Real Function Limits:Infinite - Revision history

Khanh at 22:15, 6 November 2021

2021-11-06T22:15:45Z

Lila: /* Formal Definition of Infinite Limits */

2021-10-20T19:53:49Z

Formal Definition of Infinite Limits

Lila: /* Infinite limits */

2021-10-20T19:52:51Z

Infinite limits

Lila: /* Limits at infinity */

2021-10-20T19:51:12Z

Limits at infinity

Lila: /* Informal Infinite Limits */

2021-10-20T19:50:37Z

Informal Infinite Limits

Lila: /* Examples */

2021-10-20T19:47:30Z

Examples

Lila at 19:45, 20 October 2021

2021-10-20T19:45:56Z

Lila: /* Informal Infinite Limits */

2021-10-20T19:44:42Z

Informal Infinite Limits

Lila: /* Informal Infinite Limits */

2021-10-20T19:43:36Z

Informal Infinite Limits

Lila: /* Resources */

2021-10-20T19:41:51Z

Resources

@@ Line 130: / Line 130: @@
 We only use big ''N'' and ''M'' because the connotation associated with ε and δ is that they are small numbers. Big ''N'' and ''M'' has the opposite connotation.
-==Resources==
+== Licensing ==
-* [https://en.wikibooks.org/wiki/Calculus/Infinite_Limits Infinite limits], Wikibooks: Calculus
+Content obtained and/or adapted from:
-* [https://en.wikibooks.org/wiki/Real_Analysis/Limits Limits], Wikibooks: Real Analysis
+* [https://en.wikibooks.org/wiki/Calculus/Infinite_Limits Infinite limits, Wikibooks: Calculus] under a CC BY-SA license
-* [https://en.wikipedia.org/wiki/Limit_of_a_function#Limits_at_infinity Limit of a function], Wikipedia
+* [https://en.wikibooks.org/wiki/Real_Analysis/Limits Limits, Wikibooks: Real Analysis] under a CC BY-SA license

@@ Line 89: / Line 89: @@
 :<math>\lim_{x\to\infty}\frac{x^3+x-6}{x^2-4x+3}=\infty</math> .
-==Formal Definition of Infinite Limits==
+==Formal Definitions of Limits Involving Infinity==
 [[File:Limit-at-infinity-graph.png|thumb|alt=A graph which has a hump in the beginning, then a smaller valley, as it tapers off.|A graphical example of a function converging to a limit as it approaches infinity]]

@@ Line 14: / Line 14: @@
 :<math>\lim_{x \to 0^+} \ln x = -\infty. </math>
-Limits involving infinity are connected with the concept of [[asymptote]]s.
+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 (mathematics)|interval]] [−∞,&nbsp;''c'') for some ''c''&nbsp;∈&nbsp;'''R''',
+*a neighborhood of −∞ is defined to contain an interval [−∞,&nbsp;''c'') for some ''c''&nbsp;∈&nbsp;'''R''',
 *a neighborhood of ∞ is defined to contain an interval (''c'',&nbsp;∞] where ''c''&nbsp;∈&nbsp;'''R''', and
 *a neighborhood of ''a'' ∈ '''R''' is defined in the normal way metric space '''R'''.

@@ Line 1: / Line 1: @@
-==Limits at infinity==
+==Infinite limits==
-Let <math>S\subseteq\mathbb{R}</math>, <math>x\in S</math> and <math>f:S\mapsto\mathbb{R}</math>.
+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
-'''The limit of ''f'' as ''x'' approaches infinity is ''L''''', denoted
+:<math> \lim_{x \to a} f(x) = \infty, </math>
-:<math> \lim_{x \to \infty}f(x) = L,</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>
-means that for all <math>\varepsilon > 0</math>, there exists ''c'' such that
+These ideas can be combined in a natural way to produce definitions for different combinations, such as
-Similarly, '''the limit of ''f'' as ''x'' approaches negative infinity is ''L''''', denoted
+:<math> \lim_{x \to \infty} f(x) = \infty, \lim_{x \to a^+}f(x) = -\infty. </math>
-:<math> \lim_{x \to -\infty}f(x) = L,</math>
+For example,
-means that for all <math>\varepsilon > 0</math> there exists ''c'' such that <math>|f(x) - L| < \varepsilon</math> whenever ''x''&nbsp;<&nbsp;''c''. Or, symbolically:
+Limits involving infinity are connected with the concept of [[asymptote]]s.
-For example,
+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
-:<math> \lim_{x \to -\infty}e^x = 0. \, </math>
+*a neighborhood of −∞ is defined to contain an [[Interval (mathematics)|interval]] [−∞,&nbsp;''c'') for some ''c''&nbsp;∈&nbsp;'''R''',
 ==Limits at Infinity==

@@ Line 22: / Line 22: @@
 :<math> \lim_{x \to -\infty}e^x = 0. \, </math>
-==Informal Infinite Limits==
+==Limits at Infinity==
 Another kind of limit involves looking at what happens to <math>f(x)</math> as <math>x</math> gets very big. For example, consider the function <math>f(x)=\frac{1}{x}</math> . As <math>x</math> gets very big, <math>\frac{1}{x}</math> gets very small. In fact, <math>\frac{1}{x}</math> gets closer and closer to 0 the bigger <math>x</math> gets. Without limits it is very difficult to talk about this fact, because <math>x</math> can keep getting bigger and bigger and <math>\frac{1}{x}</math> never actually gets to 0; but the language of limits exists precisely to let us talk about the behavior of a function as it approaches something - without caring about the fact that it will never get there. In this case, however, we have the same problem as before: how big does <math>x</math> have to be to be sure that <math>f(x)</math> is really going towards 0?

