limit (mathematics)#Infinity in limits of functions

In formulas, a limit of a function is usually written as

:$\backslash lim\_\{x\; \backslash to\; c\}\; f(x)\; =\; L,$

and is read as "the limit of {{math|f}} of {{mvar|x}} as {{mvar|x}} approaches {{mvar|c}} equals {{math|L}}". This means that the value of the function {{math|f}} can be made arbitrarily close to {{math|L}}, by choosing {{mvar|x}} sufficiently close to {{mvar|c}}. Alternatively, the fact that a function {{math|f}} approaches the limit {{math|L}} as {{mvar|x}} approaches {{mvar|c}} is sometimes denoted by a right arrow (→ or $\backslash rightarrow$), as in

:$f(x)\; \backslash to\; L\; \backslash text\{\; as\; \}\; x\; \backslash to\; c,$

which reads "$f$ of $x$ tends to $L$ as $x$ tends to $c$".

According to Hankel (1871), the modern concept of limit originates from Proposition X.1 of Euclid's Elements, which forms the basis of the Method of exhaustion found in Euclid and Archimedes: "Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out."{{cite book |last1=Schubring |first1=Gert |title=Conflicts between generalization, rigor, and intuition: number concepts underlying the development of analysis in 17th-19th century France and Germany |date=2005 |publisher=Springer |location=New York |isbn=0387228365 |pages=22–23}}{{cite web |title=Euclid's Elements, Book X, Proposition 1 |url=http://aleph0.clarku.edu/~djoyce/elements/bookX/propX1.html |website=aleph0.clarku.edu}}

Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."{{Cite journal |last=Van Looy |first=Herman |date=1984 |title=A chronology and historical analysis of the mathematical manuscripts of Gregorius a Sancto Vincentio (1584–1667) |journal=Historia Mathematica |language=en |volume=11 |issue=1 |pages=57–75 |doi=10.1016/0315-0860(84)90005-3|doi-access=free }}

In the Scholium to Principia in 1687, Isaac Newton had a clear definition of a limit, stating that "Those ultimate ratios... are not actually ratios of ultimate quantities, but limits... which they can approach so closely that their difference is less than any given quantity".{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=ipA4DwAAQBAJ&pg=PA28 |title=Newton and the Great World System |date=2017 |publisher=World Scientific Publishing |isbn=978-1-78634-372-7 |pages=28 |language=en |doi=10.1142/q0108}}

The modern definition of a limit goes back to Bernard Bolzano who, in 1817, developed the basics of the epsilon-delta technique to define continuous functions. However, his work remained unknown to other mathematicians until thirty years after his death.{{Citation|title=Bolzano, Cauchy, Epsilon, Delta|last=Felscher|first=Walter|journal=American Mathematical Monthly|volume=107|issue=9|pages=844–862|year=2000|doi=10.2307/2695743|jstor=2695743}}

Augustin-Louis Cauchy in 1821,{{cite book |last1=Larson |first1=Ron |title=Calculus of a single variable |last2=Edwards |first2=Bruce H. |publisher=Brooks/Cole, Cengage Learning |year=2010 |isbn=978-0-547-20998-2 |edition=Ninth |author-link1=Ron Larson (mathematician)}} followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit.

The modern notation of placing the arrow below the limit symbol is due to G. H. Hardy, who introduced it in his book A Course of Pure Mathematics in 1908.{{Citation|last=Miller|first=Jeff|title=Earliest Uses of Symbols of Calculus|date=1 December 2004|url=http://jeff560.tripod.com/calculus.html|access-date=2008-12-18|archive-date=2015-05-01|archive-url=https://web.archive.org/web/20150501123549/http://jeff560.tripod.com/calculus.html|url-status=live}}

{{main|Limit of a sequence}}

The expression 0.999... should be interpreted as the limit of the sequence 0.9, 0.99, 0.999, ... and so on. This sequence can be rigorously shown to have the limit 1, and therefore this expression is meaningfully interpreted as having the value 1.{{citation |last=Stillwell |first=John |title=Elements of algebra: geometry, numbers, equations |pages=42 |year=1994 |publisher=Springer |isbn=978-1441928399 |author-link=John Stillwell}}

