Limits at Infinity and Asymptotes - Revision history

Lila: /* Resources */

2021-10-27T20:46:28Z

Resources

Lila at 00:04, 29 September 2021

2021-09-29T00:04:44Z

Lila at 00:03, 29 September 2021

2021-09-29T00:03:52Z

Lila: /* Asymptotes */

2021-09-29T00:02:37Z

Asymptotes

Lila at 00:01, 29 September 2021

2021-09-29T00:01:26Z

Lila at 23:57, 28 September 2021

2021-09-28T23:57:42Z

Lila at 23:55, 28 September 2021

2021-09-28T23:55:06Z

Lila: /* Formal Definition of a Limit Being Infinity */

2021-09-28T23:09:09Z

Formal Definition of a Limit Being Infinity

Lila at 22:40, 28 September 2021

2021-09-28T22:40:42Z

James.kercheville: Added notes link

2020-08-24T18:02:43Z

Added notes link

← Older revision		Revision as of 20:46, 27 October 2021
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	* [https://youtu.be/sjLFl7Z8W_Io Limits at Infinity & Horizontal Asymptotes] by the Organic Chemistry Tutor		* [https://youtu.be/sjLFl7Z8W_Io Limits at Infinity & Horizontal Asymptotes] by the Organic Chemistry Tutor
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		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Infinite_Limits Infinite Limits, Wikibooks: Calculus] under a CC BY-SA license

← Older revision		Revision as of 00:04, 29 September 2021
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	When a function approaches infinity or negative infinity as <math> x\to c </math> for some finite value <math> c </math>, we say that the function has a vertical asymptote at <math> x = c </math>. A vertical asymptote is a vertical on a graph that a function approaches, but never reaches. For example, consider the function <math> f(x) = \frac{1}{x-1} </math>. <math>\lim_{x\to 1^{+}}\frac{1}{x}=\infty</math> and <math>\lim_{x\to 1^{-}}\frac{1}{x}=-\infty</math>, so there is a vertical asymptote at <math> x = 1 </math> (where the denominator goes to 0).		When a function approaches infinity or negative infinity as <math> x\to c </math> for some finite value <math> c </math>, we say that the function has a vertical asymptote at <math> x = c </math>. A vertical asymptote is a vertical on a graph that a function approaches, but never reaches. For example, consider the function <math> f(x) = \frac{1}{x-1} </math>. <math>\lim_{x\to 1^{+}}\frac{1}{x}=\infty</math> and <math>\lim_{x\to 1^{-}}\frac{1}{x}=-\infty</math>, so there is a vertical asymptote at <math> x = 1 </math> (where the denominator goes to 0).
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← Older revision		Revision as of 00:03, 29 September 2021
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	When a function approaches infinity or negative infinity as <math> x\to c </math> for some finite value <math> c </math>, we say that the function has a vertical asymptote at <math> x = c </math>. A vertical asymptote is a vertical on a graph that a function approaches, but never reaches. For example, consider the function <math> f(x) = \frac{1}{x-1} </math>. <math>\lim_{x\to 1^{+}}\frac{1}{x}=\infty</math> and <math>\lim_{x\to 1^{-}}\frac{1}{x}=-\infty</math>, so there is a vertical asymptote at <math> x = 1 </math> (where the denominator goes to 0).		When a function approaches infinity or negative infinity as <math> x\to c </math> for some finite value <math> c </math>, we say that the function has a vertical asymptote at <math> x = c </math>. A vertical asymptote is a vertical on a graph that a function approaches, but never reaches. For example, consider the function <math> f(x) = \frac{1}{x-1} </math>. <math>\lim_{x\to 1^{+}}\frac{1}{x}=\infty</math> and <math>\lim_{x\to 1^{-}}\frac{1}{x}=-\infty</math>, so there is a vertical asymptote at <math> x = 1 </math> (where the denominator goes to 0).
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	==Resources==		==Resources==

← Older revision		Revision as of 00:02, 29 September 2021
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	When a function approaches infinity or negative infinity as <math> x\to c </math> for some finite value <math> c </math>, we say that the function has a vertical asymptote at <math> x = c </math>. A vertical asymptote is a vertical on a graph that a function approaches, but never reaches. For example, consider the function <math> f(x) = \frac{1}{x-1} </math>. <math>\lim_{x\to 1^{+}}\frac{1}{x}=\infty</math> and <math>\lim_{x\to 1^{-}}\frac{1}{x}=-\infty</math>, so there is a vertical asymptote at <math> x = 1 </math> (where the denominator goes to 0).		When a function approaches infinity or negative infinity as <math> x\to c </math> for some finite value <math> c </math>, we say that the function has a vertical asymptote at <math> x = c </math>. A vertical asymptote is a vertical on a graph that a function approaches, but never reaches. For example, consider the function <math> f(x) = \frac{1}{x-1} </math>. <math>\lim_{x\to 1^{+}}\frac{1}{x}=\infty</math> and <math>\lim_{x\to 1^{-}}\frac{1}{x}=-\infty</math>, so there is a vertical asymptote at <math> x = 1 </math> (where the denominator goes to 0).
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	==Resources==		==Resources==

