Basic Calculus Teaching Guide PDF

Commission on Higher Education in collaboration with the Philippine Normal University TEACHING GUIDE FOR SENIOR HIGH SCHOOL Basic Calculus CORE SUBJECT This Teaching Guide was collaboratively developed and reviewed by educators from public and private schools, colleges, and universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Commission on Higher Education, K to 12 Transition Program Management Unit - Senior High School Support Team at [email protected]. We value your feedback and recommendations. Published by the Commission on Higher Education, 2016 Chairperson: Patricia B. Licuanan, Ph.D. Commission on Higher Education K to 12 Transition Program Management Unit Office Address: 4th Floor, Commission on Higher Education, C.P. Garcia Ave., Diliman, Quezon City Telefax: (02) 441-1143 / E-mail Address: [email protected] DEVELOPMENT TEAM Team Leader: Jose Maria P. Balmaceda, Ph.D. Writers: Carlene Perpetua P. Arceo, Ph.D. Richard S. Lemence, Ph.D. Oreste M. Ortega, Jr., M.Sc. This Teaching Guide by the Louie John D. Vallejo, Ph.D. Commission on Higher Education is licensed under a Creative Technical Editors: Commons Attribution- Jose Ernie C. Lope, Ph.D. NonCommercial-ShareAlike Marian P. Roque, Ph.D. 4.0 International License. This Copy Reader: Roderick B. Lirios means you are free to: Cover Artists: Paolo Kurtis N. Tan, Renan U. Ortiz Share — copy and redistribute the material in any medium or CONSULTANTS format THIS PROJECT WAS DEVELOPED WITH THE PHILIPPINE NORMAL UNIVERSITY. Adapt — remix, transform, and University President: Ester B. Ogena, Ph.D. build upon the material. VP for Academics: Ma. Antoinette C. Montealegre, Ph.D. The licensor, CHED, cannot VP for University Relations & Advancement: Rosemarievic V. Diaz, Ph.D. revoke these freedoms as long as Ma. Cynthia Rose B. Bautista, Ph.D., CHED you follow the license terms. Bienvenido F. Nebres, S.J., Ph.D., Ateneo de Manila University However, under the following terms: Carmela C. Oracion, Ph.D., Ateneo de Manila University Minella C. Alarcon, Ph.D., CHED Attribution — You must give Gareth Price, Sheffield Hallam University appropriate credit, provide a link Stuart Bevins, Ph.D., Sheffield Hallam University to the license, and indicate if changes were made. You may do SENIOR HIGH SCHOOL SUPPORT TEAM so in any reasonable manner, but CHED K TO 12 TRANSITION PROGRAM MANAGEMENT UNIT not in any way that suggests the Program Director: Karol Mark R. Yee licensor endorses you or your use. Lead for Senior High School Support: Gerson M. Abesamis NonCommercial — You may not use the material for Lead for Policy Advocacy and Communications: Averill M. Pizarro commercial purposes. Course Development Officers: ShareAlike — If you remix, Danie Son D. Gonzalvo, John Carlo P. Fernando transform, or build upon the material, you must distribute Teacher Training Officers: your contributions under the Ma. Theresa C. Carlos, Mylene E. Dones same license as the original. Monitoring and Evaluation Officer: Robert Adrian N. Daulat Printed in the Philippines by EC-TEC Commercial, No. 32 St. Louis Administrative Officers: Ma. Leana Paula B. Bato, Compound 7, Baesa, Quezon City, Kevin Ross D. Nera, Allison A. Danao, Ayhen Loisse B. Dalena [email protected] Introduction As the Commission supports DepEd’s implementation of Senior High School (SHS), it upholds the vision and mission of the K to 12 program, stated in Section 2 of Republic Act 10533, or the Enhanced Basic Education Act of 2013, that “every graduate of basic education be an empowered individual, through a program rooted on...the competence to engage in work and be productive, the ability to coexist in fruitful harmony with local and global communities, the capability to engage in creative and critical thinking, and the capacity and willingness to transform others and oneself.” To accomplish this, the Commission partnered with the Philippine Normal University (PNU), the National Center for Teacher Education, to develop Teaching Guides for Courses of SHS. Together with PNU, this Teaching Guide was studied and reviewed by education and pedagogy experts, and was enhanced with appropriate methodologies and strategies. Furthermore, the Commission believes that teachers are the most important partners in attaining this goal. Incorporated in this Teaching Guide is a framework that will guide them in creating lessons and assessment tools, support them in facilitating activities and questions, and assist them towards deeper content areas and competencies. Thus, the introduction of the SHS for SHS Framework. The SHS for SHS Framework The SHS for SHS Framework, which stands for “Saysay-Husay-Sarili for Senior High School,” is at the core of this book. The lessons, which combine high-quality content with flexible elements to accommodate diversity of teachers and environments, promote these three fundamental concepts: SAYSAY: MEANING HUSAY: MASTERY SARILI: OWNERSHIP Why is this important? How will I deeply understand this? What can I do with this? Through this Teaching Guide, Given that developing mastery When teachers empower teachers will be able to goes beyond memorization, learners to take ownership of facilitate an understanding of teachers should also aim for deep their learning, they develop the value of the lessons, for understanding of the subject independence and self- each learner to fully engage in matter where they lead learners direction, learning about both the content on both the to analyze and synthesize the subject matter and cognitive and affective levels. knowledge. themselves. The Parts of the Teaching Guide Pedagogical Notes This Teaching Guide is mapped and aligned to the The teacher should strive to keep a good balance DepEd SHS Curriculum, designed to be highly between conceptual understanding and facility in usable for teachers. It contains classroom activities skills and techniques. Teachers are advised to be and pedagogical notes, and integrated with conscious of the content and performance standards and of the suggested time frame for innovative pedagogies. All of these elements are each lesson, but flexibility