Editing Limit of a function (section)

===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>