Editing Limit of a function (section)

===Multiple limits at infinity===

The concept of multiple limit can extend to the limit at infinity, in a way similar to that of a single variable function. For <math>f : S \times T \to \R,</math> we say '''the double limit of {{mvar|f}} as {{mvar|x}} and {{mvar|y}} approaches infinity is {{mvar|L}}''', written
<math display=block> \lim_{ {x \to \infty} \atop {y \to \infty} } f(x, y) = L </math>

if the following condition holds:
{{block indent| For every {{math|''ε'' > 0}}, there exists a {{math|''c'' > 0}} such that for all {{mvar|x}} in {{mvar|S}} and {{mvar|y}} in {{mvar|T}}, whenever {{math|''x'' > ''c''}} and {{math|''y'' > ''c''}}, we have {{math|{{abs|''f''(''x'', ''y'') − ''L''}} < ''ε''}}.}}
<math display=block>(\forall \varepsilon > 0)\, (\exists c> 0)\, (\forall x \in S) \, (\forall y \in T)\, ( (x > c) \land (y > c) \implies |f(x, y) - L| < \varepsilon) .</math>

We say '''the double limit of {{mvar|f}} as {{mvar|x}} and {{mvar|y}} approaches minus infinity is {{mvar|L}}''', written
<math display=block> \lim_{ {x \to -\infty} \atop {y \to -\infty} } f(x, y) = L </math>

if the following condition holds:
{{block indent| For every {{math|''ε'' > 0}}, there exists a {{math|''c'' > 0}} such that {{mvar|x}} in {{mvar|S}} and {{mvar|y}} in {{mvar|T}}, whenever {{math|''x'' < −''c''}} and {{math|''y'' < −''c''}}, we have {{math|{{abs|''f''(''x'', ''y'') − ''L''}} < ''ε''}}.}}
<math display=block>(\forall \varepsilon > 0)\, (\exists c> 0)\, (\forall x \in S) \, (\forall y \in T)\, ( (x < -c) \land (y < -c) \implies |f(x, y) - L| < \varepsilon) .</math>