Editing Limit of a function (section)

==Functions of more than one variable==

===Ordinary limits===
By noting that {{math|{{abs|''x'' − ''p''}}}} represents a [[distance]], the definition of a limit can be extended to functions of more than one variable. In the case of a function <math>f : S \times T \to \R</math> defined on <math>S \times T \subseteq \R^2,</math> we defined the limit as follows: '''the limit of {{mvar|f}} as {{math|(''x'', ''y'')}} approaches {{math|(''p'', ''q'')}} is {{mvar|L}}''', written

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

if the following condition holds:
: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 display=inline>0 < \sqrt{(x-p)^2 + (y-q)^2} < \delta,</math> we have {{math|{{abs|''f''(''x'', ''y'') − ''L''}} < ''ε''}},<ref>{{citation
 | last = Stewart | first = James |author-link = James Stewart (mathematician)
 | chapter = Chapter 14.2 Limits and Continuity
 | pages = 952
 | title = Multivariable Calculus
 | year = 2020
 | publisher = Cengage Learning | edition = 9th
 | isbn = 9780357042922}}</ref>

or formally:
<math display=block>(\forall \varepsilon > 0)\, (\exists \delta > 0)\, (\forall x \in S) \, (\forall y \in T)\, (0 < \sqrt{(x-p)^2 + (y-q)^2} < \delta \implies |f(x, y) - L| < \varepsilon)).</math>

Here <math display=inline>\sqrt{(x-p)^2 + (y-q)^2}</math> is the [[Euclidean distance]] between {{math|(''x'', ''y'')}} and {{math|(''p'', ''q'')}}. (This can in fact be replaced by any [[Norm (mathematics)|norm]] {{math|{{abs|{{abs|(''x'', ''y'') − (''p'', ''q'')}}}}}}, and be extended to any number of variables.)

For example, we may say
<math display=block> \lim_{(x,y) \to (0, 0)} \frac{x^4}{x^2+y^2} = 0 </math>
because for every {{math|''ε'' > 0}}, we can take <math display=inline>\delta = \sqrt \varepsilon</math> such that for all real {{math|''x'' ≠ 0}} and real {{math|''y'' ≠ 0}}, if <math display=inline>0 < \sqrt{(x-0)^2 + (y-0)^2} < \delta,</math> then {{math|{{abs|''f''(''x'', ''y'') − 0}} < ''ε''}}.

Similar to the case in single variable, the value of {{mvar|f}} at {{math|(''p'', ''q'')}} does not matter in this definition of limit.

For such a multivariable limit to exist, this definition requires the value of {{mvar|f}} approaches {{mvar|L}} along every possible path approaching {{math|(''p'', ''q'')}}.{{sfnp|Stewart|2020|p=953}} In the above example, the function
<math display=block>f(x, y) = \frac{x^4}{x^2+y^2}</math>
satisfies this condition. This can be seen by considering the [[polar coordinates]] 
<math display=block>(x,y) = (r\cos\theta, r\sin\theta) \to (0, 0),</math>
which gives
<math display=block>\lim_{r \to 0} f(r \cos \theta, r \sin \theta) = \lim_{r \to 0} \frac{r^4 \cos^4 \theta}{r^2} = \lim_{r \to 0} r^2 \cos^4 \theta.</math>
Here {{math|1=''θ'' = ''θ''(''r'')}} is a function of ''r'' which controls the shape of the path along which {{mvar|f}} is approaching {{math|(''p'', ''q'')}}. Since {{math|cos ''θ''}} is bounded between [−1, 1], by the [[sandwich theorem]], this limit tends to 0.