Formally, suppose {{math|a₁, a₂, ...}} is a sequence of real numbers. When the limit of the sequence exists, the real number {{math|L}} is the limit of this sequence if and only if for every real number {{math|ε > 0}}, there exists a natural number {{math|N}} such that for all {{math|n > N}}, we have {{math|{{abs|a_n − L}} < ε}}.{{Cite web|last=Weisstein|first=Eric W.|title=Limit|url=https://mathworld.wolfram.com/Limit.html|access-date=2020-08-18|website=mathworld.wolfram.com|language=en|archive-date=2020-06-20|archive-url=https://web.archive.org/web/20200620203909/https://mathworld.wolfram.com/Limit.html|url-status=live}}

The common notation

$$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; a\_n\; =\; L$$

is read as:

:"The limit of a_n as n approaches infinity equals L" or "The limit as n approaches infinity of a_n equals L".

The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value {{math|{{abs|a_n − L}}}} is the distance between {{math|a_n}} and {{math|L}}.

Not every sequence has a limit. A sequence with a limit is called convergent; otherwise it is called 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 one hand, the limit as {{mvar|n}} approaches infinity of a sequence {{math|{{mset|a_n}}}} is simply the limit at infinity of a function {{math|a(n)}}—defined on the natural numbers {{math|{{mset|n}}}}. On the other hand, if X is the domain of a function {{math|f(x)}} and if the limit as {{mvar|n}} approaches infinity of {{math|f(x_n)}} is {{math|L}} for every arbitrary sequence of points {{math|{{mset|x_n}}}} in X − x₀ which converges to {{math|x₀}}, then the limit of the function {{math|f(x)}} as {{math|x}} approaches {{math|x₀}} is equal to {{math|L}}.{{harvtxt|Apostol|1974|pp=75–76}} One such sequence would be {{math|{{mset|x₀ + 1/n}}}}.

There is also a notion of having a limit "tend to infinity", rather than to a finite value $L$. A sequence $\backslash \{a\_n\backslash \}$ is said to "tend to infinity" if, for each real number $M\; >\; 0$, known as the bound, there exists an integer $N$ such that for each $n\; >\; N$,

$$a\_n\; >\; M.$$

That is, for every possible bound, the sequence eventually exceeds the bound. This is often written $\backslash lim\_\{n\backslash rightarrow\; \backslash infty\}\; a\_n\; =\; \backslash infty$ or simply $a\_n\; \backslash rightarrow\; \backslash infty$.

It is possible for a sequence to be divergent, but not tend to infinity. Such sequences are called oscillatory. An example of an oscillatory sequence is $a\_n\; =\; (-1)^n$.

There is a corresponding notion of tending to negative infinity, $\backslash lim\_\{n\backslash rightarrow\; \backslash infty\}\; a\_n\; =\; -\backslash infty$, defined by changing the inequality in the above definition to with

A sequence $\backslash \{a\_n\backslash \}$ with $\backslash lim\_\{n\backslash rightarrow\; \backslash infty\}\; |a\_n|\; =\; \backslash infty$ is called unbounded, a definition equally valid for sequences in the complex numbers, or in any metric space. Sequences which do not tend to infinity are called bounded. Sequences which do not tend to positive infinity are called bounded above, while those which do not tend to negative infinity are bounded below.

The discussion of sequences above is for sequences of real numbers. The notion of limits can be defined for sequences valued in more abstract spaces, such as metric spaces. If $M$ is a metric space with distance function $d$, and $\backslash \{a\_n\backslash \}\_\{n\; \backslash geq\; 0\}$ is a sequence in $M$, then the limit (when it exists) of the sequence is an element $a\backslash in\; M$ such that, given $\backslash varepsilon\; >\; 0$, there exists an $N$ such that for each $n\; >\; N$, we have

An equivalent statement is that $a\_n\; \backslash rightarrow\; a$ if the sequence of real numbers $d(a,\; a\_n)\; \backslash rightarrow\; 0$.

