Limits Involving Infinity - Revision history

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Khanh: Created page with "==Limits involving infinity== ===Limits at infinity=== The limit of this function at infinity exists. Let $S\subseteq\mathbb..."$

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Created page with "==Limits involving infinity== ===Limits at infinity=== thumb|300px|The limit of this function at infinity exists. Let <math>S\subseteq\mathbb..."

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==Limits involving infinity==

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

Let <math>S\subseteq\mathbb{R}</math>, <math>x\in S</math> and <math>f:S\mapsto\mathbb{R}</math>.

'''The limit of ''f'' as ''x'' approaches infinity is ''L''''', denoted

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

means that for all <math>\varepsilon > 0</math>, there exists ''c'' such that
<math>|f(x) - L| < \varepsilon</math> whenever ''x'' > ''c''. Or, symbolically:
:<math>\forall \varepsilon > 0 \; \exists c \; \forall x > c :\; |f(x) - L| < \varepsilon</math>.

Similarly, '''the limit of ''f'' as ''x'' approaches negative infinity is ''L''''', denoted

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

means that for all <math>\varepsilon > 0</math> there exists ''c'' such that <math>|f(x) - L| < \varepsilon</math> whenever ''x'' < ''c''. Or, symbolically:
:<math>\forall \varepsilon > 0 \; \exists c \; \forall x < c :\; |f(x) - L| < \varepsilon</math>.

For example,
:<math> \lim_{x \to -\infty}e^x = 0. \, </math>

===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>S\subseteq\mathbb{R}</math>, <math>x\in S</math> and <math>f:S\mapsto\mathbb{R}</math>. The statement '''the limit of ''f'' as ''x'' approaches ''a'' is infinity''', denoted

:<math> \lim_{x \to a} f(x) = \infty, </math>

means that for all <math>N > 0</math> there exists <math>\delta > 0</math> such that <math>f(x) > N</math> whenever <math>0 < |x - a| < \delta.</math>

These ideas can be combined in a natural way to produce definitions for different combinations, such as

:<math> \lim_{x \to \infty} f(x) = \infty, \lim_{x \to a^+}f(x) = -\infty. </math>

For example,
:<math>\lim_{x \to 0^+} \ln x = -\infty. </math>

Limits involving infinity are connected with the concept of asymptotes.

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 [−∞, ''c'') for some ''c'' ∈ '''R''',
*a neighborhood of ∞ is defined to contain an interval (''c'', ∞] where ''c'' ∈ '''R''', and
*a neighborhood of ''a'' ∈ '''R''' is defined in the normal way metric space '''R'''.
In this case, <span style="text-decoration: overline">'''R'''</span> is a topological space and any function of the form ''f'': ''X'' → ''Y'' with ''X'', ''Y''⊆ <span style="text-decoration: overline">'''R'''</span> 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 allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as {{nowrap|'''R''' ∪ {−∞, +∞}{{null}}}} and the projectively extended real line is '''R''' ∪ {∞} where a neighborhood of ∞ is a set of the form {{nowrap|{''x'': {{abs|''x''}} > ''c''}.}} 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: −∞, left, central, right, and +∞; three bounds: −∞, finite, or +∞). 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>\lim_{x \to 0^{+}}{1\over x} = +\infty, \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>\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 differentiation and integration, it is, for example, convenient to have 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 infinitesimals.

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

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

==Resources==
* [https://tutorial.math.lamar.edu/classes/calci/limitsatinfinityi.aspx Limits at Infinity Part I], Paul's Online Notes
* [https://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityII.aspx Limits at Infinity Part II], Paul's Online Notes
* [https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-15/v/introduction-to-limits-at-infinity Introduction to limits at infinity], Khan Academy

← Older revision		Revision as of 20:28, 3 November 2021
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	* [https://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityII.aspx Limits at Infinity Part II], Paul's Online Notes		* [https://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityII.aspx Limits at Infinity Part II], Paul's Online Notes
	* [https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-15/v/introduction-to-limits-at-infinity Introduction to limits at infinity], Khan Academy		* [https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-15/v/introduction-to-limits-at-infinity Introduction to limits at infinity], Khan Academy
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Limit_of_a_function Limit of a function, Wikipedia] under a CC BY-SA license

@@ Line 48: / Line 48: @@
 ===Alternative notation===
-Many authors allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as '''R''' ∪ {−∞, +∞} and the projectively extended real line is '''R'''&nbsp;∪&nbsp;{∞} where a neighborhood of ∞ is a set of the form {''x'': {{abs|''x''}} > ''c''}.  The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases.
+Many authors allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as '''R''' ∪ {−∞, +∞} and the projectively extended real line is '''R'''&nbsp;∪&nbsp;{∞} where a neighborhood of ∞ is a set of the form {''x'': |''x''| > ''c''}.  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):

@@ Line 48: / Line 48: @@
 ===Alternative notation===
-Many authors allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as {{nowrap|'''R''' ∪ {−∞, +∞}{{null}}}} and the projectively extended real line is '''R'''&nbsp;∪&nbsp;{∞} where a neighborhood of ∞ is a set of the form {{nowrap|{''x'': {{abs|''x''}} > ''c''}.}}  The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases.
+Many authors allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as '''R''' ∪ {−∞, +∞} and the projectively extended real line is '''R'''&nbsp;∪&nbsp;{∞} where a neighborhood of ∞ is a set of the form {''x'': {{abs|''x''}} > ''c''}.  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):

Limits Involving Infinity - Revision history

Khanh at 20:28, 3 November 2021

Khanh at 05:03, 18 September 2021

Khanh at 05:01, 18 September 2021

Khanh: Created page with "==Limits involving infinity== ===Limits at infinity=== The limit of this function at infinity exists. Let S\subseteq\mathbb..."

Khanh: Created page with "==Limits involving infinity== ===Limits at infinity=== The limit of this function at infinity exists. Let $S\subseteq\mathbb..."$