in the management of presented in the following parts: the lessons is possible. Interruptions in the class 1. INTRODUCTION schedule, or students’ poor reception or difficulty Highlight key concepts and identify the with a particular lesson, may require a teacher to essential questions extend a particular presentation or discussion. Show the big picture Computations in some topics may be facilitated by Connect and/or review prerequisite the use of calculators. This is encour- aged; knowledge however, it is important that the student understands the concepts and processes involved Clearly communicate learning competencies and objectives in the calculation. Exams for the Basic Calculus course may be designed so that calculators are not Motivate through applications and necessary. connections to real-life Because senior high school is a transition period 2. INSTRUCTION/DELIVERY for students, the latter must also be prepared for Give a demonstration/lecture/simulation/ college-level academic rigor. Some topics in hands-on activity calculus require much more rigor and precision Show step-by-step solutions to sample than topics encountered in previous mathematics problems courses, and treatment of the material may be Use multimedia and other creative tools different from teaching more elementary courses. Give applications of the theory The teacher is urged to be patient and careful in presenting and developing the topics. To avoid too Connect to a real-life problem if applicable much technical discussion, some ideas can be 3. PRACTICE introduced intuitively and informally, without Discuss worked-out examples sacrificing rigor and correctness. Provide easy-medium-hard questions The teacher is encouraged to study the guide very Give time for hands-on unguided classroom well, work through the examples, and solve work and discovery exercises, well in advance of the lesson. The development of calculus is one of humankind’s Use formative assessment to give feedback greatest achievements. With patience, motivation 4. ENRICHMENT and discipline, teaching and learning calculus Provide additional examples and effectively can be realized by anyone. The teaching applications guide aims to be a valuable resource in this Introduce extensions or generalisations of objective. concepts Engage in reflection questions Encourage analysis through higher order thinking prompts 5. EVALUATION Supply a diverse question bank for written work and exercises Provide alternative formats for student work: written homework, journal, portfolio, group/individual projects, student-directed research project On DepEd Functional Skills and CHED’s College Readiness Standards As Higher Education Institutions (HEIs) welcome the graduates of the Senior High School program, it is of paramount importance to align Functional Skills set by DepEd with the College Readiness Standards stated by CHED. The DepEd articulated a set of 21st century skills that should be embedded in the SHS curriculum across various subjects and tracks. These skills are desired outcomes that K to 12 graduates should possess in order to proceed to either higher education, employment, entrepreneurship, or middle-level skills development. On the other hand, the Commission declared the College Readiness Standards that consist of the combination of knowledge, skills, and reflective thinking necessary to participate and succeed - without remediation - in entry-level undergraduate courses in college. The alignment of both standards, shown below, is also presented in this Teaching Guide - prepares Senior High School graduates to the revised college curriculum which will initially be implemented by AY 2018-2019. College Readiness Standards Foundational Skills DepEd Functional Skills Produce all forms of texts (written, oral, visual, digital) based on: 1. Solid grounding on Philippine experience and culture; 2. An understanding of the self, community, and nation; Visual and information literacies Media literacy 3. Application of critical and creative thinking and doing processes; Critical thinking and problem solving skills 4. Competency in formulating ideas/arguments logically, scientifically, Creativity and creatively; and Initiative and self-direction 5. Clear appreciation of one’s responsibility as a citizen of a multicultural Philippines and a diverse world; Global awareness Scientific and economic literacy Systematically apply knowledge, understanding, theory, and skills Curiosity for the development of the self, local, and global communities using Critical thinking and problem solving skills prior learning, inquiry, and experimentation Risk taking Flexibility and adaptability Initiative and self-direction Global awareness Media literacy Work comfortably with relevant technologies and develop Technological literacy adaptations and innovations for significant use in local and global Creativity communities; Flexibility and adaptability Productivity and accountability Global awareness Multicultural literacy Communicate with local and global communities with proficiency, Collaboration and interpersonal skills orally, in writing, and through new technologies of communication; Social and cross-cultural skills Leadership and responsibility Media literacy Interact meaningfully in a social setting and contribute to the Multicultural literacy Global awareness fulfilment of individual and shared goals, respecting the Collaboration and interpersonal skills fundamental humanity of all persons and the diversity of groups Social and cross-cultural skills and communities Leadership and responsibility Ethical, moral, and spiritual values K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL – SCIENCE, TECHNOLOGY, ENGINEERING AND MATHEMATICS (STEM) SPECIALIZED SUBJECT Correspondence*between*the*Learning*Competencies*and*the*Topics*in*this*Learning*Guide Course*Title:"Basic"Calculus Semester:"Second"Semester No.