An important example is the space of $n$-dimensional real vectors, with elements $\backslash mathbf\{x\}\; =\; (x\_1,\; \backslash cdots,\; x\_n)$ where each of the $x\_i$ are real, an example of a suitable distance function is the Euclidean distance, defined by

$$d(\backslash mathbf\{x\},\; \backslash mathbf\{y\})\; =\; \backslash |\backslash mathbf\{x\}\; -\; \backslash mathbf\{y\}\backslash |\; =\; \backslash sqrt\{\backslash sum\_i(x\_i\; -\; y\_i)^2\}.$$

The sequence of points $\backslash \{\backslash mathbf\{x\}\_n\backslash \}\_\{n\; \backslash geq\; 0\}$ converges to $\backslash mathbf\{x\}$ if the limit exists and $\backslash |\backslash mathbf\{x\}\_n\; -\; \backslash mathbf\{x\}\backslash |\; \backslash rightarrow\; 0$.

In some sense the most abstract space in which limits can be defined are topological spaces. If $X$ is a topological space with topology $\backslash tau$, and $\backslash \{a\_n\backslash \}\_\{n\; \backslash geq\; 0\}$ is a sequence in $X$, then the limit (when it exists) of the sequence is a point $a\backslash in\; X$ such that, given a (open) neighborhood $U\backslash in\; \backslash tau$ of $a$, there exists an $N$ such that for every $n\; >\; N$,

$$a\_n\; \backslash in\; U$$

is satisfied. In this case, the limit (if it exists) may not be unique. However it must be unique if $X$ is a Hausdorff space.

This section deals with the idea of limits of sequences of functions, not to be confused with the idea of limits of functions, discussed below.

The field of functional analysis partly seeks to identify useful notions of convergence on function spaces. For example, consider the space of functions from a generic set $E$ to $\backslash mathbb\{R\}$. Given a sequence of functions $\backslash \{f\_n\backslash \}\_\{n\; >\; 0\}$ such that each is a function $f\_n:\; E\; \backslash rightarrow\; \backslash mathbb\{R\}$, suppose that there exists a function such that for each $x\; \backslash in\; E$,

$$f\_n(x)\; \backslash rightarrow\; f(x)\; \backslash text\{\; or\; equivalently\; \}\; \backslash lim\_\{n\; \backslash rightarrow\; \backslash infty\}f\_n(x)\; =\; f(x).$$

Then the sequence $f\_n$ is said to converge pointwise to $f$. However, such sequences can exhibit unexpected behavior. For example, it is possible to construct a sequence of continuous functions which has a discontinuous pointwise limit.

Another notion of convergence is uniform convergence. The uniform distance between two functions $f,g:\; E\; \backslash rightarrow\; \backslash mathbb\{R\}$ is the maximum difference between the two functions as the argument $x\; \backslash in\; E$ is varied. That is,

$$d(f,g)\; =\; \backslash max\_\{x\; \backslash in\; E\}|f(x)\; -\; g(x)|.$$

Then the sequence $f\_n$ is said to uniformly converge or have a uniform limit of $f$ if $f\_n\; \backslash rightarrow\; f$ with respect to this distance. The uniform limit has "nicer" properties than the pointwise limit. For example, the uniform limit of a sequence of continuous functions is continuous.

Many different notions of convergence can be defined on function spaces. This is sometimes dependent on the regularity of the space. Prominent examples of function spaces with some notion of convergence are Lp spaces and Sobolev space.

{{main|Limit of a function}}

File:Limit-at-infinity-graph.png is {{math|L}}. For any arbitrary distance {{mvar|ε}}, there must be a value {{math|S}} such that the function stays within {{math|L ± ε}} for all {{math|x > S}}.|300x300px]]Suppose {{math|f}} is a real-valued function and {{mvar|c}} is a real number. Intuitively speaking, the expression

:$\backslash lim\_\{x\; \backslash to\; c\}f(x)\; =\; L$

means that {{math|f(x)}} can be made to be as close to {{math|L}} as desired, by making {{mvar|x}} sufficiently close to {{mvar|c}}.{{Cite web |last=Weisstein |first=Eric W. |title=Epsilon-Delta Definition |url=https://mathworld.wolfram.com/Epsilon-DeltaDefinition.html |access-date=2020-08-18 |website=mathworld.wolfram.com |language=en |archive-date=2020-06-25 |archive-url=https://web.archive.org/web/20200625125230/https://mathworld.wolfram.com/Epsilon-DeltaDefinition.html |url-status=live }} In that case, the above equation can be read as "the limit of {{math|f}} of {{mvar|x}}, as {{mvar|x}} approaches {{mvar|c}}, is {{math|L}}".