In contrast, the function
<math display=block>f(x, y) = \frac{xy}{x^2 + y^2}</math>
does not have a limit at {{math|(0, 0)}}. Taking the path {{math|1=(''x'', ''y'') = (''t'', 0) → (0, 0)}}, we obtain
<math display=block>\lim_{t \to 0} f(t, 0) = \lim_{t \to 0} \frac{0}{t^2} = 0,</math>
while taking the path {{math|1=(''x'', ''y'') = (''t'', ''t'') → (0, 0)}}, we obtain
<math display=block>\lim_{t \to 0} f(t, t) = \lim_{t \to 0} \frac{t^2}{t^2 + t^2} = \frac{1}{2}.</math>

Since the two values do not agree, {{mvar|f}} does not tend to a single value as {{math|(''x'', ''y'')}} approaches {{math|(0, 0)}}.

===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" />

===Multiple limits at infinity===

The concept of multiple limit can extend to the limit at infinity, in a way similar to that of a single variable function. For <math>f : S \times T \to \R,</math> we say '''the double limit of {{mvar|f}} as {{mvar|x}} and {{mvar|y}} approaches infinity is {{mvar|L}}''', written
<math display=block> \lim_{ {x \to \infty} \atop {y \to \infty} } f(x, y) = L </math>

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

We say '''the double limit of {{mvar|f}} as {{mvar|x}} and {{mvar|y}} approaches minus infinity is {{mvar|L}}''', written
<math display=block> \lim_{ {x \to -\infty} \atop {y \to -\infty} } f(x, y) = L </math>

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

===Pointwise limits and uniform limits===

{{Main|Pointwise convergence|Uniform convergence}}

Let <math>f : S \times T \to \R.</math> Instead of taking limit as {{math|(''x'', ''y'') → (''p'', ''q'')}}, we may consider taking the limit of just one variable, say, {{math|''x'' → ''p''}}, to obtain a single-variable function of {{mvar|y}}, namely <math>g : T \to \R.</math> In fact, this limiting process can be done in two distinct ways. The first one is called '''pointwise limit'''. We say '''the pointwise limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|g}}''', denoted
<math display=block>\lim_{x\to p}f(x, y) = g(y),</math> or
<math display=block>\lim_{x \to p}f(x, y) = g(y) \;\; \text{pointwise}.</math>

Alternatively, we may say '''{{mvar|f}} tends to {{mvar|g}} pointwise as {{mvar|x}} approaches {{mvar|p}}''', denoted
<math display=block>f(x, y) \to g(y) \;\; \text{as} \;\; x \to p,</math> or
<math display=block>f(x, y) \to g(y) \;\; \text{pointwise} \;\; \text{as} \;\; x \to p.</math>

This limit exists if the following holds:
{{block indent| For every {{math|''ε'' > 0}} and every fixed {{mvar|y}} in {{mvar|T}}, there exists a {{math|''δ''(''ε'', ''y'') > 0}} such that for all {{mvar|x}} in {{mvar|S}}, whenever {{math|0 < {{abs|''x'' − ''p''}} < ''δ''}}, we have {{math|{{abs|''f''(''x'', ''y'') − ''g''(''y'')}} < ''ε''}}.{{sfnp|Zakon|2011|p=220}}}}
<math display=block>(\forall \varepsilon > 0)\, (\forall y \in T) \, (\exists \delta> 0)\, (\forall x \in S)\, ( 0 < |x-p| < \delta \implies |f(x, y) - g(y)| < \varepsilon) .</math>

Here, {{math|1=''δ'' = ''δ''(''ε'', ''y'')}} is a function of both {{mvar|ε}} and {{mvar|y}}. Each {{mvar|δ}} is chosen for a ''specific point'' of {{mvar|y}}. Hence we say the limit is pointwise in {{mvar|y}}. For example,
<math display=block>f(x, y) = \frac{x}{\cos y}</math>
has a pointwise limit of constant zero function
<math display=block>\lim_{x \to 0}f(x, y) = 0(y) \;\; \text{pointwise}</math>
because for every fixed {{mvar|y}}, the limit is clearly 0. This argument fails if {{mvar|y}} is not fixed: if {{mvar|y}} is very close to {{math|''π''/2}}, the value of the fraction may deviate from 0.

This leads to another definition of limit, namely the '''uniform limit'''. We say '''the uniform limit of {{mvar|f}} on {{mvar|T}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|g}}''', denoted
<math display=block>\underset{ {x \to p} \atop {y \in T} }{\mathrm{unif} \lim \;} f(x, y) = g(y),</math> or
<math display=block>\lim_{x \to p}f(x, y) = g(y) \;\; \text{uniformly on} \; T.</math>

Alternatively, we may say '''{{mvar|f}} tends to {{mvar|g}} uniformly on {{mvar|T}} as {{mvar|x}} approaches {{mvar|p}}''', denoted
<math display=block>f(x, y) \rightrightarrows g(y) \; \text{on} \; T \;\; \text{as} \;\; x \to p,</math> or
<math display=block>f(x, y) \to g(y) \;\; \text{uniformly on}\; T \;\; \text{as} \;\; x \to p.</math>

This limit exists if the following holds:
{{block indent| For every {{math|''ε'' > 0}}, there exists a {{math|''δ''(''ε'') > 0}} such that for all {{mvar|x}} in {{mvar|S}} and {{mvar|y}} in {{mvar|T}}, whenever {{math|0 < {{abs|''x'' − ''p''}} < ''δ''}}, we have {{math|{{abs|''f''(''x'', ''y'') − ''g''(''y'')}} < ''ε''}}.{{sfnp|Zakon|2011|p=220}}}}
<math display=block>(\forall \varepsilon > 0) \, (\exists \delta > 0)\, (\forall x \in S)\, (\forall y \in T)\, ( 0 < |x-p| < \delta \implies |f(x, y) - g(y)| < \varepsilon) .</math>

Here, {{math|1=''δ'' = ''δ''(''ε'')}} is a function of only {{mvar|ε}} but not {{mvar|y}}. In other words, ''δ'' is ''uniformly applicable'' to all {{mvar|y}} in {{mvar|T}}. Hence we say the limit is uniform in {{mvar|y}}. For example,
<math display=block>f(x, y) = x \cos y</math>
has a uniform limit of constant zero function
<math display=block>\lim_{x \to 0}f(x, y) = 0(y) \;\; \text{ uniformly on}\; \R</math>
because for all real {{mvar|y}}, {{math|cos ''y''}} is bounded between {{math|[−1, 1]}}. Hence no matter how {{mvar|y}} behaves, we may use the [[sandwich theorem]] to show that the limit is 0.

===Iterated limits===

{{Main|Iterated limits}}

Let <math>f : S \times T \to \R.</math> We may consider taking the limit of just one variable, say, {{math|''x'' → ''p''}}, to obtain a single-variable function of {{mvar|y}}, namely <math>g : T \to \R,</math> and then take limit in the other variable, namely {{math|''y'' → ''q''}}, to get a number {{mvar|L}}. Symbolically,
<math display=block>\lim_{y \to q} \lim_{x \to p} f(x, y) = \lim_{y \to q} g(y) = L.</math>

This limit is known as '''iterated limit''' of the multivariable function.{{sfnp|Zakon|2011|p=223}} The order of taking limits may affect the result, i.e.,

<math display=block>\lim_{y \to q} \lim_{x \to p} f(x,y) \ne \lim_{x \to p} \lim_{y \to q} f(x, y)</math> in general.

A sufficient condition of equality is given by the [[Moore-Osgood theorem]], which requires the limit <math>\lim_{x \to p}f(x, y) = g(y)</math> to be uniform on {{mvar|T}}.<ref>{{citation
 | last1 = Taylor | first1 = Angus E.
 | title = General Theory of Functions and Integration
 | year = 2012
 | publisher = Dover Books on Mathematics Series
 | isbn = 9780486152141
 | pages = 139–140}}</ref>