Editing Limit of a function (section)

===More general definition using limit points and subsets===
{{further|Limit point}}
Limits can also be defined by approaching from subsets of the domain.

In general:<ref>{{harv|Bartle|Sherbert|2000}}</ref> Let <math>f : S \to \R</math> be a real-valued function defined on some <math>S \subseteq \R.</math> Let {{mvar|p}} be a [[limit point]] of some <math>T \subset S</math>&mdash;that is, {{mvar|p}} is the limit of some sequence of elements of {{mvar|T}} distinct from {{mvar|p}}.  Then we say '''the limit of {{mvar|f}}, as {{mvar|x}} approaches {{mvar|p}} from values in {{mvar|T}}, is {{mvar|L}}''', written
<math display=block>\lim_{ {x \to p} \atop {x \in T} } f(x) = L</math>
if the following holds:
{{block indent|For every {{math|''ε'' > ''0''}}, there exists a {{math|''δ'' > ''0''}} such that for all {{math|''x'' ∈ ''T''}}, {{math|0 < {{abs|''x'' − ''p''}} < ''δ''}} implies that {{math|{{abs|''f''(''x'') − ''L''}} < ''ε''}}.}}
<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in T)\, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

Note, {{mvar|T}} can be any subset of {{mvar|S}}, the domain of {{mvar|f}}. And the limit might depend on the selection of {{mvar|T}}. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking {{mvar|T}} to be an open interval of the form {{math|(–∞, ''a'')}}), and right-handed limits (e.g., by taking {{mvar|T}} to be an open interval of the form {{math|(''a'', ∞)}}). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the [[square root function]] <math>f(x) = \sqrt x</math> can have limit 0 as {{mvar|x}} approaches 0 from above:
<math display=block>\lim_{ {x\to 0} \atop {x\in [0, \infty)} } \sqrt{x} = 0</math>
since for every {{math|''ε'' > 0}}, we may take {{math|1=''δ'' = ''ε''{{sup|2}}}} such that for all {{math|''x'' ≥ 0}}, if {{math|0 < {{abs|''x'' − 0}} < ''δ''}}, then {{math|{{abs|''f''(''x'') − 0}} < ''ε''}}.

This definition allows a limit to be defined at limit points of the domain {{mvar|S}}, if a suitable subset {{mvar|T}} which has the same limit point is chosen.

Notably, the previous two-sided definition works on <math>\operatorname{int} S \cup \operatorname{iso} S^c,</math> which is a subset of the limit points of {{mvar|S}}.

For example, let <math>S = [0,1)\cup (1, 2].</math> The previous two-sided definition would work at <math>1 \in \operatorname{iso} S^c = \{1\},</math> but it wouldn't work at 0 or 2, which are limit points of {{mvar|S}}.