Formally, the definition of the "limit of $f(x)$ as $x$ approaches $c$" is given as follows. The limit is a real number $L$ so that, given an arbitrary real number $\backslash varepsilon\; >\; 0$ (thought of as the "error"), there is a $\backslash delta\; >\; 0$ such that, for any $x$ satisfying , it holds that . This is known as the (ε, δ)-definition of limit.

The inequality is used to exclude $c$ from the set of points under consideration, but some authors do not include this in their definition of limits, replacing with simply . This replacement is equivalent to additionally requiring that $f$ be continuous at $c$.

It can be proven that there is an equivalent definition which makes manifest the connection between limits of sequences and limits of functions.{{cite web

|url=https://dec41.user.srcf.net/h/IA_L/analysis_i

|title=Analysis I (based on a course given by Timothy Gowers)

|last=Chua

|first=Dexter

|website=Notes from the Mathematical Tripos

}} The equivalent definition is given as follows. First observe that for every sequence $\backslash \{x\_n\backslash \}$ in the domain of $f$, there is an associated sequence $\backslash \{f(x\_n)\backslash \}$, the image of the sequence under $f$. The limit is a real number $L$ so that, for all sequences $x\_n\; \backslash rightarrow\; c$, the associated sequence $f(x\_n)\; \backslash rightarrow\; L$.

{{Main article | One-sided limit}}

It is possible to define the notion of having a "left-handed" limit ("from below"), and a notion of a "right-handed" limit ("from above"). These need not agree. An example is given by the positive indicator function, $f:\; \backslash mathbb\{R\}\; \backslash rightarrow\; \backslash mathbb\{R\}$, defined such that $f(x)\; =\; 0$ if $x\; \backslash leq\; 0$, and $f(x)\; =\; 1$ if $x\; >\; 0$. At $x\; =\; 0$, the function has a "left-handed limit" of 0, a "right-handed limit" of 1, and its limit does not exist. Symbolically, this can be stated as, for this example,

$\backslash lim\_\{x\; \backslash to\; c^-\}f(x)\; =\; 0$, and $\backslash lim\_\{x\; \backslash to\; c^+\}f(x)\; =\; 1$, and from this it can be deduced $\backslash lim\_\{x\; \backslash to\; c\}f(x)$ doesn't exist, because $\backslash lim\_\{x\; \backslash to\; c^-\}f(x)\; \backslash neq\; \backslash lim\_\{x\; \backslash to\; c^+\}f(x)$.

It is possible to define the notion of "tending to infinity" in the domain of $f$,

$$\backslash lim\_\{x\; \backslash rightarrow\; +\backslash infty\}\; f(x)\; =\; L.$$

This could be considered equivalent to the limit as a reciprocal tends to 0:

$$\backslash lim\_\{x\text{'}\; \backslash rightarrow\; 0^+\}\; f(1/x\text{'})\; =\; L.$$

or it can be defined directly: the "limit of $f$ as $x$ tends to positive infinity" is defined as a value $L$ such that, given any real $\backslash varepsilon\; >\; 0$, there exists an $M\; >\; 0$ so that for all $x\; >\; M$, . The definition for sequences is equivalent: As $n\; \backslash rightarrow\; +\backslash infty$, we have $f(x\_n)\; \backslash rightarrow\; L$.

In these expressions, the infinity is normally considered to be signed ($+\backslash infty$ or $-\backslash infty$) and corresponds to a one-sided limit of the reciprocal. A two-sided infinite limit can be defined, but an author would explicitly write $\backslash pm\backslash infty$ to be clear.

It is also possible to define the notion of "tending to infinity" in the value of $f$,

$$\backslash lim\_\{x\; \backslash rightarrow\; c\}\; f(x)\; =\; \backslash infty.$$

Again, this could be defined in terms of a reciprocal:

$$\backslash lim\_\{x\; \backslash rightarrow\; c\}\; \backslash frac\{1\}\{f(x)\}\; =\; 0.$$

Or a direct definition can be given as follows: given any real number $M>0$, there is a $\backslash delta\; >\; 0$ so that for , the absolute value of the function $|f(x)|\; >\; M$. A sequence can also have an infinite limit: as $n\; \backslash rightarrow\; \backslash infty$, the sequence $f(x\_n)\; \backslash rightarrow\; \backslash infty$.

