Editing Limit of a function (section)

===Multiple limits===

Although less commonly used, there is another type of limit for a multivariable function, known as the '''multiple limit'''. For a two-variable function, this is the '''double limit'''.<ref name="Zakon_219">{{citation
 | last = Zakon |first = Elias
 | chapter = Chapter 4. Function Limits and Continuity
 | pages = 219–220
 | title = Mathematical Anaylysis, Volume I
 | year = 2011
 |publisher = University of Windsor
 | isbn = 9781617386473}}</ref> Let <math>f : S \times T \to \R</math> be defined on <math>S \times T \subseteq \R^2,</math> we say '''the double limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} and {{mvar|y}} approaches {{mvar|q}} is {{mvar|L}}''', written

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

if the following condition holds:
{{block indent| For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that for all {{mvar|x}} in {{mvar|S}} and {{mvar|y}} in {{mvar|T}}, whenever {{math|0 < {{abs|''x'' − ''p''}} < ''δ''}} and {{math|0 < {{abs|''y'' − ''q''}} < ''δ''}}, we have {{math|{{abs|''f''(''x'', ''y'') − ''L''}} < ''ε''}}.<ref name="Zakon_219" />}}
<math display=block>(\forall \varepsilon > 0)\, (\exists \delta > 0)\, (\forall x \in S) \, (\forall y \in T)\, ( (0 < |x-p| < \delta) \land (0 < |y-q| < \delta) \implies |f(x, y) - L| < \varepsilon) .</math>

For such a double limit to exist, this definition requires the value of {{mvar|f}} approaches {{mvar|L}} along every possible path approaching {{math|(''p'', ''q'')}}, excluding the two lines {{math|1=''x'' = ''p''}} and {{math|1=''y'' = ''q''}}. As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equals {{mvar|L}}, then the multiple limit exists and also equals {{mvar|L}}. The converse is not true: the existence of the multiple limits does not imply the existence of the ordinary limit. Consider the example
<math display=block>f(x,y) = \begin{cases}
1 \quad \text{for} \quad xy \ne 0 \\
0 \quad \text{for} \quad xy = 0
\end{cases}</math>
where
<math display=block> \lim_{ {x \to 0} \atop {y \to 0} } f(x, y) = 1 </math>
but 
<math display=block>\lim_{(x, y) \to (0, 0)} f(x, y)</math> does not exist.

If the domain of {{mvar|f}} is restricted to <math>(S\setminus\{p\}) \times (T\setminus\{q\}),</math> then the two definitions of limits coincide.<ref name="Zakon_219" />