The Limit Laws - Revision history

Khanh at 23:29, 15 January 2022

2022-01-15T23:29:03Z

Khanh: /* Scalar Product Rule for Limits */

2022-01-15T23:27:48Z

Scalar Product Rule for Limits

Lila: /* Resources */

2021-10-27T20:31:00Z

Resources

Lila at 21:45, 28 September 2021

2021-09-28T21:45:06Z

James.kercheville: changed video links

2020-08-21T20:29:51Z

changed video links

James.kercheville: Added video links

2020-08-20T02:23:57Z

Added video links

James.kercheville: Edit to link title

2020-08-11T14:47:17Z

Edit to link title

James.kercheville at 14:46, 11 August 2020

2020-08-11T14:46:40Z

James.kercheville: Created page and two links

2020-08-11T14:46:14Z

Created page and two links

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214-2.3TheLimitLawsPwPt.pptx The Limit Laws] PowerPoint file created by Dr. Sara Shirinkam, UTSA.

* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214_2.3TheLimitLawsWS1.pdf The Limit Laws] PowerPoint file created by Dr. Sara Shirinkam, UTSA.

@@ Line 20: / Line 20: @@
 ===Sum Rule for Limits===
 : Suppose that <math>\lim_{x\to c}f(x)=L</math> and <math>\lim_{x\to c}g(x)=M</math>. Then,
-:: <math>\lim_{x\to c}\Big[f(x)+g(x)\Big]=\lim_{x\to c}f(x)+\lim_{x\to c}g(x)=L+M</math>}}.
+:: <math>\lim_{x\to c}\Big[f(x)+g(x)\Big]=\lim_{x\to c}f(x)+\lim_{x\to c}g(x)=L+M</math>.
 Proof: Since we are given that <math>\lim_{x\to c}f(x)=L</math> and <math>\lim_{x\to c}g(x)=M</math> , there must be functions, call them <math>\delta_f(\varepsilon)</math> and <math>\delta_g(\varepsilon)</math> , such that for all <math>\varepsilon>0</math> , <math>\Big|f(x)-L\Big|<\varepsilon</math> whenever <math>|x-c|<\delta_f(\varepsilon)</math> , and <math>\Big|g(x)-M\Big|<\varepsilon</math> whenever <math>|x-c|<\delta_{g}(\varepsilon)</math> .<br/>
@@ Line 27: / Line 27: @@
 ===Difference Rule for Limits===
 : Suppose that <math>\lim_{x\to c}f(x)=L</math> and <math>\lim_{x\to c}g(x)=M</math>. Then
-:: <math>\lim_{x\to c}\Big[f(x)-g(x)\Big]=\lim_{x\to c}f(x)-\lim_{x\to c}g(x)=L-M</math>}}
+:: <math>\lim_{x\to c}\Big[f(x)-g(x)\Big]=\lim_{x\to c}f(x)-\lim_{x\to c}g(x)=L-M</math>
 Proof: Define <math>h(x)=-g(x)</math> . By the Scalar Product Rule for Limits, <math>\lim_{x\to c}h(x)=-M</math> . Then by the Sum Rule for Limits, <math>\lim_{x\to c}\Big[f(x)-g(x)\Big]=\lim_{x\to c}\Big[f(x)+h(x)\Big]=L-M</math>.
@@ Line 33: / Line 33: @@
 ===Product Rule for Limits===
 : Suppose that <math>\lim_{x\to c}f(x)=L</math> and <math>\lim_{x\to c}g(x)=M</math> . Then
-:: <math>\lim_{x\to c}\Big[f(x)\cdot g(x)\Big]=\lim_{x\to c}f(x)\cdot\lim_{x\to c}g(x)=L\cdot M</math>}}
+:: <math>\lim_{x\to c}\Big[f(x)\cdot g(x)\Big]=\lim_{x\to c}f(x)\cdot\lim_{x\to c}g(x)=L\cdot M</math>
 Proof: Let <math>\varepsilon</math> be any positive number. The assumptions imply the existence of the positive numbers <math>\delta_1,\delta_2,\delta_3</math> such that<br/>
 :<math>(1)\qquad\Big|f(x)-L\Big|<\frac{\varepsilon}{2(1+|M|)}</math> when <math>0<|x-c|<\delta_1</math>