This direct definition is easier to extend to one-sided infinite limits. While mathematicians do talk about functions approaching limits "from above" or "from below", there is not a standard mathematical notation for this as there is for one-sided limits.

In non-standard analysis (which involves a hyperreal enlargement of the number system), the limit of a sequence $(a\_n)$ can be expressed as the standard part of the value $a\_H$ of the natural extension of the sequence at an infinite hypernatural index n=H. Thus,

:$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; a\_n\; =\; \backslash operatorname\{st\}(a\_H)\; .$

Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is infinitesimal). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal $a=[a\_n]$ represented in the ultrapower construction by a Cauchy sequence $(a\_n)$, is simply the limit of that sequence:

:$\backslash operatorname\{st\}(a)=\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; a\_n\; .$

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

Let $\backslash \{a\_n\backslash \}\_\{n\; >\; 0\}$ be a sequence in a topological space $X$. For concreteness, $X$ can be thought of as $\backslash mathbb\{R\}$, but the definitions hold more generally. The limit set is the set of points such that if there is a convergent subsequence $\backslash \{a\_\{n\_k\}\backslash \}\_\{k\; >0\}$ with $a\_\{n\_k\}\backslash rightarrow\; a$, then $a$ belongs to the limit set. In this context, such an $a$ is sometimes called a limit point.

A use of this notion is to characterize the "long-term behavior" of oscillatory sequences. For example, consider the sequence $a\_n\; =\; (-1)^n$. Starting from n=1, the first few terms of this sequence are $-1,\; +1,\; -1,\; +1,\; \backslash cdots$. It can be checked that it is oscillatory, so has no limit, but has limit points $\backslash \{-1,\; +1\backslash \}$.

This notion is used in dynamical systems, to study limits of trajectories. Defining a trajectory to be a function $\backslash gamma:\; \backslash mathbb\{R\}\; \backslash rightarrow\; X$, the point $\backslash gamma(t)$ is thought of as the "position" of the trajectory at "time" $t$. The limit set of a trajectory is defined as follows. To any sequence of increasing times $\backslash \{t\_n\backslash \}$, there is an associated sequence of positions $\backslash \{x\_n\backslash \}\; =\; \backslash \{\backslash gamma(t\_n)\backslash \}$. If $x$ is the limit set of the sequence $\backslash \{x\_n\backslash \}$ for any sequence of increasing times, then $x$ is a limit set of the trajectory.

Technically, this is the $\backslash omega$-limit set. The corresponding limit set for sequences of decreasing time is called the $\backslash alpha$-limit set.

An illustrative example is the circle trajectory: $\backslash gamma(t)\; =\; (\backslash cos(t),\; \backslash sin(t))$. This has no unique limit, but for each $\backslash theta\; \backslash in\; \backslash mathbb\{R\}$, the point $(\backslash cos(\backslash theta),\; \backslash sin(\backslash theta))$ is a limit point, given by the sequence of times $t\_n\; =\; \backslash theta\; +\; 2\backslash pi\; n$. But the limit points need not be attained on the trajectory. The trajectory $\backslash gamma(t)\; =\; t/(1\; +\; t)(\backslash cos(t),\; \backslash sin(t))$ also has the unit circle as its limit set.

Limits are used to define a number of important concepts in analysis.

{{Main article|series (mathematics)}}

A particular expression of interest which is formalized as the limit of a sequence is sums of infinite series. These are "infinite sums" of real numbers, generally written as

$$\backslash sum\_\{n\; =\; 1\}^\backslash infty\; a\_n.$$

This is defined through limits as follows: given a sequence of real numbers $\backslash \{a\_n\backslash \}$, the sequence of partial sums is defined by

$$s\_n\; =\; \backslash sum\_\{i\; =\; 1\}^n\; a\_i.$$

If the limit of the sequence $\backslash \{s\_n\backslash \}$ exists, the value of the expression $\backslash sum\_\{n\; =\; 1\}^\backslash infty\; a\_n$ is defined to be the limit. Otherwise, the series is said to be divergent.

