Editing Limit of a function (section)

==Functions of a single variable==
=== {{math|1=(''ε'', ''δ'')}}-definition of limit ===
[[File:Epsilon-delta limit.svg|thumb|For the depicted {{mvar|f}}, {{mvar|a}}, and {{mvar|b}}, we can ensure that the value {{math|''f''(''x'')}} is within an arbitrarily small interval {{math|(''b'' – ε, ''b'' + ε)}} by restricting {{mvar|x}} to a sufficiently small interval {{math|(''a'' – δ, ''a'' + δ).}} Hence {{math|''f''(''x'') → ''b''}} as {{math|''x'' → ''a''}}.]]
Suppose <math>f: \R \rightarrow \R</math> is a function defined on the [[real line]], and there are two real numbers {{mvar|p}} and {{mvar|L}}. One would say: '''The limit of {{mvar|f}} of {{mvar|x}}, as {{mvar|x}} approaches {{mvar|p}}, exists, and it equals {{mvar|L}}'''. and write,<ref name="swokowski">{{citation
 | last = Swokowski | first = Earl W.
 | title = Calculus with Analytic Geometry
 | url = https://books.google.com/books?id=gJlAOiCZRnwC&pg=PA58
 | year = 1979
 | edition = 2nd
 | publisher = Taylor & Francis
 | page = 58| isbn = 978-0-87150-268-1
 }}</ref>

<math display=block> \lim_{x \to p} f(x) = L,</math>

or alternatively, say '''{{math|''f''(''x'')}} tends to {{mvar|L}} as {{mvar|x}} tends to {{mvar|p}}''', and write,

<math display=block> f(x) \to L \text{ as } x \to p,</math>

if the following property holds: for every real {{math|''ε'' > 0}}, there exists a real {{math|''δ'' > 0}} such that for all real {{mvar|x}}, {{math| 0 < {{!}}''x'' − ''p''{{!}} < ''δ''}} implies {{math|{{!}}''f''(''x'') − ''L''{{!}} < ''ε''}}.<ref name="swokowski" /> Symbolically,
<math display=block>(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in \R) \, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

For example, we may say
<math display=block>\lim_{x \to 2} (4x + 1) = 9</math>
because for every real {{math|''ε'' > 0}}, we can take {{math|1=''δ'' = ''ε''/4}}, so that for all real {{mvar|x}}, if {{math|0 < {{abs|''x'' − 2}} < ''δ''}}, then {{math|{{abs|4''x'' + 1 − 9}} < ''ε''}}.

A more general definition applies for functions defined on [[subset]]s of the real line. Let {{mvar|S}} be a subset of {{tmath|\R.}} Let <math>f: S \to \R</math> be a [[real-valued function]]. Let {{mvar|p}} be a point such that there exists some open interval {{math|(''a'', ''b'')}} containing {{mvar|p}} with <math>(a,p)\cup (p,b) \subset S.</math> It is then said that the limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|L}}, if:

{{block indent|For every real {{math|''ε'' > 0}}, there exists a real {{math|''δ'' > 0}} such that for all {{math|''x'' ∈ (''a'', ''b'')}}, {{math|0 < {{!}}''x'' − ''p''{{!}} < ''δ''}}  implies that {{math|{{!}}''f''(''x'') − ''L''{{!}} < ''ε''}}.}}

Or, symbolically:
<math display=block>(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in (a, b)) \, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

For example, we may say
<math display=block>\lim_{x \to 1} \sqrt{x+3} = 2</math>
because for every real {{math|''ε'' > 0}}, we can take {{math|1=''δ'' = ''ε''}}, so that for all real {{math|''x'' ≥ −3}}, if {{math|0 < {{abs|''x'' − 1}} < ''δ''}}, then {{math|{{abs|''f''(''x'') − 2}} < ''ε''}}. In this example, {{math|1=''S'' = [−3, ∞)}} contains open intervals around the point 1 (for example, the interval (0, 2)).

Here, note that the value of the limit does not depend on {{mvar|f}} being defined at {{mvar|p}}, nor on the value {{math|''f''(''p'')}}—if it is defined. For example, let <math>f: [0,1)\cup (1,2] \to \R, f(x) =  \tfrac{2x^2 - x - 1}{x-1}.</math>
<math display=block>\lim_{x \to 1} f(x) = 3</math>
because for every {{math|''ε'' > 0}}, we can take {{math|1=''δ'' = ''ε''/2}}, so that for all real {{math|''x'' ≠ 1}}, if {{math|0 < {{abs|''x'' − 1}} < ''δ''}}, then {{math|{{abs|''f''(''x'') − 3}} < ''ε''}}. Note that here {{math|''f''(1)}} is undefined.