*of*Hours/Semester:"80"hrs/sem Prerequisite:"Pre8Calculus Subject Description: At the end of the course, the students must know how to determine the limit of a function, differentiate, and integrate algebraic, exponential, logarithmic, and trigonometric functions in one variable, and to formulate and solve problems involving continuity, extreme values, related rates, population models, and areas of plane regions. CONTENT PERFORMANCE TOPIC CONTENT LEARNING COMPETENCIES CODE STANDARDS STANDARDS NUMBER Limits and The learners The learners shall be able The learners… Continuity demonstrate an to... STEM_BC11LC-IIIa-1 1.1 understanding of... 1. illustrate the limit of a function using a table of formulate and solve values and the graph of the function the basic concepts accurately real-life 2. distinguish between limx→cf(x)!and f(c) STEM_BC11LC-IIIa-2 1.2 of limit and problems involving 3. illustrate the limit laws STEM_BC11LC-IIIa-3 1.3 continuity of a continuity of functions 4. apply the limit laws in evaluating the limit of function algebraic functions (polynomial, rational, and STEM_BC11LC-IIIa-4 1.4 radical) 5. compute the limits of exponential, logarithmic,and trigonometric functions using tables of values and STEM_BC11LC-IIIb-1 2.1 graphs of the functions 6. evaluate limits involving the expressions (sint)/t , STEM_BC11LC-IIIb-2 2.2 (1-cost)/t and (et - 1)/t using tables of values 7. illustrate continuity of a function at a number STEM_BC11LC-IIIc-1 8. determine whether a function is continuous at a 3.1 STEM_BC11LC-IIIc-2 number or not 9. illustrate continuity of a function on an interval STEM_BC11LC-IIIc-3 10. determine whether a function is continuous on an 3.2 STEM_BC11LC-IIIc-4 interval or not. K to 12 BASIC EDUCATION CURRICULUM !SENIOR HIGH SCHOOL – SCIENCE, TECHNOLOGY, ENGINEERING AND MATHEMATICS (STEM) SPECIALIZED SUBJECT CONTENT PERFORMANCE TOPIC CONTENT LEARNING COMPETENCIES CODE STANDARDS STANDARDS NUMBER 11. illustrate different types of discontinuity STEM_BC11LC-IIId-1 4.1 (hole/removable, jump/essential, asymptotic/infinite) 12. illustrate the Intermediate Value and Extreme STEM_BC11LC-IIId-2 4.2 Value Theorems 13. solves problems involving continuity of a function STEM_BC11LC-IIId-3 4.3 Derivatives basic concepts of 1. formulate and solve 1. illustrate the tangent line to the graph of a function STEM_BC11D-IIIe-1 5.1 derivatives accurately situational at a given point problems involving 2. applies the definition of the derivative of a function STEM_BC11D-IIIe-2 5.3 extreme values at a given number 3. relate the derivative of a function to the slope of STEM_BC11D-IIIe-3 5.2 the tangent line 4. determine the relationship between differentiability STEM_BC11D -IIIf-1 6.1 and continuity of a function 5. derive the differentiation rules STEM_BC11D-IIIf-2 6. apply the differentiation rules in computing the 6.2 derivative of an algebraic, exponential, and STEM_BC11D-IIIf-3 trigonometric functions 7. solve optimization problems STEM_BC11D-IIIg-1 7.1 2. formulate and solve 8. compute higher-order derivatives of functions STEM_BC11D-IIIh-1 8.1 accurately situational 9. illustrate the Chain Rule of differentiation STEM_BC11D-IIIh-2 problems involving 8.2 10. solve problems using the Chain Rule STEM_BC11D-IIIh-i-1 related rates 11. illustrate implicit differentiation STEM_BC11D-IIIi-2 12. solve problems (including logarithmic, and inverse 9.1 trigonometric functions) using implicit STEM_BC11D-IIIi-j-1 differentiation 13. solve situational problems involving related rates STEM_BC11D-IIIj-2 10.1 K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL – SCIENCE, TECHNOLOGY, ENGINEERING AND MATHEMATICS (STEM) SPECIALIZED SUBJECT CONTENT PERFORMANCE TOPIC CONTENT LEARNING COMPETENCIES CODE STANDARDS STANDARDS NUMBER Integration antiderivatives and 1. formulate and solve 1. illustrate an antiderivative of a function STEM_BC11I-IVa-1 11.1 Riemann integral accurately situational 2. compute the general antiderivative of problems involving polynomial, radical, exponential, and trigonometric STEM_BC11I-IVa-b-1 11.2J11.4 population models functions 3. compute the antiderivative of a function using substitution rule and table of integrals (includingthose STEM_BC11I-IVb-c-1 12.1 whose antiderivatives involve logarithmic and inverse trigonometric functions) 4. solve separable differential equations using STEM_BC11I-IVd-1 13.1 antidifferentiation 5. solve situational problems involving exponential growth and decay, bounded growth, and logistic growth STEM_BC11I-IVe-f-1 14.1 2. formulate and solve 6. approximate the area of a region under a curve accurately real-life using Riemann sums: (a) left, (b) right, and (c) STEM_BC11I-IVg-1 15.1 problems involving areas midpoint of plane regions 7. define the definite integral as the limit of the STEM_BC11I-IVg-2 15.2 Riemann sums 8. illustrate the Fundamental Theorem of Calculus STEM_BC11I-IVh-1 16.1 9. compute the definite integral of a function using STEM_BC11I-IVh-2 16.2 the Fundamental Theorem of Calculus 10. illustrates the substitution rule STEM_BC11I-IVi-1 11. compute the definite integral of a function using 17.1 STEM_BC11I-IVi-2 the substitution rule 12. compute the area of a plane region using the STEM_BC11I-IVi-j-1 18.1 definite integral 13. solve problems involving areas of plane regions STEM_BC11I-IVj-2 18.2 Contents 1 Limits and Continuity 1 Lesson 1: The Limit of a Function: Theorems and Examples............... 2 Topic 1.1: The Limit of a Function.......................... 3 Topic 1.2: The Limit of a Function at c versus the Value of the Function at c.. 17 Topic 1.3: Illustration of Limit Theorems...................... 22 Topic 1.4: Limits of Polynomial, Rational, and Radical Functions......... 28 Lesson 2: Limits of Some Transcendental Functions and Some Indeterminate Forms.. 38 Topic 2.1: Limits of Exponential, Logarithmic, and Trigonometric Functions... 39 Topic 2.2: Some Special Limits............................ 46 Lesson 3: Continuity of Functions.............................. 52 Topic 3.1: Continuity at a Point............................ 53 Topic 3.2: Continuity on an Interval......................... 58 Lesson 4: More on Continuity................................ 64 Topic 4.1: Diﬀerent Types of Discontinuities..................... 65 Topic 4.2: The Intermediate Value and the Extreme Value Theorems....... 