A classic example is the Basel problem, where $a\_n\; =\; 1/n^2$. Then

$$\backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{n^2\}\; =\; \backslash frac\{\backslash pi^2\}\{6\}.$$

However, while for sequences there is essentially a unique notion of convergence, for series there are different notions of convergence. This is due to the fact that the expression $\backslash sum\_\{n\; =\; 1\}^\backslash infty\; a\_n$ does not discriminate between different orderings of the sequence $\backslash \{a\_n\backslash \}$, while the convergence properties of the sequence of partial sums can depend on the ordering of the sequence.

A series which converges for all orderings is called unconditionally convergent. It can be proven to be equivalent to absolute convergence. This is defined as follows. A series is absolutely convergent if $\backslash sum\_\{n\; =\; 1\}^\backslash infty\; |a\_n|$ is well defined. Furthermore, all possible orderings give the same value.

Otherwise, the series is conditionally convergent. A surprising result for conditionally convergent series is the Riemann series theorem: depending on the ordering, the partial sums can be made to converge to any real number, as well as $\backslash pm\; \backslash infty$.

{{Main article|Power series}}

A useful application of the theory of sums of series is for power series. These are sums of series of the form

$$f(z)\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; c\_n\; z^n.$$

Often $z$ is thought of as a complex number, and a suitable notion of convergence of complex sequences is needed. The set of values of $z\backslash in\; \backslash mathbb\{C\}$ for which the series sum converges is a circle, with its radius known as the radius of convergence.

The definition of continuity at a point is given through limits.

The above definition of a limit is true even if $f(c)\; \backslash neq\; L$. Indeed, the function {{math|f}} need not even be defined at {{mvar|c}}. However, if $f(c)$ is defined and is equal to $L$, then the function is said to be continuous at the point $c$.

Equivalently, the function is continuous at $c$ if $f(x)\; \backslash rightarrow\; f(c)$ as $x\; \backslash rightarrow\; c$, or in terms of sequences, whenever $x\_n\; \backslash rightarrow\; c$, then $f(x\_n)\; \backslash rightarrow\; f(c)$.

An example of a limit where $f$ is not defined at $c$ is given below.

Consider the function

$$f(x)\; =\; \backslash frac\{x^2\; -\; 1\}\{x\; -\; 1\}.$$

then {{math|f(1)}} is not defined (see Indeterminate form), yet as {{mvar|x}} moves arbitrarily close to 1, {{math|f(x)}} correspondingly approaches 2:{{Cite web |title=limit {{!}} Definition, Example, & Facts |url=https://www.britannica.com/science/limit-mathematics |access-date=2020-08-18 |website=Encyclopedia Britannica |language=en |archive-date=2021-05-09 |archive-url=https://web.archive.org/web/20210509211046/https://www.britannica.com/science/limit-mathematics |url-status=live }}

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

In other words,

$$\backslash lim\_\{x\; \backslash to\; 1\}\; \backslash frac\{x^2-1\}\{x-1\}\; =\; 2.$$

This can also be calculated algebraically, as $\backslash frac\{x^2-1\}\{x-1\}\; =\; \backslash frac\{(x+1)(x-1)\}\{x-1\}\; =\; x+1$ for all real numbers {{math|x ≠ 1}}.

Now, since {{math|x + 1}} is continuous in {{mvar|x}} at 1, we can now plug in 1 for {{mvar|x}}, leading to the equation

$$\backslash lim\_\{x\; \backslash to\; 1\}\; \backslash frac\{x^2-1\}\{x-1\}\; =\; 1+1\; =\; 2.$$

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

$$f(x)\; =\; \backslash frac\{2x-1\}\{x\}$$

where:

• {{math|1=f(100) = 1.9900}}
• {{math|1=f(1000) = 1.9990}}
• {{math|1=f(10000) = 1.9999}}

As {{mvar|x}} becomes extremely large, the value of {{math|f(x)}} approaches {{math|2}}, and the value of {{math|f(x)}} can be made as close to {{math|2}} as one could wish—by making {{mvar|x}} sufficiently large. So in this case, the limit of {{math|f(x)}} as {{mvar|x}} approaches infinity is {{math|2}}, or in mathematical notation,$$\backslash lim\_\{x\backslash to\backslash infty\}\backslash frac\{2x-1\}\{x\}\; =\; 2.$$

An important class of functions when considering limits are continuous functions. These are precisely those functions which preserve limits, in the sense that if $f$ is a continuous function, then whenever $a\_n\; \backslash rightarrow\; a$ in the domain of $f$, then the limit $f(a\_n)$ exists and furthermore is $f(a)$.

