Editing Limit of a function (section)

===Pointwise limits and uniform limits===

{{Main|Pointwise convergence|Uniform convergence}}

Let <math>f : S \times T \to \R.</math> Instead of taking limit as {{math|(''x'', ''y'') → (''p'', ''q'')}}, 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> In fact, this limiting process can be done in two distinct ways. The first one is called '''pointwise limit'''. We say '''the pointwise limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|g}}''', denoted
<math display=block>\lim_{x\to p}f(x, y) = g(y),</math> or
<math display=block>\lim_{x \to p}f(x, y) = g(y) \;\; \text{pointwise}.</math>

Alternatively, we may say '''{{mvar|f}} tends to {{mvar|g}} pointwise as {{mvar|x}} approaches {{mvar|p}}''', denoted
<math display=block>f(x, y) \to g(y) \;\; \text{as} \;\; x \to p,</math> or
<math display=block>f(x, y) \to g(y) \;\; \text{pointwise} \;\; \text{as} \;\; x \to p.</math>

This limit exists if the following holds:
{{block indent| For every {{math|''ε'' > 0}} and every fixed {{mvar|y}} in {{mvar|T}}, there exists a {{math|''δ''(''ε'', ''y'') > 0}} such that for all {{mvar|x}} in {{mvar|S}}, whenever {{math|0 < {{abs|''x'' − ''p''}} < ''δ''}}, we have {{math|{{abs|''f''(''x'', ''y'') − ''g''(''y'')}} < ''ε''}}.{{sfnp|Zakon|2011|p=220}}}}
<math display=block>(\forall \varepsilon > 0)\, (\forall y \in T) \, (\exists \delta> 0)\, (\forall x \in S)\, ( 0 < |x-p| < \delta \implies |f(x, y) - g(y)| < \varepsilon) .</math>

Here, {{math|1=''δ'' = ''δ''(''ε'', ''y'')}} is a function of both {{mvar|ε}} and {{mvar|y}}. Each {{mvar|δ}} is chosen for a ''specific point'' of {{mvar|y}}. Hence we say the limit is pointwise in {{mvar|y}}. For example,
<math display=block>f(x, y) = \frac{x}{\cos y}</math>
has a pointwise limit of constant zero function
<math display=block>\lim_{x \to 0}f(x, y) = 0(y) \;\; \text{pointwise}</math>
because for every fixed {{mvar|y}}, the limit is clearly 0. This argument fails if {{mvar|y}} is not fixed: if {{mvar|y}} is very close to {{math|''π''/2}}, the value of the fraction may deviate from 0.

This leads to another definition of limit, namely the '''uniform limit'''. We say '''the uniform limit of {{mvar|f}} on {{mvar|T}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|g}}''', denoted
<math display=block>\underset{ {x \to p} \atop {y \in T} }{\mathrm{unif} \lim \;} f(x, y) = g(y),</math> or
<math display=block>\lim_{x \to p}f(x, y) = g(y) \;\; \text{uniformly on} \; T.</math>

Alternatively, we may say '''{{mvar|f}} tends to {{mvar|g}} uniformly on {{mvar|T}} as {{mvar|x}} approaches {{mvar|p}}''', denoted
<math display=block>f(x, y) \rightrightarrows g(y) \; \text{on} \; T \;\; \text{as} \;\; x \to p,</math> or
<math display=block>f(x, y) \to g(y) \;\; \text{uniformly on}\; T \;\; \text{as} \;\; x \to p.</math>

This limit exists if the following holds:
{{block indent| For every {{math|''ε'' > 0}}, there exists a {{math|''δ''(''ε'') > 0}} such that for all {{mvar|x}} in {{mvar|S}} and {{mvar|y}} in {{mvar|T}}, whenever {{math|0 < {{abs|''x'' − ''p''}} < ''δ''}}, we have {{math|{{abs|''f''(''x'', ''y'') − ''g''(''y'')}} < ''ε''}}.{{sfnp|Zakon|2011|p=220}}}}
<math display=block>(\forall \varepsilon > 0) \, (\exists \delta > 0)\, (\forall x \in S)\, (\forall y \in T)\, ( 0 < |x-p| < \delta \implies |f(x, y) - g(y)| < \varepsilon) .</math>

Here, {{math|1=''δ'' = ''δ''(''ε'')}} is a function of only {{mvar|ε}} but not {{mvar|y}}. In other words, ''δ'' is ''uniformly applicable'' to all {{mvar|y}} in {{mvar|T}}. Hence we say the limit is uniform in {{mvar|y}}. For example,
<math display=block>f(x, y) = x \cos y</math>
has a uniform limit of constant zero function
<math display=block>\lim_{x \to 0}f(x, y) = 0(y) \;\; \text{ uniformly on}\; \R</math>
because for all real {{mvar|y}}, {{math|cos ''y''}} is bounded between {{math|[−1, 1]}}. Hence no matter how {{mvar|y}} behaves, we may use the [[sandwich theorem]] to show that the limit is 0.