@@ Line 12: / Line 12: @@
 ===Scalar Product Rule for Limits===
-: Suppose that <math>\lim_{x\to a}f(x)=L</math> for finite <math>L</math> and that <math>c</math> is constant. Then <math>\lim_{x\to a}c\cdot f(x)=c\cdot\lim_{x\to a}f(x)=c\cdot L</math>}}.
+: Suppose that <math>\lim_{x\to a}f(x)=L</math> for finite <math>L</math> and that <math>c</math> is constant. Then <math>\lim_{x\to a}c\cdot f(x)=c\cdot\lim_{x\to a}f(x)=c\cdot L</math>.
 Proof: Given the limit above, there exists in particular a <math>\delta>0</math> such that <math>\Big|f(x)-L\Big|<\frac{\varepsilon}{k}</math> whenever <math>0<|x-a|<\delta</math> , for some <math>k>0</math> such that <math>|c|<k</math> . Hence

← Older revision		Revision as of 20:31, 27 October 2021
Line 97:		Line 97:

	* [https://youtu.be/KeAgfb3NZLc Limit Laws to Evaluate a Limit , Example 3] by patrickJMT		* [https://youtu.be/KeAgfb3NZLc Limit Laws to Evaluate a Limit , Example 3] by patrickJMT
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Proofs_of_Some_Basic_Limit_Rules Proofs of Some Basic Limit Rules, Wikibooks: Calculus] under a CC BY-SA license

@@ Line 1: / Line 1: @@
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214-2.3TheLimitLawsPwPt.pptx  The Limit Laws] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214_2.3TheLimitLawsWS1.pdf  The Limit Laws Worksheet]
 * [https://youtu.be/P7G7F3NzPw0 Properties of Limits] by The Organic Chemistry Tutor

@@ Line 4: / Line 4: @@
-* [https://youtu.be/mQpU2nYG3Jw The Limit Laws Part 1] Video Lecture by Instructor Sharon, UTSA.
+* [https://youtu.be/P7G7F3NzPw0 Properties of Limits] by The Organic Chemistry Tutor
-* [https://youtu.be/hquujjjX-3A The Limit Laws Part 2] Video Lecture by Instructor Sharon, UTSA.
+* [https://youtu.be/v_Nz6UUQ4HQ Limit Laws to Evaluate a Limit , Example 1] by patrickJMT
-* [https://youtu.be/DvND8hoNtHE The Limit Laws Part 3] Video Lecture by Instructor Sharon, UTSA.
+* [https://youtu.be/3d7qLvgZp0s Limit Laws to Evaluate a Limit , Example 2] by patrickJMT
-* [https://youtu.be/KoxjCjrTrik The Limit Laws Part 4] Video Lecture by Instructor Sharon, UTSA.
+* [https://youtu.be/KeAgfb3NZLc Limit Laws to Evaluate a Limit , Example 3] by patrickJMT

← Older revision		Revision as of 02:23, 20 August 2020
Line 2:		Line 2:

	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214_2.3TheLimitLawsWS1.pdf The Limit Laws Worksheet]		* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214_2.3TheLimitLawsWS1.pdf The Limit Laws Worksheet]
		+
		+
		+	* [https://youtu.be/mQpU2nYG3Jw The Limit Laws Part 1] Video Lecture by Instructor Sharon, UTSA.
		+
		+	* [https://youtu.be/hquujjjX-3A The Limit Laws Part 2] Video Lecture by Instructor Sharon, UTSA.
		+
		+	* [https://youtu.be/DvND8hoNtHE The Limit Laws Part 3] Video Lecture by Instructor Sharon, UTSA.
		+
		+	* [https://youtu.be/KoxjCjrTrik The Limit Laws Part 4] Video Lecture by Instructor Sharon, UTSA.
		+
		+	* [https://youtu.be/jJsQxqU50Lw The Limit Laws Part 5] Video Lecture by Instructor Sharon, UTSA.

← Older revision		Revision as of 14:47, 11 August 2020
Line 1:		Line 1:
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214-2.3TheLimitLawsPwPt.pptx The Limit Laws] PowerPoint file created by Dr. Sara Shirinkam, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214-2.3TheLimitLawsPwPt.pptx The Limit Laws] PowerPoint file created by Dr. Sara Shirinkam, UTSA.

−	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214_2.3TheLimitLawsWS1.pdf The Limit Laws]	+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214_2.3TheLimitLawsWS1.pdf The Limit Laws Worksheet]

← Older revision		Revision as of 14:46, 11 August 2020
Line 1:		Line 1:
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214-2.3TheLimitLawsPwPt.pptx The Limit Laws] PowerPoint file created by Dr. Sara Shirinkam, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214-2.3TheLimitLawsPwPt.pptx The Limit Laws] PowerPoint file created by Dr. Sara Shirinkam, UTSA.

−	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214_2.3TheLimitLawsWS1.pdf The Limit Laws] ~~PowerPoint file created by Dr. Sara Shirinkam, UTSA.~~	+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Limit%20Laws/MAT1214_2.3TheLimitLawsWS1.pdf The Limit Laws]