Editing Limit of a function (section)

==Limits involving infinity==

===Limits at infinity===
{{^|[[Limit at infinity]] redirects here}}
[[File:Limit Infinity SVG.svg|thumb|300px|The limit of this function at infinity exists]]

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

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

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

Similarly, '''the limit of {{mvar|f}} as {{mvar|x}} approaches minus infinity is {{mvar|L}}''', denoted

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

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

For example, 
<math display=block> \lim_{x \to \infty} \left(-\frac{3\sin x}{x} + 4\right) = 4</math>
because for every {{math|''ε'' > 0}}, we can take {{math|1=''c'' = 3/''ε''}} such that for all real {{mvar|x}}, if {{math|''x'' > ''c''}}, then {{math|{{abs|''f''(''x'') − 4}} < ''ε''}}.

Another example is that
<math display=block> \lim_{x \to -\infty}e^{x} = 0</math>
because for every {{math|''ε'' > 0}}, we can take {{math|1=''c'' = max{1, −ln(''ε'')} }} such that for all real {{mvar|x}}, if {{math|''x'' < −''c''}}, then {{math|{{abs|''f''(''x'') − 0}} < ''ε''}}.

===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.

===Alternative notation===
Many authors<ref>For example, [https://encyclopediaofmath.org/wiki/Limit Limit] at ''[[Encyclopedia of Mathematics]]''</ref> allow for the [[projectively extended real line]] to be used as a way to include infinite values as well as [[Extended real number line|extended real line]]. With this notation, the extended real line is given as {{tmath|\R \cup \{-\infty, +\infty\} }} and the projectively extended real line is {{tmath|\R \cup \{\infty\} }} where a neighborhood of ∞ is a set of the form <math>\{x: |x| > c\}.</math>  The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases.
As presented above, for a completely rigorous account, we would need to consider 15 separate cases for each combination of infinities (five directions: −&infin;, left, central, right, and +&infin;; three bounds: −&infin;, finite, or +&infin;).  There are also noteworthy pitfalls.  For example, when working with the extended real line, <math>x^{-1}</math> does not possess a central limit (which is normal):

<math display=block>\lim_{x \to 0^{+}}{1\over x} = +\infty, \quad \lim_{x \to 0^{-}}{1\over x} = -\infty.</math>

In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit ''does'' exist in that context:

<math display=block>\lim_{x \to 0^{+}}{1\over x} = \lim_{x \to 0^{-}}{1\over x} = \lim_{x \to 0}{1\over x} = \infty.</math>

In fact there are a plethora of conflicting formal systems in use.
In certain applications of [[Numerical analysis|numerical differentiation and integration]], it is, for example, convenient to have [[Negative zero|signed zeroes]].  
A simple reason has to do with the converse of <math>\lim_{x \to 0^{-}}{x^{-1}} = -\infty,</math> namely, it is convenient for <math>\lim_{x \to -\infty}{x^{-1}} = -0</math> to be considered true.
Such zeroes can be seen as an approximation to [[infinitesimal]]s.

===Limits at infinity for rational functions===
[[File:Tamasol SVG.svg|thumb|300px|Horizontal asymptote about {{math|1=''y'' = 4}}]]
There are three basic rules for evaluating limits at infinity for a [[rational function]] <math>f(x) = \tfrac{p(x)}{q(x)}</math> (where {{mvar|p}} and {{mvar|q}} are polynomials):
*If the [[Degree of a polynomial|degree]] of {{mvar|p}} is greater than the degree of {{mvar|q}}, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
*If the degree of {{mvar|p}} and {{mvar|q}} are equal, the limit is the leading coefficient of {{mvar|p}} divided by the leading coefficient of {{mvar|q}};
*If the degree of {{mvar|p}} is less than the degree of {{mvar|q}}, the limit is 0.

If the limit at infinity exists, it represents a horizontal asymptote at {{math|1=''y'' = ''L''}}. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.