In fact, a limit can exist in <math>\{p\in \R\, |\, \exists (a,b) \subset \R : \, p \in (a,b) \text{ and } (a,p)\cup (p, b) \subset S\},</math> which equals <math>\operatorname{int} S \cup \operatorname{iso} S^c,</math> where {{math|int ''S''}} is the [[interior (topology)|interior]] of {{mvar|S}}, and {{math|iso ''S{{sup|c}}''}} are the [[isolated point]]s of the complement of {{mvar|S}}. In our previous example where <math>S = [0,1) \cup (1,2],</math> <math>\operatorname{int} S = (0,1) \cup (1,2),</math> <math>\operatorname{iso} S^c = \{1\}.</math> We see, specifically, this definition of limit allows a limit to exist at 1, but not 0 or 2.

The letters {{mvar|ε}} and {{mvar|δ}} can be understood as "error" and "distance". In fact, Cauchy used {{mvar|ε}} as an abbreviation for "error" in some of his work,<ref name="Grabiner1983" /> though in his definition of continuity, he used an infinitesimal <math>\alpha</math> rather than either {{mvar|ε}} or {{mvar|δ}} (see ''[[Cours d'Analyse]]''). In these terms, the error (''ε'') in the measurement of the value at the limit can be made as small as desired, by reducing the distance (''δ'') to the limit point. As discussed below, this definition also works for functions in a more general context. The idea that {{mvar|δ}} and {{mvar|ε}} represent distances helps suggest these generalizations.

===Existence and one-sided limits===
{{Main|One-sided limit}}
[[Image:Upper semi.svg|thumb|The limit as <math>x \to x_0^+</math> differs from that as <math>x \to x_0^-.</math> Therefore, the limit as {{math|''x'' → ''x''<sub>0</sub>}} does not exist.]]

Alternatively, {{mvar|x}} may approach {{mvar|p}} from above (right) or below (left), in which case the limits may be written as

<math display=block> \lim_{x \to p^+}f(x) = L </math>

or

<math display=block> \lim_{x \to p^-}f(x) = L </math>
[[File:Undefined limit examples.png|thumb|The first three functions have points for which the limit does not exist, while the function<math display="block"> f(x) = \frac{\sin(x)}{x} </math>is not defined at <math>x = 0</math>, but its limit does exist.]]
respectively. If these limits exist at p and are equal there, then this can be referred to as ''the'' limit of {{math|''f''(''x'')}} at {{mvar|p}}.{{sfnp|Swokowski|1979|p=72–73}} If the one-sided limits exist at {{mvar|p}}, but are unequal, then there is no limit at {{mvar|p}} (i.e., the limit at {{mvar|p}} does not exist). If either one-sided limit does not exist at {{mvar|p}}, then the limit at {{mvar|p}} also does not exist.

A formal definition is as follows. The '''limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} from above is {{mvar|L}}''' if:
:For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that whenever {{math|0 < ''x'' − ''p'' < ''δ''}}, we have {{math|{{abs|''f''(''x'') − ''L''}} < ''ε''}}.
<math display=block>(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in (a,b))\, (0 < x - p < \delta \implies |f(x) - L| < \varepsilon).</math>

The '''limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} from below is {{mvar|L}}''' if:
:For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that whenever {{math|0 < ''p'' − ''x'' < ''δ''}}, we have {{math|{{abs|''f''(''x'') − ''L''}} < ''ε''}}.
<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in (a,b)) \, (0 < p - x < \delta \implies |f(x) - L| < \varepsilon).</math>

If the limit does not exist, then the [[Oscillation of a function at a point|oscillation]] of {{mvar|f}} at {{mvar|p}} is non-zero.

===More general definition using limit points and subsets===
{{further|Limit point}}
Limits can also be defined by approaching from subsets of the domain.

In general:<ref>{{harv|Bartle|Sherbert|2000}}</ref> Let <math>f : S \to \R</math> be a real-valued function defined on some <math>S \subseteq \R.</math> Let {{mvar|p}} be a [[limit point]] of some <math>T \subset S</math>&mdash;that is, {{mvar|p}} is the limit of some sequence of elements of {{mvar|T}} distinct from {{mvar|p}}.  Then we say '''the limit of {{mvar|f}}, as {{mvar|x}} approaches {{mvar|p}} from values in {{mvar|T}}, is {{mvar|L}}''', written
<math display=block>\lim_{ {x \to p} \atop {x \in T} } f(x) = L</math>
if the following holds:
{{block indent|For every {{math|''ε'' > ''0''}}, there exists a {{math|''δ'' > ''0''}} such that for all {{math|''x'' ∈ ''T''}}, {{math|0 < {{abs|''x'' − ''p''}} < ''δ''}} implies that {{math|{{abs|''f''(''x'') − ''L''}} < ''ε''}}.}}
<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in T)\, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

Note, {{mvar|T}} can be any subset of {{mvar|S}}, the domain of {{mvar|f}}. And the limit might depend on the selection of {{mvar|T}}. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking {{mvar|T}} to be an open interval of the form {{math|(–∞, ''a'')}}), and right-handed limits (e.g., by taking {{mvar|T}} to be an open interval of the form {{math|(''a'', ∞)}}). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the [[square root function]] <math>f(x) = \sqrt x</math> can have limit 0 as {{mvar|x}} approaches 0 from above:
<math display=block>\lim_{ {x\to 0} \atop {x\in [0, \infty)} } \sqrt{x} = 0</math>
since for every {{math|''ε'' > 0}}, we may take {{math|1=''δ'' = ''ε''{{sup|2}}}} such that for all {{math|''x'' ≥ 0}}, if {{math|0 < {{abs|''x'' − 0}} < ''δ''}}, then {{math|{{abs|''f''(''x'') − 0}} < ''ε''}}.

