Editing Limit of a function (section)

=== {{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.