75 Topic 4.3: Problems Involving Continuity...................... 85 2 Derivatives 89 Lesson 5: The Derivative as the Slope of the Tangent Line............... 90 Topic 5.1: The Tangent Line to the Graph of a Function at a Point........ 91 Topic 5.2: The Equation of the Tangent Line.................... 100 Topic 5.3: The Definition of the Derivative...................... 107 Lesson 6: Rules of Diﬀerentiation.............................. 119 Topic 6.1: Diﬀerentiability Implies Continuity.................... 120 Topic 6.2: The Diﬀerentiation Rules and Examples Involving Algebraic, Expo- nential, and Trigonometric Functions..................... 126 Lesson 7: Optimization................................... 141 Topic 7.1: Optimization using Calculus........................ 142 Lesson 8: Higher-Order Derivatives and the Chain Rule................. 156 Topic 8.1: Higher-Order Derivatives of Functions.................. 157 Topic 8.2: The Chain Rule............................... 162 Lesson 9: Implicit Diﬀerentiation.............................. 168 Topic 9.1: What is Implicit Diﬀerentiation?..................... 169 Lesson 10: Related Rates.................................. 180 Topic 10.1: Solutions to Problems Involving Related Rates............. 181 3 Integration 191 Lesson 11: Integration.................................... 192 Topic 11.1: Illustration of an Antiderivative of a Function............. 193 Topic 11.2: Antiderivatives of Algebraic Functions................. 196 Topic 11.3: Antiderivatives of Functions Yielding Exponential Functions and Logarithmic Functions............................. 199 Topic 11.4: Antiderivatives of Trigonometric Functions............... 202 Lesson 12: Techniques of Antidiﬀerentiation........................ 204 Topic 12.1: Antidiﬀerentiation by Substitution and by Table of Integrals..... 205 Lesson 13: Application of Antidiﬀerentiation to Diﬀerential Equations......... 217 Topic 13.1: Separable Diﬀerential Equations..................... 218 Lesson 14: Application of Diﬀerential Equations in Life Sciences............. 224 Topic 14.1: Situational Problems Involving Growth and Decay Problems..... 225 Lesson 15: Riemann Sums and the Definite Integral.................... 237 Topic 15.1: Approximation of Area using Riemann Sums.............. 238 Topic 15.2: The Formal Definition of the Definite Integral............. 253 Lesson 16: The Fundamental Theorem of Calculus.................... 268 Topic 16.1: Illustration of the Fundamental Theorem of Calculus......... 269 Topic 16.2: Computation of Definite Integrals using the Fundamental Theorem of Calculus................................... 273 Lesson 17: Integration Technique: The Substitution Rule for Definite Integrals.... 280 Topic 17.1: Illustration of the Substitution Rule for Definite Integrals....... 281 Lesson 18: Application of Definite Integrals in the Computation of Plane Areas.... 292 Topic 18.1: Areas of Plane Regions Using Definite Integrals............ 293 Topic 18.2: Application of Definite Integrals: Word Problems........... 304 Biographical Notes 309 Chapter 1 Limits and Continuity LESSON 1: The Limit of a Function: Theorems and Examples TIME FRAME: 4 hours LEARNING OUTCOMES: At the end of the lesson, the learner shall be able to: 1. Illustrate the limit of a function using a table of values and the graph of the function; 2. Distinguish between lim f (x) and f (c); x!c 3. Illustrate the limit theorems; and 4. Apply the limit theorems in evaluating the limit of algebraic functions (polynomial, ratio- nal, and radical). LESSON OUTLINE: 1. Evaluation of limits using a table of values 2. Illustrating the limit of a function using the graph of the function 3. Distinguishing between lim f (x) and f (c) using a table of values x!c 4. Distinguishing between lim f (x) and f (c) using the graph of y = f (x) x!c 5. Enumeration of the eight basic limit theorems 6. Application of the eight basic limit theorems on simple examples 7. Limits of polynomial functions 8. Limits of rational functions 9. Limits of radical functions 10. Intuitive notions of infinite limits 2 TOPIC 1.1: The Limit of a Function DEVELOPMENT OF THE LESSON (A) ACTIVITY In order to find out what the students’ idea of a limit is, ask them to bring cutouts of news items, articles, or drawings which for them illustrate the idea of a limit. These may be posted on a wall so that they may see each other’s homework, and then have each one explain briefly why they think their particular cutout represents a limit. (B) INTRODUCTION Limits are the backbone of calculus, and calculus is called the Mathematics of Change. The study of limits is necessary in studying change in great detail. The evaluation of a particular limit is what underlies the formulation of the derivative and the integral of a function. For starters, imagine that you are going to watch a basketball game. When you choose seats, you would want to be as close to the action as possible. You would want to be as close to the players as possible and have the best view of the game, as if you were in the basketball court yourself. Take note that you cannot actually be in the court and join the players, but you will be close enough to describe clearly what is happening in the game. This is how it is with limits of functions. We will consider functions of a single variable and study the behavior of the function as its variable approaches a particular value (a constant). The variable can only take values very, very close to the constant, but it cannot equal the constant itself. However, the limit will be able to describe clearly what is happening to the function near that constant. (C) LESSON PROPER Consider a function f of a single variable x. Consider a constant c which the variable x will approach (c may or may not be in the domain of f ). The limit, to be denoted by L, is the unique real value that f (x) will approach as x approaches c. In symbols, we write this process as lim f (x) = L. x!c This is read, ‘ ‘The limit of f (x) as x approaches c is L.” 3 LOOKING AT A TABLE OF VALUES To illustrate, let us consider lim (1 + 3x). x!2 Here, f (x) = 1 + 3x and the constant