Editing Limit of a function (section)

==Functions on metric spaces==
Suppose {{mvar|M}} and {{mvar|N}} are subsets of [[metric spaces]] {{mvar|A}} and {{mvar|B}}, respectively, and {{math|''f'' : ''M'' → ''N''}} is defined between {{mvar|M}} and {{mvar|N}}, with {{math|''x'' ∈ ''M''}}, {{mvar|p}} a [[limit point]] of {{mvar|M}} and {{math|''L'' ∈ ''N''}}. It is said that '''the limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|L}}''' and write

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

if the following property holds:

{{block indent|For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that for all points {{math|''x'' ∈ ''M''}}, {{math|0 < ''d<sub>A</sub>''(''x'', ''p'') < ''δ''}} implies {{math|''d<sub>B</sub>''(''f''(''x''), ''L'') < ''ε''}}.<ref>{{citation
 | last = Rudin | first = W.
 | url = http://worldcat.org/oclc/962920758
 | title = Principles of mathematical analysis
 | date = 1986
 | publisher = McGraw - Hill Book C
 | pages = 84
 | oclc = 962920758}}</ref>}}
<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in M) \,(0 < d_A(x, p) < \delta \implies d_B(f(x), L) < \varepsilon).</math>

Again, note that {{mvar|p}} need not be in the domain of {{mvar|f}}, nor does {{mvar|L}} need to be in the range of {{mvar|f}}, and even if {{math|''f''(''p'')}} is defined it need not be equal to {{mvar|L}}.

===Euclidean metric===
The limit in [[Euclidean space]] is a direct generalization of limits to [[vector-valued functions]]. For example, we may consider a function <math>f:S \times T \to \R^3</math> such that
<math display=block>f(x, y) = (f_1(x, y), f_2(x, y), f_3(x, y) ).</math>
Then, under the usual [[Euclidean metric]],
<math display=block>\lim_{(x, y) \to (p, q)} f(x, y) = (L_1, L_2, L_3)</math>
if the following 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}}, <math display=inline>0 < \sqrt{(x-p)^2 + (y-q)^2} < \delta</math> implies <math display=inline>\sqrt{(f_1-L_1)^2 + (f_2-L_2)^2 + (f_3-L_3)^2} < \varepsilon.</math><ref name="Hartman">{{citation
 | last = Hartman | first = Gregory
 | url = https://math.libretexts.org/Courses/Georgia_State_University_-_Perimeter_College/MATH_2215%3A_Calculus_III/13%3A_Vector-valued_Functions/The_Calculus_of_Vector-Valued_Functions_II
 | date = 2019
 | title = The Calculus of Vector-Valued Functions II
 | language = en
 | access-date = 2022-10-31}}</ref>}}
<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in S) \, (\forall y \in T)\, \left(0 < \sqrt{(x-p)^2+(y-q)^2} < \delta \implies \sqrt{(f_1-L_1)^2 + (f_2-L_2)^2 + (f_3-L_3)^2} < \varepsilon \right).</math>

In this example, the function concerned are finite-[[Dimension (vector space)|dimension]] vector-valued function. In this case, the '''limit theorem for vector-valued function''' states that if the limit of each component exists, then the limit of a vector-valued function equals the vector with each component taken the limit:<ref name="Hartman" />
<math display=block>\lim_{(x, y) \to (p, q)} \Bigl(f_1(x, y), f_2(x, y), f_3(x, y)\Bigr) = \left(\lim_{(x, y) \to (p, q)}f_1(x, y), \lim_{(x, y) \to (p, q)}f_2(x, y), \lim_{(x, y) \to (p, q)}f_3(x, y)\right).</math>

===Manhattan metric===

One might also want to consider spaces other than Euclidean space. An example would be the Manhattan space. Consider <math>f:S \to \R^2</math> such that
<math display=block>f(x) = (f_1(x), f_2(x)).</math>
Then, under the [[Manhattan metric]],
<math display=block>\lim_{x \to p} f(x) = (L_1, L_2)</math>
if the following holds:

{{block indent|For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that for all {{mvar|x}} in {{mvar|S}}, {{math|0 < {{!}}''x'' − ''p''{{!}} < ''δ''}} implies {{math|{{!}}''f''<sub>1</sub> − ''L''<sub>1</sub>{{!}} + {{!}}''f''<sub>2</sub> − ''L''<sub>2</sub>{{!}} < ''ε''}}.}}
<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in S) \,(0 < |x - p| < \delta \implies |f_1 - L_1| + |f_2 - L_2| < \varepsilon).</math>

Since this is also a finite-dimension vector-valued function, the limit theorem stated above also applies.{{sfnp|Zakon|2011|p=172}}

===Uniform metric===

Finally, we will discuss the limit in [[function space]], which has infinite dimensions. Consider a function {{math|''f''(''x'', ''y'')}} in the function space <math>S \times T \to \R.</math> We want to find out as {{mvar|x}} approaches {{mvar|p}}, how {{math|''f''(''x'', ''y'')}} will tend to another function {{math|''g''(''y'')}}, which is in the function space <math>T \to \R.</math> The "closeness" in this function space may be measured under the [[uniform metric]].<ref>{{citation
 | last = Rudin | first = W
 | url = http://worldcat.org/oclc/962920758
 | title = Principles of mathematical analysis
 | date = 1986
 | publisher = McGraw - Hill Book C
 | pages = 150–151
 | oclc = 962920758}}</ref> Then, we will say '''the uniform limit of {{mvar|f}} on {{mvar|T}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|g}}''' and write
<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>

if the following holds:

{{block indent|For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that for all {{mvar|x}} in {{mvar|S}}, {{math|0 < {{!}}''x'' − ''p''{{!}} < ''δ''}} implies <math>\sup_{y \in T}|f(x,y) - g(y)| < \varepsilon.</math>}}
<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in S) \,(0 < |x-p| < \delta \implies \sup_{y \in T} | f(x, y) - g(y) | < \varepsilon).</math>

In fact, one can see that this definition is equivalent to that of the uniform limit of a multivariable function introduced in the previous section.