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 ==Asymptotes==
 When a function has a positive, finite limit L as <math>x\to\infty</math> or <math>x\to-\infty</math>, we say that the function has a horizontal asymptote at <math> y = L </math>. A horizontal asymptote is a horizontal line on a graph that a function approaches as <math> x </math> goes to <math> \pm\infty </math>. For example, consider the function <math> f(x) = \frac{1}{x-1} </math>. <math>\lim_{x\to\infty}\frac{1}{x}=0</math>, so there is a horizontal asymptote at <math> y = 0 </math>.
 When a function approaches infinity or negative infinity as <math> x\to c </math> for some finite value <math> c </math>, we say that the function has a vertical asymptote at <math> x = c </math>. A vertical asymptote is a vertical on a graph that a function approaches, but never reaches. For example, consider the function <math> f(x) = \frac{1}{x-1} </math>. <math>\lim_{x\to 1^{+}}\frac{1}{x}=\infty</math> and <math>\lim_{x\to 1^{-}}\frac{1}{x}=-\infty</math>, so there is a vertical asymptote at <math> x = 1 </math> (where the denominator goes to 0).
-[[File:Horizontal and vertical asymptote.png |thumb| <math> f(x) = \frac{1}{x-1} </math> has a vertical asymptote at <math> x = 1</math> and a horizontal asymptote at <math> y = 0</math>]]
 ==Resources==

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 We look at the terms with the highest exponents; for the numerator, it is <math>x^3</math>, while for the denominator it is <math>x^2</math>. Since the exponent on the numerator is higher, we know the limit at <math>\infty</math> will be <math>\infty</math>. We can see by factoring out and canceling the highest power that the denominator will go to 0, while the top will be a positive 1. So,
 :<math>\lim_{x\to\infty}\frac{x^3+x-6}{x^2-4x+3}=+\infty</math>.
 ==Resources==

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-==Formal Definition of a Limit Being Infinity==
+==Infinite Limits==
-Let <math>f(x)</math> be a function defined on an open interval <math>D</math> that contains <math>c</math> , except possibly at <math>x=c</math> . Then we say that
+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.
-When this holds we write
+===Definition of a limit at infinity===
-:<math>\lim_{x\to c}f(x)=\infty</math>
+: 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'''.
 or
-:<math>f(x)\to\infty</math> as <math>x\to c</math>
+:<math>f(x)\to L\quad\mbox{as}\quad x\to\infty</math>
-Similarly, we say that
+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'''.
-:<math>\lim_{x\to c}f(x)=-\infty</math>
-if, for every <math>\varepsilon</math> , there exists a <math>\delta>0</math> such that for all <math>x\in D</math> with
+When this holds we write
-:<math>0<|x-c|<\delta</math>
+:<math>\lim_{x\to-\infty}f(x)=L</math>
 or
-:<math>f(x)\to-\infty</math> as <math>x\to c</math> .
+:<math>f(x)\to L\quad\mbox{as}\quad x\to-\infty</math>
 ==Resources==

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 * [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20at%20Infinity/MAT1214-4.6LimitsAtInfinityPwPt.pptx  Limits at Infinity and Asymptotes] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
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 * [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20at%20Infinity/MAT1214-4.6LimitsAtInfinityNotes.pdf Limits at Infinity and Asymptotes] notes created by Instructor Beatty,UTSA.
 * [https://youtu.be/FVJNuukADeQ Limits at Infinity - Basic Idea and Shortcuts!] by patrickJMT

← Older revision		Revision as of 18:02, 24 August 2020
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	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20at%20Infinity/MAT1214-4.6LimitsAtInfinityWS1.pdf Limits at Infinity and Asymptotes Worksheet 1]		* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20at%20Infinity/MAT1214-4.6LimitsAtInfinityWS1.pdf Limits at Infinity and Asymptotes Worksheet 1]
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		+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20at%20Infinity/MAT1214-4.6LimitsAtInfinityNotes.pdf Limits at Infinity and Asymptotes] notes created by Instructor Beatty,UTSA.