c, which x will approach, is 2. To evaluate the given limit, we will make use of a table to help us keep track of the eﬀect that the approach of x toward 2 will have on f (x). Of course, on the number line, x may approach 2 in two ways: through values on its left and through values on its right. We first consider approaching 2 from its left or through values less than 2. Remember that the values to be chosen should be close to 2. x f (x) 1 4 1.4 5.2 1.7 6.1 1.9 6.7 1.95 6.85 1.997 6.991 1.9999 6.9997 1.9999999 6.9999997 Now we consider approaching 2 from its right or through values greater than but close to 2. x f (x) 3 10 2.5 8.5 2.2 7.6 2.1 7.3 2.03 7.09 2.009 7.027 2.0005 7.0015 2.0000001 7.0000003 Observe that as the values of x get closer and closer to 2, the values of f (x) get closer and closer to 7. This behavior can be shown no matter what set of values, or what direction, is taken in approaching 2. In symbols, lim (1 + 3x) = 7. x!2 4 EXAMPLE 1: Investigate lim (x2 + 1) x! 1 by constructing tables of values. Here, c = 1 and f (x) = x2 + 1. We start again by approaching 1 from the left. x f (x) 1.5 3.25 1.2 2.44 1.01 2.0201 1.0001 2.00020001 Now approach 1 from the right. x f (x) 0.5 1.25 0.8 1.64 0.99 1.9801 0.9999 1.99980001 The tables show that as x approaches 1, f (x) approaches 2. In symbols, lim (x2 + 1) = 2. x! 1 EXAMPLE 2: Investigate lim |x| through a table of values. x!0 Approaching 0 from the left and from the right, we get the following tables: x |x| x |x| 0.3 0.3 0.3 0.3 0.01 0.01 0.01 0.01 0.00009 0.00009 0.00009 0.00009 0.00000001 0.00000001 0.00000001 0.00000001 Hence, lim |x| = 0. x!0 5 EXAMPLE 3: Investigate x2 5x + 4 lim x!1 x 1 x2 5x + 4 by constructing tables of values. Here, c = 1 and f (x) =. x 1 Take note that 1 is not in the domain of f , but this is not a problem. In evaluating a limit, remember that we only need to go very close to 1; we will not go to 1 itself. We now approach 1 from the left. x f (x) 1.5 2.5 1.17 2.83 1.003 2.997 1.0001 2.9999 Approach 1 from the right. x f (x) 0.5 3.5 0.88 3.12 0.996 3.004 0.9999 3.0001 The tables show that as x approaches 1, f (x) approaches 3. In symbols, x2 5x + 4 lim = 3. x!1 x 1 EXAMPLE 4: Investigate through a table of values lim f (x) x!4 if 8 > x+3 if x < 1 x!0 x2 + 2 < h. lim f (x) if f (x) = 2x if x = 1 1 x!1 > > e. lim :p5x 1 > if x > 1 x!1 x + 1 12 4. Consider the function f (x) whose graph is shown below. y 6 Determine the following: 5 a. lim f (x) 4 x! 3 3 b. lim f (x) x! 1 2 c. lim f (x) x!1 1 d. lim f (x) x!3 5 4 3 2 1 1 2 3 4 5 6 x 1 e. lim f (x) x!5 2 5. Consider the function f (x) whose graph is shown below. y 6 What can be said about the limit of f (x) 5 4 a. at c = 1, 2, 3, and 4? 3 b. at integer values of c? 2 c. at c = 0.4, 2. 3, 4.7, and 5.5? 1 d. at non-integer values of c? 0 1 2 3 4 5 6 x 13 6. Consider the function f (x) whose graph is shown below. y 6 5 4 Determine the following: 3 a. lim f (x) 2 x! 1.5 1 b. lim f (x) x!0 5 4 3 2 1 0 1 2 3 4 5 6 x c. lim f (x) x!2 1 d. lim f (x) x!4 2 3 4 5 Teaching Tip Test how well the students have understood limit evaluation. It is hoped that by now they have observed that for polynomial and rational functions f , if c is in the domain of f , then to evaluate lim f (x) they just need to substitute the x!c value of c for every x in f (x). However, this is not true for general functions. Ask the students if they can give an example or point out an earlier example of a case where c is in the domain of f , but lim f (x) 6= f (c). x!c 7. Without a table of values and without graphing f (x), give the values of the following limits and explain how you arrived at your evaluation. a. lim (3x 5) x! 1 x2 9 b. lim where c = 0, 1, 2 x!c x2 4x + 3 x2 9 ?c. lim 2 x!3 x 4x + 3 14 1 ?8. Consider the function f (x) = whose graph is shown below. x y 5 4 3 1 2 f (x) = x 1 4 3 2 1 0 1 2 3 4 x 1 2 3 4 5 What can be said about lim f (x)? Does it exist or not? Why? x!0 Answer: The limit does not exist. From the graph itself, as x-values approach 0, the arrows move in opposite directions. If tables of values are constructed, one for x-values approaching 0 through negative values and another through positive values, it is easy to observe that the closer the x-values are to 0, the more negatively and positively large the corresponding f (x)-values become. ?9. Consider the function f (x) whose graph is y shown below. What can be said about lim f (x)? x!0 Does it exist or not? Why? 8 7 Answer: The limit does not exist. Although as 6 x-values approach 0, the arrows seem to move 5 in the same direction, they will not “stop” at a 4 limiting value. In the absence of such a definite 3 limiting value, we still say the limit does not 1 2 f (x) = x2 exist. (We will revisit this function in the lesson 1 about infinite limits where we will discuss more 4 3 2 1 0 1 2 3 4 x about its behavior near 0.) 15 ?10. Sketch one possible graph of a function f (x) defined on R that satisfies all the listed conditions. a. lim f (x) = 1 e. f (2) = 0 x!0 b. lim f (x) DNE x!1 f. f (4) = 5 c. lim f (x) = 0 x!2 d. f (1) = 2 g. lim f (x) = 5 for all c > 4. x!c Possible answer (there are many other possibilities): y 6 5 4 3 2 1 2 1 0 1 2 3 4 5 6 x 16 TOPIC 1.2: The Limit of a Function at c versus the Value of the Function at c DEVELOPMENT OF THE LESSON (A) INTRODUCTION Critical to the study of limits is the understanding that the value of lim f (x) x!c may be distinct from the value of the function at x = c, that is, f (c). As seen in previous examples, the limit may be evaluated at values not included in the domain of f. Thus, it must be clear to a student of calculus that the exclusion of a value from the domain of a function does not prohibit the evaluation of the limit of that function at that excluded value, provided of course that f is defined at the points near c. In fact, these cases are actually the more interesting ones to investigate and evaluate. Furthermore, the awareness of this distinction will help the student understand the concept of continuity, which will be tackled in Lessons 3 and 4. (B) LESSON PROPER We will mostly recall our discussions and examples