In the most general setting of topological spaces, a short proof is given below:

Let $f:\; X\backslash rightarrow\; Y$ be a continuous function between topological spaces $X$ and $Y$. By definition, for each open set $V$ in $Y$, the preimage $f^\{-1\}(V)$ is open in $X$.

Now suppose $a\_n\; \backslash rightarrow\; a$ is a sequence with limit $a$ in $X$. Then $f(a\_n)$ is a sequence in $Y$, and $f(a)$ is some point.

Choose a neighborhood $V$ of $f(a)$. Then $f^\{-1\}(V)$ is an open set (by continuity of $f$) which in particular contains $a$, and therefore $f^\{-1\}(V)$ is a neighborhood of $a$. By the convergence of $a\_n$ to $a$, there exists an $N$ such that for $n\; >\; N$, we have $a\_n\; \backslash in\; f^\{-1\}(V)$.

Then applying $f$ to both sides gives that, for the same $N$, for each $n\; >\; N$ we have $f(a\_n)\; \backslash in\; V$. Originally $V$ was an arbitrary neighborhood of $f(a)$, so $f(a\_n)\; \backslash rightarrow\; f(a)$. This concludes the proof.

In real analysis, for the more concrete case of real-valued functions defined on a subset $E\; \backslash subset\; \backslash mathbb\{R\}$, that is, $f:\; E\; \backslash rightarrow\; \backslash mathbb\{R\}$, a continuous function may also be defined as a function which is continuous at every point of its domain.

In topology, limits are used to define limit points of a subset of a topological space, which in turn give a useful characterization of closed sets.

In a topological space $X$, consider a subset $S$. A point $a$ is called a limit point if there is a sequence $\backslash \{a\_n\backslash \}$ in $S\backslash backslash\backslash \{a\backslash \}$ such that $a\_n\; \backslash rightarrow\; a$.

The reason why $\backslash \{a\_n\backslash \}$ is defined to be in $S\backslash backslash\backslash \{a\backslash \}$ rather than just $S$ is illustrated by the following example. Take $X\; =\; \backslash mathbb\{R\}$ and $S\; =\; [0,1]\; \backslash cup\; \backslash \{2\backslash \}$. Then $2\; \backslash in\; S$, and therefore is the limit of the constant sequence $2,\; 2,\; \backslash cdots$. But $2$ is not a limit point of $S$.

A closed set, which is defined to be the complement of an open set, is equivalently any set $C$ which contains all its limit points.

{{Main article|derivative}}

The derivative is defined formally as a limit. In the scope of real analysis, the derivative is first defined for real functions $f$ defined on a subset $E\; \backslash subset\; \backslash mathbb\{R\}$. The derivative at $x\; \backslash in\; E$ is defined as follows. If the limit of

$$\backslash frac\{f(x+h)\; -\; f(x)\}\{h\}$$

as $h\; \backslash rightarrow\; 0$ exists, then the derivative at $x$ is this limit.

Equivalently, it is the limit as $y\; \backslash rightarrow\; x$ of

$$\backslash frac\{f(y)\; -\; f(x)\}\{y-x\}.$$

If the derivative exists, it is commonly denoted by $f\text{'}(x)$.

For sequences of real numbers, a number of properties can be proven. Suppose $\backslash \{a\_n\backslash \}$ and $\backslash \{b\_n\backslash \}$ are two sequences converging to $a$ and $b$ respectively.

• Sum of limits is equal to limit of sum

$$a\_n\; +\; b\_n\; \backslash rightarrow\; a\; +\; b.$$

• Product of limits is equal to limit of product

$$a\_n\; \backslash cdot\; b\_n\; \backslash rightarrow\; a\; \backslash cdot\; b.$$

• Inverse of limit is equal to limit of inverse (as long as $a\; \backslash neq\; 0$)

$$\backslash frac\{1\}\{a\_n\}\; \backslash rightarrow\; \backslash frac\{1\}\{a\}.$$

Equivalently, the function $f(x)\; =\; 1/x$ is continuous about nonzero $x$.

