Editing Limit of a function (section)

===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>