in Lesson 1. Let us again consider lim (1 + 3x). x!2 Recall that its tables of values are: x f (x) x f (x) 1 4 3 10 1.4 5.2 2.5 8.5 1.7 6.1 2.2 7.6 1.9 6.7 2.1 7.3 1.95 6.85 2.03 7.09 1.997 6.991 2.009 7.027 1.9999 6.9997 2.0005 7.0015 1.9999999 6.9999997 2.0000001 7.0000003 and we had concluded that lim (1 + 3x) = 7. x!2 17 In comparison, f (2) = 7. So, in this example, lim f (x) and f (2) are equal. Notice that the x!2 same holds for the next examples discussed: lim f (x) f (c) x!c lim (x2 + 1) = 2 f ( 1) = 2 x! 1 lim |x| = 0 f (0) = 0 x!0 This, however, is not always the case. Let us consider the function 8 0. Therefore, by the Radical/Root Rule, x!1 p q p lim x= lim x = 1 = 1. x!1 x!1. p EXAMPLE 10: Evaluate lim x + 4. x!0 Solution. Note that lim (x + 4) = 4 > 0. Hence, by the Radical/Root Rule, x!0 p q p lim x+4= lim (x + 4) = 4 = 2. x!0 x!0. p 3 EXAMPLE 11: Evaluate lim x2 + 3x 6. x! 2 Solution. Since the index of the radical sign is odd, we do not have to worry that the limit of the radicand is negative. Therefore, the Radical/Root Rule implies that p r p p x + 3x 6 = 3 lim (x2 + 3x 6) = 3 4 6 6 = 3 8 = 2. 3 2 lim x! 2 x! 2. 32 p 2x + 5 EXAMPLE 12: Evaluate lim. x!2 1 3x Solution. First, note that lim (1 3x) = 5 6= 0. Moreover, lim (2x + 5) = 9 > 0. Thus, x!2 x!2 using the Division and Radical Rules of Theorem 1, we obtain p q p lim 2x + 5 lim (2x + 5) p 2x + 5 x!2 9 3 lim = x!2 = = =. x!2 1 3x lim 1 3x 5 5 5 x!2. INTUITIVE NOTIONS OF INFINITE LIMITS f (x) We investigate the limit at a point c of a rational function of the form where f and g g(x) are polynomial functions with f (c) 6= 0 and g(c) = 0. Note that Theorem 3 does not cover this because it assumes that the denominator is nonzero at c. y 8 1 7 Now, consider the function f (x) = 2. x 6 Note that the function is not defined at 5 x = 0 but we can check the behavior of the 4 function as x approaches 0 intuitively. We 3 first consider approaching 0 from the left. 1 2 f (x) = x2 1 4 3 2 1 0 1 2 3 4 x x f (x) 0.9 1.2345679 0.5 4 0.1 100 0.01 10, 000 0.001 1, 000, 000 0.0001 100, 000, 000 Observe that as x approaches 0 from the left, the value of the function increases without bound. When this happens, we say that the limit of f (x) as x approaches 0 from the left is positive infinity, that is, lim f (x) = +1. x!0 33 x f (x) 0.9 1.2345679 0.5 4 0.1 100 0.01 10, 000 0.001 1, 000, 000 0.0001 100, 000, 000 Again, as x approaches 0 from the right, the value of the function increases without bound, so, lim f (x) = +1. x!0+ Since lim f (x) = +1 and lim f (x) = +1, we may conclude that lim f (x) = +1. x!0 x!0+ x!0 4 3 2 1 0 1 2 3 4 x 1 1 1 2 f (x) = Now, consider the function f (x) =. x2 x2 3 Note that the function is not defined at x = 4 0 but we can still check the behavior of the 5 function as x approaches 0 intuitively. We 6 first consider approaching 0 from the left. 7 8 y x f (x) 0.9 1.2345679 0.5 4 0.1 100 0.01 10, 000 0.001 1, 000, 000 0.0001 100, 000, 000 This time, as x approaches 0 from the left, the value of the function decreases without bound. So, we say that the limit of f (x) as x approaches 0 from the left is negative infinity, that is, lim f (x) = 1. x!0 34 x f (x) 0.9 1.2345679 0.5 4 0.1 100 0.01 10, 000 0.001 1, 000, 000 0.0001 100, 000, 000 As x approaches 0 from the right, the value of the function also decreases without bound, that is, lim f (x) = 1. x!0+ Since lim f (x) = 1 and lim f (x) = 1, we are able to conclude that lim f (x) = 1. x!0 x!0+ x!0 We now state the intuitive definition of infinite limits of functions: The limit of f (x) as x approaches c is positive infinity, denoted by, lim f (x) = +1 x!c if the value of f (x) increases without bound whenever the values of x get closer and closer to c. The limit of f (x) as x approaches c is negative infinity, denoted by, lim f (x) = 1 x!c if the value of f (x) decreases without bound whenever the values of x get closer and closer to c. 1 y Let us consider f (x) =. The graph on x 5 the right suggests that 4 1 3f (x) = lim f (x) = 1 x x!0 2 1 while lim f (x) = +1. 4 3 2 1 0 1 2 3 4 x x!0+ 1 Because the one-sided limits are not the 2 3 same, we say that 4 5 lim f (x) DNE. x!0 Remark 1: Remember that 1 is NOT a number. It holds no specific value. So, lim f (x) = x!c +1 or lim f (x) = 1 describes the behavior of the function near x = c, but it does not x!c exist as a real number. 35 Remark 2: Whenever lim f (x) = ±1 or lim f (x) = ±1, we normally see the dashed x!c+ x!c vertical line x = c. This is to indicate that the graph of y = f (x) is asymptotic to x = c, meaning, the graphs of y = f (x) and x = c are very close to each other near c. In this case, we call x = c a vertical asymptote of the graph of y = f (x). Teaching Tip Computing infinite limits is not a learning objective of this course, however, we will be needing this notion for the discussion on infinite essential discontinuity, which will be presented in Topic 4.1. It is enough that the student determines that the limit at the point c is +1 or 1 from the behavior of the graph, or the trend of the y-coordinates in a table of values. (C) EXERCISES I. Evaluate the following limits. p 1. lim (1 + 3 w)(2 w2 + 3w3 ) 4 3y 2 y 3 w!1 5. lim y! 2 6 y y2 t2 1 3 7x2 + 14x 8 2. lim 6. lim x t! 2 t2 + 3t 1 2 p 2x 3x 4 x! 1 ✓ 2 ◆3 2z + z 2 x +3 2 3. lim 7. lim z!2 z2 + 4 2 p x + 1 x! 1 p x2 x 2 2x 6 x 4. lim 8. lim x!0 x3 6x2 7x + 1 x!2 4 + x2 II. Complete the following tables. x 5 x x 5 x x x x 3 x2 6x + 9 x 3 x2 6x + 9 2.5 3.5 2.8 3.2 2.9 3.1 2.99 3.01 2.999 3.001 2.9999 3.0001 From the table, determine the following limits. x 5 x 1. lim 4. lim x!3 x 3 x!3 x2 6x + 9 x 5 x 2. lim 5. lim x!3+ x 3 x!3+ x2 6x + 9 x 5 x 3. lim 6. lim x!3 x 3 x!3 x2 6x + 9 36 III. Recall the graph of y = csc x. From the behavior of the graph of the cosecant function, determine if the following limits evaluate to +1 or to 1. 1. lim csc x 3. lim csc x x!0 x!⇡ 2. lim csc x 4. lim csc x x!0+ x!