@@ Line 69: / Line 69: @@
 Note that, if the numerator or denominator is a constant (including 1, as above), then this is the same as <math>x^0</math> . Also, a straight power of <math>x</math> , like <math>x^3</math> , has coefficient 1, since it is the same as <math>1x^3</math> .
-====Examples====
+===Examples===
 ;Example 1
 Find <math>\lim_{x\to\infty}\frac{x-5}{x-3}</math> .

@@ Line 52: / Line 52: @@
 because making <math>x</math> very negative also forces <math>\frac{1}{x}</math> to be close to <math>0</math> .
-'''Notice''', however, that [[/Infinity is not a number|infinity is not a number]]; it's just shorthand for saying "no matter how big." Thus, this is not the same as the regular limits we learned about in the last two chapters.
+'''Notice''', however, that infinity is not a number; it's just shorthand for saying "no matter how big." Thus, this is not the same as the regular limits we learned about in the last two chapters.
 ===Limits at Infinity of Rational Functions===
@@ Line 102: / Line 102: @@
 So, the definition needs to be rewritten, which is done in the following chart. The definitions for when either the limit as ''x'' ''approaches'' positive or negative infinity; or the limit as ƒ(x) ''converges'' to positive or negative infinity are as follows:
-{{Helpful hint|title=Note|hint=Yes, the ''approaching'' and ''converging'' distinction is important. You can look at it as either referencing the delta or the epsilon, respectively.}}
+'''Note:''' Yes, the ''approaching'' and ''converging'' distinction is important. You can look at it as either referencing the delta or the epsilon, respectively.}}
 {| class="wikitable" style="margin: auto;"

@@ Line 30: / Line 30: @@
 '''Definition: (Definition of a limit at infinity)'''
 :We call <math>L</math> the '''limit of <math>f(x)</math> as <math>x</math> approaches infinity''' if <math>f(x)</math> becomes '''arbitrarily close''' to <math>L</math> '''whenever''' <math>x</math> is '''sufficiently large'''.
-When this holds we write
+:When this holds we write
-:<math>\lim_{x\to\infty}f(x)=L</math>
+::<math>\lim_{x\to\infty}f(x)=L</math>
 or
-:<math>f(x)\to L\quad\mbox{as}\quad x\to\infty</math>
+::<math>f(x)\to L\quad\mbox{as}\quad x\to\infty</math>
-Similarly, we call <math>L</math> the '''limit of <math>f(x)</math> as <math>x</math> approaches negative infinity''' if <math>f(x)</math> becomes '''arbitrarily close''' to <math>L</math> '''whenever''' <math>x</math> is '''sufficiently negative'''.
+:Similarly, we call <math>L</math> the '''limit of <math>f(x)</math> as <math>x</math> approaches negative infinity''' if <math>f(x)</math> becomes '''arbitrarily close''' to <math>L</math> '''whenever''' <math>x</math> is '''sufficiently negative'''.
-When this holds we write
+:When this holds we write
-:<math>\lim_{x\to-\infty}f(x)=L</math>
+::<math>\lim_{x\to-\infty}f(x)=L</math>
 or
-:<math>f(x)\to L\quad\mbox{as}\quad x\to-\infty</math>
+::<math>f(x)\to L\quad\mbox{as}\quad x\to-\infty</math>

@@ Line 27: / Line 27: @@
 In this case, we want to say that, however close we want <math>f(x)</math> to get to 0, for <math>x</math> big enough <math>f(x)</math> is guaranteed to get that close. So we have yet another definition.
-{{Calculus/Def
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
-|title=Definition: (Definition of a limit at infinity)
+'''Definition: (Definition of a limit at infinity)'''
-|text=We call <math>L</math> the '''limit of <math>f(x)</math> as <math>x</math> approaches infinity''' if <math>f(x)</math> becomes '''arbitrarily close''' to <math>L</math> '''whenever''' <math>x</math> is '''sufficiently large'''.
+:We call <math>L</math> the '''limit of <math>f(x)</math> as <math>x</math> approaches infinity''' if <math>f(x)</math> becomes '''arbitrarily close''' to <math>L</math> '''whenever''' <math>x</math> is '''sufficiently large'''.
 When this holds we write
@@ Line 41: / Line 42: @@
 or
 :<math>f(x)\to L\quad\mbox{as}\quad x\to-\infty</math>
-}}
 So, in this case, we write:

← Older revision		Revision as of 19:41, 20 October 2021
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	* [https://en.wikibooks.org/wiki/Calculus/Infinite_Limits Infinite limits], Wikibooks: Calculus		* [https://en.wikibooks.org/wiki/Calculus/Infinite_Limits Infinite limits], Wikibooks: Calculus
	* [https://en.wikibooks.org/wiki/Real_Analysis/Limits Limits], Wikibooks: Real Analysis		* [https://en.wikibooks.org/wiki/Real_Analysis/Limits Limits], Wikibooks: Real Analysis
		+	* [https://en.wikipedia.org/wiki/Limit_of_a_function#Limits_at_infinity Limit of a function], Wikipedia