Editing Limit of a function (section)

===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>