Editing Limit of a function (section)

===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>