This definition allows a limit to be defined at limit points of the domain {{mvar|S}}, if a suitable subset {{mvar|T}} which has the same limit point is chosen.

Notably, the previous two-sided definition works on <math>\operatorname{int} S \cup \operatorname{iso} S^c,</math> which is a subset of the limit points of {{mvar|S}}.

For example, let <math>S = [0,1)\cup (1, 2].</math> The previous two-sided definition would work at <math>1 \in \operatorname{iso} S^c = \{1\},</math> but it wouldn't work at 0 or 2, which are limit points of {{mvar|S}}.

===Deleted versus non-deleted limits===
The definition of limit given here does not depend on how (or whether) {{mvar|f}} is defined at {{mvar|p}}. Bartle<ref name="Bartle 1967">{{harvtxt|Bartle|1967}}</ref> refers to this as a ''deleted limit'', because it excludes the value of {{mvar|f}} at {{mvar|p}}. The corresponding '''non-deleted limit''' does depend on the value of {{mvar|f}} at {{mvar|p}}, if {{mvar|p}} is in the domain of {{mvar|f}}. Let <math>f : S \to \R</math> be a real-valued function. '''The non-deleted limit of {{mvar|f}}, as {{mvar|x}} approaches {{mvar|p}}, is {{mvar|L}}''' if

{{block indent|For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that for all {{math|''x'' ∈ ''S''}}, {{math|{{!}}''x'' − ''p''{{!}} < ''δ''}} implies {{math|{{!}}''f''(''x'') − ''L''{{!}} < ''ε''}}.}}
<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in S)\, (|x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

The definition is the same, except that the neighborhood {{math|{{!}}''x'' − ''p''{{!}} < ''δ''}} now includes the point {{mvar|p}}, in contrast to the [[deleted neighborhood]] {{math|0 < {{!}}''x'' − ''p''{{!}} < ''δ''}}. This makes the definition of a non-deleted limit less general. One of the advantages of working with non-deleted limits is that they allow to state the [[#Limits of compositions of functions|theorem about limits of compositions]] without any constraints on the functions (other than the existence of their non-deleted limits).<ref>{{harvtxt|Hubbard|2015}}</ref>

Bartle<ref name="Bartle 1967"/> notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular.<ref>For example, {{harvtxt|Apostol|1974}}, {{harvtxt|Courant|1924}}, {{harvtxt|Hardy|1921}}, {{harvtxt|Rudin|1964}}, {{harvtxt|Whittaker|Watson|1904}} all take "limit" to mean the deleted limit.</ref>

===Examples===

====Non-existence of one-sided limit(s)====
[[Image:Discontinuity essential.svg|thumb|right|Function without a limit at an [[Classification of discontinuities|essential discontinuity]] ]]
The function  
<math display=block>f(x)=\begin{cases}
\sin\frac{5}{x-1} & \text{ for } x<1 \\
0 & \text{ for } x=1 \\[2pt]
\frac{1}{10x-10}& \text{ for } x>1
\end{cases}</math>
has no limit at {{math|1=''x''{{sub|0}} = 1}} (the left-hand limit does not exist due to the oscillatory nature of the sine function, and the right-hand limit does not exist due to the asymptotic behaviour of the reciprocal function, see picture), but has a limit at every other {{mvar|x}}-coordinate.

The function 
<math display=block>f(x)=\begin{cases}
1 &  x \text{ rational } \\
0 &  x \text{ irrational }
\end{cases}</math>
(a.k.a., the [[Dirichlet function]]) has no limit at any {{mvar|x}}-coordinate.

====Non-equality of one-sided limits====
The function 
<math display=block>f(x)=\begin{cases}
1 & \text{ for } x < 0 \\
2 &  \text{ for } x \ge 0
\end{cases}</math>
has a limit at every non-zero {{mvar|x}}-coordinate (the limit equals 1 for negative {{mvar|x}} and equals 2 for positive {{mvar|x}}). The limit at {{math|1=''x'' = 0}} does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).

====Limits at only one point====
The functions
<math display=block>f(x)=\begin{cases}
x &  x \text{ rational } \\
0 &  x \text{ irrational }
\end{cases}</math>
and
<math display=block>f(x)=\begin{cases}
|x| &  x \text{ rational } \\
0 &  x \text{ irrational }
\end{cases}</math>
both have a limit at {{math|1=''x'' = 0}} and it equals 0.

====Limits at countably many points====
The function 
<math display=block>f(x)=\begin{cases}
\sin x &  x \text{ irrational } \\
 1     &  x \text{ rational }
\end{cases}</math>
has a limit at any {{mvar|x}}-coordinate of the form <math>\tfrac{\pi}{2} + 2n\pi,</math> where {{mvar|n}} is any integer.