Editing Limit of a function (section)

===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}}