⇡ + IV. Recall the graph of y = tan x. 1. Find the value of c 2 (0, ⇡) such that lim tan x = +1. x!c 2. Find the value of d 2 (⇡, 2⇡) such that lim tan x = 1. x!d+ 37 LESSON 2: Limits of Some Transcendental Functions and Some Indeterminate Forms TIME FRAME: 4 hours LEARNING OUTCOMES: At the end of the lesson, the learner shall be able to: 1. Compute the limits of exponential, logarithmic, and trigonometric functions using tables of values and graphs of the functions; sin t 1 cos t et 1 2. Evaluate the limits of expressions involving , , and using tables of t t t values; and “ 0” 3. Evaluate the limits of expressions resulting in the indeterminate form. 0 LESSON OUTLINE: 1. Exponential functions 2. Logarithmic functions 3. Trigonometric functions sin t 4. Evaluating lim t!0 t 1 cos t 5. Evaluating lim t!0 t et 1 6. Evaluating lim t!0 t “ 0” 7. Indeterminate form 0 38 TOPIC 2.1: Limits of Exponential, Logarithmic, and Trigonomet- ric Functions DEVELOPMENT OF THE LESSON (A) INTRODUCTION Real-world situations can be expressed in terms of functional relationships. These func- tional relationships are called mathematical models. In applications of calculus, it is quite important that one can generate these mathematical models. They sometimes use functions that you encountered in precalculus, like the exponential, logarithmic, and trigonometric functions. Hence, we start this lesson by recalling these functions and their corresponding graphs. (a) If b > 0, b 6= 1, the exponential function with base b is defined by f (x) = bx , x 2 R. (b) Let b > 0, b 6= 1. If by = x then y is called the logarithm of x to the base b, denoted y = logb x. Teaching Tip Allow students to use their calculators. (B) LESSON PROPER EVALUATING LIMITS OF EXPONENTIAL FUNCTIONS First, we consider the natural exponential function f (x) = ex , where e is called the Euler number, and has value 2.718281.... EXAMPLE 1: Evaluate the lim ex. x!0 Solution. We will construct the table of values for f (x) = ex. We start by approaching the number 0 from the left or through the values less than but close to 0. Teaching Tip Some students may not be familiar with the natural number e on their scientific calculators. Demonstrate to them how to properly input powers of e on their calculators. 39 x f (x) 1 0.36787944117 0.5 0.60653065971 0.1 0.90483741803 0.01 0.99004983374 0.001 0.99900049983 0.0001 0.999900049983 0.00001 0.99999000005 Intuitively, from the table above, lim ex = 1. Now we consider approaching 0 from its x!0 right or through values greater than but close to 0. x f (x) 1 2.71828182846 0.5 1.6487212707 0.1 1.10517091808 0.01 1.01005016708 0.001 1.00100050017 0.0001 1.000100005 0.00001 1.00001000005 From the table, as the values of x get closer and closer to 0, the values of f (x) get closer and closer to 1. So, lim ex = 1. Combining the two one-sided limits allows us to conclude x!0+ that lim ex = 1. x!0. We can use the graph of f (x) = ex to determine its limit as x approaches 0. The figure below is the graph of f (x) = ex. y Looking at Figure 1.1, as the values of x approach 0, either from the right or the left, the values of f (x) will get closer and closer to 1. We also have the following: (a) lim ex = e = 2.718... x!1 y = ex (b) lim ex = e2 = 7.389... x!2 (c) lim ex = e 1 = 0.367... x! 1 1 3 2 1 0 1 2 3 x 40 EVALUATING LIMITS OF LOGARITHMIC FUNCTIONS Now, consider the natural logarithmic function f (x) = ln x. Recall that ln x = loge x. Moreover, it is the inverse of the natural exponential function y = ex. EXAMPLE 2: Evaluate lim ln x. x!1 Solution. We will construct the table of values for f (x) = ln x. We first approach the number 1 from the left or through values less than but close to 1. x f (x) 0.1 2.30258509299 0.5 0.69314718056 0.9 0.10536051565 0.99 0.01005033585 0.999 0.00100050033 0.9999 0.000100005 0.99999 0.00001000005 Intuitively, lim ln x = 0. Now we consider approaching 1 from its right or through values x!1 greater than but close to 1. x f (x) 2 0.69314718056 1.5 0.4054651081 1.1 0.0953101798 1.01 0.00995033085 1.001 0.00099950033 1.0001 0.000099995 1.00001 0.00000999995 Intuitively, lim ln x = 0. As the values of x get closer and closer to 1, the values of f (x) x!1+ get closer and closer to 0. In symbols, lim ln x = 0. x!1. We now consider the common logarithmic function f (x) = log10 x. Recall that f (x) = log10 x = log x. 41 EXAMPLE 3: Evaluate lim log x. x!1 Solution. We will construct the table of values for f (x) = log x. We first approach the number 1 from the left or through the values less than but close to 1. x f (x) 0.1 1 0.5 0.30102999566 0.9 0.04575749056 0.99 0.0043648054 0.999 0.00043451177 0.9999 0.00004343161 0.99999 0.00000434296 Now we consider approaching 1 from its right or through values greater than but close to 1. x f (x) 2 0.30102999566 1.5 0.17609125905 1.1 0.04139268515 1.01 0.00432137378 1.001 0.00043407747 1.0001 0.00004342727 1.00001 0.00000434292 As the values of x get closer and closer to 1, the values of f (x) get closer and closer to 0. In symbols, lim log x = 0. x!1. Consider now the graphs of both the natural and common logarithmic functions. We can use the following graphs to determine their limits as x approaches 1.. 42 f (x) = ln x f (x) = log x 0 1 2 3 4 5 6 7 x The figure helps verify our observations that lim ln x = 0 and lim log x = 0. Also, based x!1 x!1 on the figure, we have (a) lim ln x = 1 (d) lim log x = log 3 = 0.47... x!e x!3 (b) lim log x = 1 (e) lim ln x = 1 x!10 x!0+ (c) lim ln x = ln 3 = 1.09... (f) lim log x = 1 x!3 x!0+ TRIGONOMETRIC FUNCTIONS EXAMPLE 4: Evaluate lim sin x. x!0 Solution. We will construct the table of values for f (x) = sin x. We first approach 0 from the left or through the values less than but close to 0. x f (x) 1 0.8414709848 0.5 0.4794255386 0.1 0.09983341664 0.01 0.00999983333 0.001 0.00099999983 0.0001 0.00009999999 0.00001 0.00000999999 Now we consider approaching 0 from its right or through values greater than but close to 0. 43 x f (x) 1 0.8414709848 0.5 0.4794255386 0.1 0.09983341664 0.01 0.00999983333 0.001 0.00099999983 0.0001 0.00009999999 0.00001 0.00000999999 As the values of x get closer and closer to 1, the values of f (x) get closer and closer to 0. In symbols, lim sin x = 0. x!0. We can also find lim sin x by using the graph of the sine function. Consider the graph of x!0 f (x) = sin x. 1 ⇡ ⇡ ⇡ ⇡ 3⇡ 2⇡ 5⇡ 3⇡ 2 2 2 2 1 The graph validates our observation in Example 4 that lim sin x = 0. Also, using the x!0 graph, we have the following: (a) lim⇡ sin x = 1. (c) lim⇡ sin x = 1. x! 2 x! 2 (b) lim sin x = 0. (d) lim sin x = 0. x!