{{See also | Cauchy sequence}}

A property of convergent sequences of real numbers is that they are Cauchy sequences. The definition of a Cauchy sequence $\backslash \{a\_n\backslash \}$ is that for every real number $\backslash varepsilon\; >\; 0$, there is an $N$ such that whenever $m,\; n\; >\; N$,

Informally, for any arbitrarily small error $\backslash varepsilon$, it is possible to find an interval of diameter $\backslash varepsilon$ such that eventually the sequence is contained within the interval.

Cauchy sequences are closely related to convergent sequences. In fact, for sequences of real numbers they are equivalent: any Cauchy sequence is convergent.

In general metric spaces, it continues to hold that convergent sequences are also Cauchy. But the converse is not true: not every Cauchy sequence is convergent in a general metric space. A classic counterexample is the rational numbers, $\backslash mathbb\{Q\}$, with the usual distance. The sequence of decimal approximations to $\backslash sqrt\{2\}$, truncated at the $n$th decimal place is a Cauchy sequence, but does not converge in $\backslash mathbb\{Q\}$.

A metric space in which every Cauchy sequence is also convergent, that is, Cauchy sequences are equivalent to convergent sequences, is known as a complete metric space.

One reason Cauchy sequences can be "easier to work with" than convergent sequences is that they are a property of the sequence $\backslash \{a\_n\backslash \}$ alone, while convergent sequences require not just the sequence $\backslash \{a\_n\backslash \}$ but also the limit of the sequence $a$.

Beyond whether or not a sequence $\backslash \{a\_n\backslash \}$ converges to a limit $a$, it is possible to describe how fast a sequence converges to a limit. One way to quantify this is using the order of convergence of a sequence.

A formal definition of order of convergence can be stated as follows. Suppose $\backslash \{a\_n\backslash \}\_\{n\; >\; 0\}$ is a sequence of real numbers which is convergent with limit $a$. Furthermore, $a\_n\; \backslash neq\; a$ for all $n$. If positive constants $\backslash lambda$ and $\backslash alpha$ exist such that

$$\backslash lim\_\{n\; \backslash to\; \backslash infty\; \}\; \backslash frac\{\; \backslash left|\; a\_\{n+1\}\; -\; a\; \backslash right|\; \}\{\; \backslash left|\; a\_n\; -\; a\; \backslash right|\; ^\backslash alpha\; \}\; =\; \backslash lambda$$

then $a\_n$ is said to converge to $a$ with order of convergence $\backslash alpha$. The constant $\backslash lambda$ is known as the asymptotic error constant.

Order of convergence is used for example the field of numerical analysis, in error analysis.

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.{{Cite book |last=Soare |first=Robert I. |url=https://www.worldcat.org/oclc/1154894968 |title=Recursively enumerable sets and degrees : a study of computable functions and computably generated sets |date=2014 |publisher=Springer-Verlag |isbn=978-3-540-66681-3 |location=Berlin |oclc=1154894968}}

There are several theorems or tests that indicate whether the limit exists. These are known as convergence tests. Examples include the ratio test and the squeeze theorem. However they may not tell how to compute the limit.

• Asymptotic analysis: a method of describing limiting behavior
• Big O notation: used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity
• Banach limit defined on the Banach space $\backslash ell^\backslash infty$ that extends the usual limits.
• Convergence of random variables
• Convergent matrix
• Limit in category theory
• Direct limit
• Inverse limit
• Limit of a function
• One-sided limit: either of the two limits of functions of a real variable x, as x approaches a point from above or below
• List of limits: list of limits for common functions
• Squeeze theorem: finds a limit of a function via comparison with two other functions
• Limit superior and limit inferior
• Modes of convergence
• An annotated index

{{reflist}}

• {{ citation | last1 = Apostol | first1 = Tom M. | year = 1974 | lccn = 72011473 | title = Mathematical Analysis | edition = 2nd | publisher = Addison-Wesley | location = Menlo Park }}

{{wikibooks|Calculus|Limits}}

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