Editing Limit of a function (section)

== Other characterizations ==

=== In terms of sequences ===
For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences.  (This definition is usually attributed to [[Eduard Heine]].) In this setting:
<math display=block>\lim_{x\to a}f(x)=L</math>
if, and only if, for all sequences {{mvar|x{{sub|n}}}} (with, for all {{mvar|n}}, {{mvar|x{{sub|n}}}} not equal to {{mvar|a}}) converging to {{mvar|a}} the sequence {{math|''f''(''x{{sub|n}}'')}} converges to {{mvar|L}}.  It was shown by [[Sierpiński]] in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the [[axiom of choice]].  Note that defining what it means for a sequence {{mvar|x{{sub|n}}}} to converge to {{mvar|a}} requires the [[epsilon, delta method]].

Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on [[subset]]s of the real line. Let {{mvar|f}} be a real-valued function with the domain {{math|''Dm''(''f'' )}}.  Let {{mvar|a}} be the limit of a sequence of elements of {{math|''Dm''(''f'' ) \ {''a''}.}}  Then the limit (in this sense) of {{mvar|f}} is {{mvar|L}} as {{mvar|x}} approaches {{mvar|a}}  
if for every sequence {{math|''x{{sub|n}}'' ∈ ''Dm''(''f'' ) \ {''a''} }} (so that for all {{mvar|n}}, {{mvar|x{{sub|n}}}} is not equal to {{mvar|a}}) that converges to {{mvar|a}}, the sequence {{math|''f''(''x{{sub|n}}'')}} converges to {{mvar|L}}.  This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset {{math|''Dm''(''f'' )}} of {{tmath|\R}} as a metric space with the induced metric.

===In non-standard calculus===
In non-standard calculus the limit of a function is defined by:
<math display=block>\lim_{x\to a}f(x)=L</math>
if and only if for all <math>x\in \R^*,</math> <math>f^*(x)-L</math> is infinitesimal whenever {{math|''x'' &minus; ''a''}} is infinitesimal.  Here <math>\R^*</math> are the [[hyperreal number]]s and {{mvar|f*}} is the natural extension of {{mvar|f}} to the non-standard real numbers. [[Howard Jerome Keisler|Keisler]] proved that such a hyperreal [[Non-standard calculus#Limit|definition of limit]] reduces the quantifier complexity by two quantifiers.<ref>{{citation|last1=Keisler|first1=H. Jerome|chapter=Quantifiers in limits|title=Andrzej Mostowski and foundational studies|pages=151–170|publisher=IOS, Amsterdam|year=2008|contribution-url=http://www.math.wisc.edu/~keisler/limquant7.pdf}}</ref> On the other hand, Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the ε-δ method, and claims that, from the pedagogical point of view, the hope that non-standard calculus could be done without ε-δ methods cannot be realized in full.<ref>{{citation|last1=Hrbacek|first1=K.|editor1-first=I.|editor2-last=Neves|editor2-first=V.| chapter=Stratified Analysis?|title=The Strength of Nonstandard Analysis|publisher=Springer|year=2007|editor-last=Van Den Berg}}</ref> 
Bŀaszczyk et al. detail the usefulness of [[microcontinuity]] in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".<ref>{{citation
 | last1 = Bŀaszczyk | first1 = Piotr
 | last2 = Katz | first2 = Mikhail | author2-link = Mikhail Katz
 | last3 = Sherry | first3 = David
 | year = 2012
 | title = Ten misconceptions from the history of analysis and their debunking
 | journal = [[Foundations of Science]]
 | arxiv = 1202.4153
 | doi = 10.1007/s10699-012-9285-8
 | volume = 18
 | issue = 1
 | pages = 43–74
| s2cid = 119134151
 }}</ref>

=== In terms of nearness ===
At the 1908 international congress of mathematics [[Frigyes Riesz|F. Riesz]] introduced an alternate way defining limits and continuity in concept called "nearness".<ref>{{citation|contribution=Stetigkeitsbegriff und abstrakte Mengenlehre (The Concept of Continuity and Abstract Set Theory)|author=F. Riesz|date=7 April 1908|title=[[International Congress of Mathematicians|1908 International Congress of Mathematicians]]}}</ref> A point {{mvar|x}} is defined to be near a set <math>A\subseteq \R</math> if for every {{math|''r'' > 0}} there is a point {{math|''a'' &in; ''A''}} so that {{math|{{abs|''x'' &minus; ''a''}} < ''r''}}.  In this setting the 
<math display=block>\lim_{x\to a} f(x)=L</math>
if and only if for all <math>A\subseteq \R,</math> {{mvar|L}} is near {{math|''f''(''A'')}} whenever {{mvar|a}} is near {{mvar|A}}.
Here {{math|''f''(''A'')}} is the set <math>\{f(x) | x \in A\}.</math>  This definition can also be extended to metric and topological spaces.