keeps the graph above the line \( y = M \); the larger \( M \) is, the smaller \( \delta \) generally must be. A worked example is \( \lim_{x \to 0} \frac{1}{x^2} = \infty \).<sup>[11](https://tutorial.math.lamar.edu/classes/calci/DefnOfLimit.aspx)</sup>

**Limits at infinity.** For \( \lim_{x \to \infty} f(x) = L \), the roles reverse: for any chosen \( \varepsilon \), there is a number \( M \) such that whenever \( x > M \), the graph stays within \( \varepsilon \) of \( L \). Here \( x \) must simply be taken large enough rather than close to a finite point.<sup>[11](https://tutorial.math.lamar.edu/classes/calci/DefnOfLimit.aspx)</sup>

One set of analysis notes counts fifteen distinct combinations across three two-sided and six each of the one-sided limits, depending on whether \( a \), \( L \), or both are finite or infinite.<sup>[4](https://www.math.uci.edu/~ndonalds/math140b/notes.pdf)</sup>

## The sequential criterion

An equivalent formulation replaces intervals with sequences. A point \( q \) is a limit of \( f \) at \( p \) if and only if, for every sequence \( \{x_m\} \) of points of the domain other than \( p \) with \( x_m \to p \), the image sequence satisfies \( f(x_m) \to q \). The restriction to terms other than \( p \) mirrors the punctured interval \( 0 < \lvert x - p \rvert \) in the ε-δ definition.<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_%28Zakon%29/04%3A_Function_Limits_and_Continuity/4.02%3A_Some_General_Theorems_on_Limits_and_Continuity)</sup> The same theorem gives the sequential criterion for continuity: \( f \) is continuous at \( p \) exactly when \( x_m \to p \) implies \( f(x_m) \to f(p) \).<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_%28Zakon%29/04%3A_Function_Limits_and_Continuity/4.02%3A_Some_General_Theorems_on_Limits_and_Continuity)</sup> Some introductory analysis courses prefer the sequential route because it is easier to absorb than the ε-δ definitions, proving the two equivalent once continuity has been developed.<sup>[12](https://www.math.iitb.ac.in/~srg/acicara/CourseProposals-Paolella.pdf)</sup>

## Role in calculus

Limits are the foundation on which the rest of calculus is built: continuity, derivatives, and integrals are all defined in terms of limits.<sup>[2](https://www.geeksforgeeks.org/maths/real-life-applications-of-limits/)</sup> Continuity at a point is the special case in which the limit equals the function's value there, \( \lim_{x \to a} f(x) = f(a) \).<sup>[13](https://www.amsi.org.au/backup_ESA_2018.01.12/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html)</sup> In applied work, limits define instantaneous velocity and speed as rates of change of position, and techniques such as [L'Hôpital's rule](https://www.edgepedia.org/lhopitals-rule) use limits to evaluate otherwise indeterminate forms.<sup>[2](https://www.geeksforgeeks.org/maths/real-life-applications-of-limits/)</sup><sup> • </sup><sup>[14](https://www.khanacademy.org/math/differential-calculus/dc-context-app)</sup>

The concept also anchors school and university curricula. In the AP Calculus AB course, students explore "change, limits, and the analysis of functions" in Unit 1, which carries a 10%–15% multiple-choice exam weighting, before differentiation is treated in later units.<sup>[7](https://apcentral.collegeboard.org/courses/ap-calculus-ab)</sup>

## History

Before the nineteenth century, mathematicians such as Euler experimented with limiting processes, sometimes reaching correct answers for wrong reasons and sometimes incorrect ones; the lack of rigor produced paradoxes of the infinite.<sup>[13](https://www.amsi.org.au/backup_ESA_2018.01.12/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html)</sup> In the early nineteenth century the need for a formal approach was recognized by [Augustin-Louis Cauchy](https://www.edgepedia.org/augustin-louis-cauchy) (1789–1857) and later Karl Weierstrass (1815–1897).<sup>[13](https://www.amsi.org.au/backup_ESA_2018.01.12/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html)</sup>

Cauchy introduced proofs using the \( \varepsilon \) notation in his 1821 *Cours d'Analyse*, and the symbols ε and δ were first introduced by him in 1823, though he never specified a functional relationship expressing \( \delta \) in terms of \( \varepsilon \).<sup>[13](https://www.amsi.org.au/backup_ESA_2018.01.12/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html)</sup><sup> • </sup><sup>[6](https://arxiv.org/abs/1502.06942)</sup> Around the same time, Bernard Bolzano (1781–1848) gave the first rigorous ε-δ definition of a limit, but his work was not widely disseminated at the time.<sup>[13](https://www.amsi.org.au/backup_ESA_2018.01.12/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html)</sup> It was only in 1861 that the ε-δ method appeared in full in Weierstrass' definition of a limit.<sup>[6](https://arxiv.org/abs/1502.06942)</sup> Weierstrass, generally regarded as the father of modern analysis, gave the first rigorous definition of continuity of a function at a point, a definition that says exactly that \( \lim_{x \to a} f(x) = f(a) \).<sup>[13](https://www.amsi.org.au/backup_ESA_2018.01.12/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html)</sup>

## References

1. [Section 1.2: Epsilon-Delta Definition of a Limit, APEX Calculus (University of Lethbridge)](https://opentext.uleth.ca/apex-calculus/sec_limit_def.html)
2. [Real Life Applications of Limits, GeeksforGeeks](https://www.geeksforgeeks.org/maths/real-life-applications-of-limits/)
3. [World Web Math: Formal Definition of Limits, MIT](https://web.mit.edu/wwmath/calculus/limits/formal.html)
4. [1.20 Limits of Functions, UC Irvine lecture notes](https://www.math.uci.edu/~ndonalds/math140b/notes.pdf)
5. [4.2: Some General Theorems on Limits and Continuity, Mathematics LibreTexts (Zakon, Mathematical Analysis)](https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_%28Zakon%29/04%3A_Function_Limits_and_Continuity/4.02%3A_Some_General_Theorems_on_Limits_and_Continuity)
6. [On the history of epsilontics, arXiv:1502.06942](https://arxiv.org/abs/1502.06942)
7. [AP Calculus AB Course, AP Central, College Board](https://apcentral.collegeboard.org/courses/ap-calculus-ab)
8. [2.5 The Precise Definition of a Limit, Calculus Volume 1, OpenStax](https://openstax.org/books/calculus-volume-1/pages/2-5-the-precise-definition-of-a-limit)
9. [1.2 Epsilon-Delta Definition of a Limit, APEX Calculus I, University of North Dakota](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch1.S2.html)
10. [MATH 409 Advanced Calculus I, Lecture 23, Texas A&M University](https://people.tamu.edu/~yvorobets/MATH409-2020A/Lect3-05web.pdf)
11. [Calculus I: The Definition of the Limit, Paul's Online Math Notes, Lamar University](https://tutorial.math.lamar.edu/classes/calci/DefnOfLimit.aspx)
12. [Course proposal citing sequential versus ε-δ definitions, IIT Bombay](https://www.math.iitb.ac.in/~srg/acicara/CourseProposals-Paolella.pdf)
13. [History and applications: Cauchy and Weierstrass, Australian Mathematical Sciences Institute](https://www.amsi.org.au/backup_ESA_2018.01.12/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html)
14. [Applications of Derivatives, Differential Calculus, Khan Academy](https://www.khanacademy.org/math/differential-calculus/dc-context-app)

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