⇡ x! ⇡ Teaching Tip Ask the students what they have observed about the limit of the functions above and their functional value at a point. Lead them to the fact that if f is either exponential, logarithmic or trigonometric, and if c is a real number which is in the domain of f , then lim f (x) = f (c). x!c This property is also shared by polynomials and rational functions, as discussed in Topic 1.4. 44 (C) EXERCISES I. Evaluate the following limits by constructing the table of values. 1. lim 3x 5. lim tan x x!1 x!0 x 2. lim 5 x!2 ?6. lim cos x Answer: -1 3. lim log x x!⇡ x!4 4. lim cos x ?7. lim sin x Answer: 0 x!0 x!⇡ II. Given the graph below, evaluate the following limits: y y = bx 1 x 1. lim bx 2. lim bx 3. lim bx x!0 x!1.2 x! 1 III. Given the graph of the cosine function f (x) = cos x, evaluate the following limits: 1 ⇡ ⇡ ⇡ ⇡ 3⇡ 2⇡ 5⇡ 3⇡ 2 2 2 2 1 1. lim cos x 2. lim cos x 3. lim⇡ cos x x!0 x!⇡ x! 2 45 TOPIC 2.2: Some Special Limits DEVELOPMENT OF THE LESSON (A) INTRODUCTION sin t We will determine the limits of three special functions; namely, f (t) = , g(t) = t 1 cos t et 1 , and h(t) =. These functions will be vital to the computation of the t t derivatives of the sine, cosine, and natural exponential functions in Chapter 2. (B) LESSON PROPER THREE SPECIAL FUNCTIONS sin t We start by evaluating the function f (t) =. t sin t EXAMPLE 1: Evaluate lim. t!0 t sin t Solution. We will construct the table of values for f (t) =. We first approach the t number 0 from the left or through values less than but close to 0. t f (t) 1 0.84147099848 0.5 0.9588510772 0.1 0.9983341665 0.01 0.9999833334 0.001 0.9999998333 0.0001 0.99999999983 Now we consider approaching 0 from the right or through values greater than but close to 0. t f (t) 1 0.8414709848 0.5 0.9588510772 0.1 0.9983341665 0.01 0.9999833334 0.001 0.9999998333 0.0001 0.9999999983 46 sin t sin t Since lim and lim are both equal to 1, we conclude that t!0 t t!0+ t sin t lim = 1. t!0 t. sin t The graph of f (t) = below confirms that the y-values approach 1 as t approaches 0. t sin t 1 y= t 0.5 8 6 4 2 0 2 4 6 8 1 cos t Now, consider the function g(t) =. t 1 cos t EXAMPLE 2: Evaluate lim. t!0 t 1 cos t Solution. We will construct the table of values for g(t) =. We first approach the t number 1 from the left or through the values less than but close to 0. t g(t) 1 0.4596976941 0.5 0.2448348762 0.1 0.04995834722 0.01 0.0049999583 0.001 0.0004999999 0.0001 0.000005 Now we consider approaching 0 from the right or through values greater than but close to 0. t g(t) 1 0.4596976941 0.5 0.2448348762 0.1 0.04995834722 0.01 0.0049999583 0.001 0.0004999999 0.0001 0.000005 47 1 cos t 1 cos t Since lim = 0 and lim = 0, we conclude that t!0 t t!0+ t 1 cos t lim = 0. t!0 t. 1 cos t Below is the graph of g(t) =. We see that the y-values approach 0 as t tends to 0. t 1 1 cos t y= t 0.5 8 6 4 2 0 2 4 6 8 0.5 et 1 We now consider the special function h(t) =. t et 1 EXAMPLE 3: Evaluate lim. t!0 t et 1 Solution. We will construct the table of values for h(t) =. We first approach the t number 0 from the left or through the values less than but close to 0. t h(t) 1 0.6321205588 0.5 0.7869386806 0.1 0.9516258196 0.01 0.9950166251 0.001 0.9995001666 0.0001 0.9999500016 Now we consider approaching 0 from the right or through values greater than but close to 0. t h(t) 1 1.718281828 0.5 1.297442541 0.1 1.051709181 0.01 1.005016708 0.001 1.000500167 0.0001 1.000050002 48 et 1 et 1 Since lim = 1 and lim = 1, we conclude that x!0 t x!0+ t et 1 lim = 1. x!0 t. et 1 The graph of h(t) = below confirms that lim h(t) = 1. t t!0 1.5 1 et 1 y= t 0.5 8 6 4 2 0 2 “ 0” INDETERMINATE FORM 0 There are functions whose limits cannot be determined immediately using the Limit The- orems we have so far. In these cases, the functions must be manipulated so that the limit, if it exists, can be calculated. We call such limit expressions indeterminate forms. “ 0” In this lesson, we will define a particular indeterminate form, , and discuss how to 0 evaluate a limit which will initially result in this form. “ 0” Definition of Indeterminate Form of Type 0 f (x) If lim f (x) = 0 and lim g(x) = 0, then lim is called an indeterminate form x!c x!c x!c g(x) “ 0” of type. 0 “ 0” Remark 1: A limit that is indeterminate of type may exist. To find the actual 0 value, one should find an expression equivalent to the original. This is commonly done by factoring or by rationalizing. Hopefully, the expression that will emerge after factoring or rationalizing will have a computable limit. x2 + 2x + 1 EXAMPLE 4: Evaluate lim. x! 1 x+1 Solution. The limit of both the numerator and the denominator as x approaches 1 is 0. 0 Thus, this limit as currently written is an indeterminate form of type. However, observe 0 that (x + 1) is a factor common to the numerator and the denominator, and x2 + 2x + 1 (x + 1)2 = = x + 1, when x 6= 1. x+1 x+1 49 Therefore, x2 + 2x + 1 lim = lim (x + 1) = 0. x! 1 x+1 x! 1. x2 1 EXAMPLE 5: Evaluate lim p. x!1 x 1 p x2 1 Solution. Since lim x2 1 = 0 and lim x 1 = 0, then lim p is an indeterminate x!1 x!1 x!1 x 1 “ 0” form of type. To find the limit, observe that if x 6= 1, then 0 p p x2 1 x+1 (x 1)(x + 1)( x + 1) p p ·p = = (x + 1)( x + 1). x 1 x+1 x 1 So, we have x2 1 p lim p = lim (x + 1)( x + 1) = 4. x!1 x + 1 x!1. Teaching Tip In solutions of evaluating limits, it is a common mistake among students to forget to write the “lim" operator. They will write x2 1 p lim p = (x + 1)( x + 1) = 4, x!1 x+1 instead of always writing the limit operator until such time that they are already substituting the value x = 1. Of course, mathematically, the equation above does p not make sense since (x + 1)( x + 1) is not always equal to 4. Please stress the importance of the “lim" operator. Remark 2: We note here that the three limits discussed in Part 1 of this

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