Editing Limit of a function (section)

===Infinite limits===

For a function whose values grow without bound, the function diverges and the usual limit does not exist.  However, in this case one may introduce limits with infinite values.

Let <math>f:S \to\mathbb{R}</math> be a function defined on <math>S\subseteq\mathbb{R}.</math> The statement '''the limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} is infinity''', denoted

<math display=block> \lim_{x \to p} f(x) = \infty, </math>

means that:
{{block indent| For every {{math|''N'' > 0}}, there exists a {{math|''δ'' > 0}} such that whenever {{math|0 < {{abs|''x'' − ''p''}} < ''δ''}}, we have {{math|''f''(''x'') > ''N''}}.}}
<math display=block>(\forall N > 0)\, (\exists \delta > 0)\, (\forall x \in S)\, (0 < | x-p | < \delta \implies f(x) > N) .</math>

The statement '''the limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} is minus infinity''', denoted

<math display=block> \lim_{x \to p} f(x) = -\infty, </math>

means that:
{{block indent| For every {{math|''N'' > 0}}, there exists a {{math|''δ'' > 0}} such that whenever {{math|0 < {{abs|''x'' − ''p''}} < ''δ''}}, we have {{math|''f''(''x'') < −''N''}}.}}
<math display=block>(\forall N > 0) \, (\exists \delta > 0) \, (\forall x \in S)\, (0 < | x-p | < \delta \implies f(x) < -N) .</math>

For example,
<math display=block>\lim_{x \to 1} \frac{1}{(x-1)^2} = \infty</math>
because for every {{math|''N'' > 0}}, we can take <math display="inline">\delta =  \tfrac{1}{\sqrt{N}\delta} = \tfrac{1}{\sqrt N}</math> such that for all real {{math|''x'' > 0}}, if {{math|0 < ''x'' − 1 < ''δ''}}, then {{math|''f''(''x'') > ''N''}}.

These ideas can be used together to produce definitions for different combinations, such as

<math display=block> \lim_{x \to \infty} f(x) = \infty,</math> or <math> \lim_{x \to p^+}f(x) = -\infty.</math>

For example,
<math display=block>\lim_{x \to 0^+} \ln x = -\infty</math>
because for every {{math|''N'' > 0}}, we can take {{math|1=''δ'' = ''e''<sup>−''N''</sup>}} such that for all real {{math|''x'' > 0}}, if {{math|0 < ''x'' − 0 < ''δ''}}, then {{math|''f''(''x'') < −''N''}}.

Limits involving infinity are connected with the concept of [[asymptote]]s.

These notions of a limit attempt to provide a metric space interpretation to limits at infinity.  In fact, they are consistent with the topological space definition of limit if
*a neighborhood of −∞ is defined to contain an [[Interval (mathematics)|interval]] {{math|[−∞, ''c'')}} for some {{tmath|c \in \R,}}
*a neighborhood of ∞ is defined to contain an interval {{math|(''c'', ∞]}} where {{tmath|c \in \R,}} and
*a neighborhood of {{tmath|a \in \R}} is defined in the normal way metric space {{tmath|\R.}}
In this case, {{tmath|\overline \R}} is a topological space and any function of the form <math>f : X \to Y</math> with <math>X, Y \subseteq \overline \R</math> is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense.