chapter 2: limits and continuity pdf

489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 Verify the value of the limit of a function at a point using the definition of the limit; 2. The Mathematica command Plot[f[x],{x,a,b}] /LastChar 196 >> 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 491.3 383.7 615.2 517.4 762.5 598.1 525.2 494.2 349.5 400.2 673.4 531.3 295.1 0 0 0000036468 00000 n CHAPTER 2 Limits and Continuity 2.1 Limits If the values of f (x) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a) but x ≠ a. endobj (a) Does not exist. 15 0 obj 0000002186 00000 n "proofs) limit laws multiplying by a clever form of 1 to nd a limit Sandwich Theorem one-sided limits, including the theorem relating one- and two-sided limits limits involving absolute value >> §1.2 Limits of Functions The concept of limit is the cornerstone on which the development of Calculus rests. 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 There is no single number L that all the values g(x) get arbitrarily close to as x 1. 1); (x;y. �� ^.%�X��*.``0vX�mV掸�Ȩ̟�˟d�߸�~��Yb��3��.��D�I�.�THZ�/��fޜ�\L&�Ŷ�f(yH. >> 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 Determine whether a function is continuous at a number. 2.5 Continuity 1 Chapter 2. stream << 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 CHAPTER 2 LIMITS AND CONTINUITY 2.1 RATES OF CHANGE AND LIMITS 1. 1 lim Actual limit is 4. x→ 3 1 x 2 x 3 0.25 x 3.1 3.01 3.001 2.999 2.99 2.9 f x 0.2485 0.2498 0.2500 0.2500 0.2502 0.2516 5. (a) −0.25 −0.1 −0.001 −0.0001 0.0001 0.001 In this section, we develop the notion of limit using some common language and illustrate the idea with some simple examples. A.P. 16. /FontDescriptor 11 0 R œ_ if for every positive number B (or negative number B) there exists a corresponding number 0 such $ that for all x, x x x f(x) B.!! For B 0, B 0 x . The following are the daily homework assignments for Chapter 2 – Limits and Continuity Section Pages Topics Assignment 2.1 Day 1 9/15 p.63-72 Limits of function values, limits by tables and graphs, definition of a limit, two-sided limits p.72-75: # 5 – 21 odd Worksheet 2.1-1 2.1 Day 2 9/16 p.63-72 Rules of Calculating Limits~ Part 1 You cannot use substitution because the expression 1 x2 is not defined at x = 0. /Name/F3 Multiple Choice _____ 1. (a) The limit of a sum is equal to the sum of the limits, namely lim x!+1 f(x) = L and lim x!+1 g(x) = M =) lim x!+1 [f(x)+g(x)] = L+M: (b) The limit of a product is equal to the product of the limits, namely lim x!+1 f(x) = L and lim x!+1 g(x) = M =) lim x!+1 0000043923 00000 n Using the language of left and right hand limits, we may say that the left (respectively right) hand limit of f at 0 is 1 (respectively 2). 80 Chapter 2 Limits and Continuity (c) We say that f(x) approaches minus infinity as x approaches x from the left, and write lim f(x) ,! /Name/F1 0000029398 00000 n /LastChar 196 29 0 obj 2.5 Continuity 1 Chapter 2. Chapter 2 Limits and Continuity In this chapter, we develop the concept of and skills for calculating something called a limit, which is the genesis of calculus. We saw in Section 2.2 that, by Dr. Bob’s Anthropomorphic Deﬁnition of Limit, a function f deﬁned on an open interval about c, except possibly at c itself, has a limit L, lim x→c f(x) = L, if the graph of y = … << 0000039855 00000 n 0000049502 00000 n 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 0000020216 00000 n As x approaches 1 from the left, g(x) approaches 1. /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 14 CHAPTER 2. 0000023014 00000 n endobj 46 Chapter 2 (c) −0.25 −0.1 −0.001 −0.0001 −7.4641 −19.487 −1999.5 −20000 0-100-0.25 0 The limit is −∞. 46 Chapter 2 Limits and Continuity Q Slope of PQ œ? /FontDescriptor 17 0 R /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 ©2007 Pearson Education Asia Limits Limits (Continued) Continuity Continuity Applied to Inequalities 10.1) 10.2) 10.3) Chapter 10: Limits and Continuity Chapter OutlineChapter Outline 10.4) 6. A continuous process is one that takes place gradually—without << 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 0000001416 00000 n A t Q (5 3.9) 2.1 gal/day" ßœ 3.9 6 54 Q (4.5 4.8) 2.4 gal/day# ßœ 4.8 6 4.5 4 Q (4.1 5.7) 3 gal/day$ ßœ 5.7 6 18 0 obj LESSON 2: CONTINUITY AND LIMITS PREPARED BY: ENGR. 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 A limit is defined as a number approached by the function as an independent function’s variable approaches a particular value. Since 1 x2 becomes arbitrarily large as Chapter 2. /LastChar 196 �o��X�n��R��~m�lk#b� >^�0&&��pM�zJJ��c�%I�C�\� � j��ڔ7(P��. 777.8 777.8 777.8 777.8 777.8 1000 1000 777.8 666.7 555.6 540.3 540.3 429.2] View CHAPTER-2-LIMITS-AND-CONTINUITY.pdf from ENG 18 at Mariano Marcos State University. 0000023910 00000 n 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x ≈ 2.526 Chapter 2 Limits and Continuity Section 2.1 Rates of Change and Limits (pp. x�b```f``�� ]� Ȁ �@16��8��h46 666.7 722.2 722.2 1000 722.2 722.2 666.7 1888.9 2333.3 1888.9 2333.3 0 555.6 638.9 (a) Does not exist. 0000040143 00000 n /LastChar 196 2.1 Rates of Change and Tangents to Curves 1 Chapter 2. 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 We can express this as a limit . Chapter 2 Limits and Continuity in Higher Dimensions This topic mentions about limits and Chapter 4: More Derivatives. 0000035071 00000 n /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 As x approaches 1 from the left, g(x) approaches 1. 0000046028 00000 n 0000040214 00000 n >> Mathematic I Prepared by Dr. Sahar abdul Hadi 1Page LIMITS AND CONTINUITY CHAPTER TWO LIMITS AND CONTINUITY LIMITS OF FUNCTION: Y-axis Definitions: L y=f(x) ܔܑܕ࢞→ࢇࢌ(࢞) = ࡸ ܯ݁ܽ݊ ݐℎܽݐ ݓℎ݁݊ ܽ … 46 Chapter 2 Limits and Continuity Q Slope of PQ œ? /Length 1580 Both procedures are based on the fundamental concept of the limit of a function. 0000002270 00000 n 0000004809 00000 n 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 chapter 2 limits and continuity 2.1 rates of change and tangents to curves 1. %PDF-1.2 CHAPTER 2 LIMITS AND CONTINUITY 2.1 RATES OF CHANGE AND LIMITS 1. 46 Chapter 2 Limits and Continuity Q Slope of PQ œ? c. Looking at the graph, describe this trip in words. /Subtype/Type1 View Multivariable Analysis Lecture(Page 13-22).pdf from INFORMATIQ 123 at Université de Strasbourg. Chapter 2 LIMITS AND CONTINUITY The concept of limit of a function is one of the fundamental ideas that distinguishes calculus from algebra and trigonometry. Choose . Hence we say that the limit of x x f x sin ( ) , x is in radians and x 0 … trailer <<332CAB9DD8D111DCB4C90016CBCA733C>]>> startxref 0 %%EOF 298 0 obj<>stream 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 /FontDescriptor 26 0 R 0000032275 00000 n Calculus Test Chapter 2 Limits and Continuity Name _____ I. MARK LYNDON D. BRILLANTES LESSON 2: 2 Limits and Continuity Outline 2.1 Notion of a Limit 2.2 Properties of Limits 2.3 Operations and Evaluation with Limits 2.4 Unbounded Functions 2.5 One-Sided Limits 2.6 Continuity Learning Outcomes 1. /FirstChar 33 This chapter deals with the concepts of limit and continuity for real functions of one real variable. /Widths[777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 Chapter 2 Limits and Continuity Exercise 2.2 27E The following function has been given in the problem: Chapter 2 Limits and Continuity Exercise 2.2 28E. This underlying concept is the thread that binds together virtually all of the calculus you are about to study. Chapter 2 Limits and Continuity Exercise 2.2 26E. y = f(x) y = f(x) x a y x a y x a y y = f(x) (a) (b) (c) 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 Chapter 2 Limits and Continuity in Higher Dimensions This topic mentions about limits and /Name/F5 Chapter 2 Limits and Continuity In this chapter, we develop the concept of and skills for calculating something called a limit, which is the genesis of calculus. This underlying concept is the thread that binds together virtually all of the calculus you are about to study. 2.1 Rates of change and tangents to curves How do the values of a function f(x) behave when its argument xgets closer and closer to a given There is no single number L that all the values g(x) get arbitrarily close to as x 1.Ä (b) 1 (c) 0 2. /Subtype/Type1 46 Chapter 2 (c) −0.25 −0.1 −0.001 −0.0001 −7.4641 −19.487 −1999.5 −20000 0-100-0.25 0 The limit is −∞. 0000006949 00000 n 777.8 777.8 777.8 500 277.8 222.2 388.9 611.1 722.2 611.1 722.2 777.8 777.8 777.8 << ��%�� (a) 0 (b) 1 (c) Does not exist. /Name/F6 /BaseFont/ZNAVNQ+CMMI8 /Type/Font 0000015425 00000 n 2 Now we want to know instantaneous velocityat t = 2 seconds, for example. January 27, 2005 11:43 L24-ch02 Sheet number 1 Page number 49 black CHAPTER 2 Limits and Continuity EXERCISE SET 2.1 1. 243 0 obj <> endobj xref 243 56 0000000016 00000 n 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 9 0 obj /LastChar 196 Chapter 6: The Definite Integral. View Multivariable Analysis Lecture(Page 13-22).pdf from INFORMATIQ 123 at Université de Strasbourg. Limits and Continuity Note. ��n�K��6d2�oBPU�A�#�Tp�Ä�_�6�PaJAc��/5"��H]�@L��$�PR~�+�h�D\�oe�� fU,P��]y}�+�.X�U_5rDי]ոs�MQ��}m$[m��gkC2&t�@F$�I��t'��w�y�o�JW~k�@8��cE�ˇ�Pę3�^eև ֆ�G�)?�B��l��e�[W��^��9P��q4Ad!h��a�/�Q}��{Ү�ٰS�Y"��`�\I��[�Vn1|�*�׻��ʭ(*CXf��|{��knv L�\��0G@a��B��ӈ(�%E��U��5��=ܜ.��#��;U�4QeJ/�P xxÄ! Chapter 5: Applications of Derivatives. 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 /FirstChar 33 04�8b1``It�`h�v,*�!��5�l�46�W��2:�l��f4��%-?�Ap endobj /LastChar 127 /Type/Font Chapter 5 CONTINUITY AND DIFFERENTIABILITY Sir Issac Newton (1642-1727) Fig 5.1 148 MATHEMATICS 0.001, the value of the function is 2. 16 2 h 2 16 2 2 h 60 Chapter 2 Limits and Continuity Figure 2.1 (a) A graph and (b) table of values for f x sin x x that suggest the limit of f as x approaches 0 is 1. 27 0 obj Functions are the major tools for describing the real world in mathematical terms. 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] 0000043569 00000 n 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 (a) 0 (b) 1 (c) Does not exist. /FirstChar 33 59–69) ... defined at points near x = –2, the limit does not exist. As x approaches 1 from the right, g(x) approaches 0. 0000028794 00000 n /FontDescriptor 20 0 R Chapter 2 LIMITS AND CONTINUITY The concept of limit of a function is one of the fundamental ideas that distinguishes calculus from algebra and trigonometry. Intuitively speaking, the limit process involves examining the behavior of a function f(x) as x approaches a number c that may or may not be in the domain of f. 6/69. /BaseFont/XPAEFQ+CMMI12 Chapter 2 Limits and Continuity 2.1 Tangent lines and velocity Tangent lines Deﬁnition 2.1.1.Let f beafunctiondeﬁnedonanopeninterval I containinga >> So the one-sided limits exist but do not agree. 0000006029 00000 n You already probably have an intuitive idea of what it means for a function to be continuous. Table 2.1 Average Speeds over Short Time Intervals Starting at y t Length of Average Speed Time Interval, for Interval h (sec) y t (ft/sec) 180 0.1 65.6 0.01 64.16 0.001 64.016 0.0001 64.0016 LIMITS AND CONTINUITY Proposition 2.27 (Properties of limits). 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 Limits and Continuity Note. Limits are used to deﬁne all of the topics covered in limits_1_the_limit_and_computing_the_limit_completed_v2.pdf: File Size: 5596 kb: File Type: pdf 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 /Type/Font 0000038350 00000 n In this chapter, we develop the most fundamental idea behind calculus, that of a limit. Limits and continuity concept is one of the most crucial topics in calculus. ©2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.1 Limits10.1 Limits Example 1 – Estimating a Limit from a Graph • The limit of f(x) as x approaches a is the number L, written as a. 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 obj 0000006486 00000 n CHAPTER 2 LIMITS AND CONTINUITY 2.1 RATE OF CHANGE AND LIMITS 1. (a) −0.25 −0.1 −0.001 −0.0001 0.0001 0.001 << 0000002460 00000 n Chapter 2 (Calculus) – Limits and Continuity 2.1 Rates of Change and Limits Average and Instantaneous Speed A moving body’s average speed during an interval of time is found by _____ _____ Example 1 – Finding an Average Speed Use the fact that a dense solid … In later chapters you’ll see that the derivative, the integral and inﬁnite series—all of calculus depends on it, so your understanding of the topic is critical. CHAPTER 2 LIMITS AND CONTINUITY 2.1 RATES OF CHANGE AND TANGENTS TO CURVES 1. 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.9 885.4 806.2 736.8 Limits are used to deﬁne all of the topics covered in Calculus 1, 2, and 3 … Limits and Continuity 2.5 Continuity Note. View LESSON 2 - CONTINUITY AND LIMITS.pdf from BSCE 1 at Western Institute of Technology - Lapaz, Iloilo City. >> endobj /Name/F4 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 CHAPTER 2 LIMITS AND CONTINUITY 2.1 RATE OF CHANGE AND LIMITS 1. /FirstChar 33 2 We already know how to nd the average velocity over an interval of time. 2 Now we want to know instantaneous velocityat t = 2 seconds, for example. 21. /Subtype/Type1 (a) 19 (b) 1???? 0000020009 00000 n f289 f 20 x3 1 x 1(1) ... œœœœÄÄÊˆ‰ˆ‰ab2h 3 2 344hh 31 4hh 2 … /BaseFont/ADPNHM+CMBX12 0000040508 00000 n Chapter 2: Limits and Continuity. 0000039634 00000 n 2.1 Rates of change and tangents to curves How do the values of a function f(x) behave when its argument xgets closer and closer to a given 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 Calculus: Graphical, Numerical, Algebraic, 3rd Edition Answers Ch 2 Limits and Continuity Ex 2.1 Calculus: Graphical, Numerical, Algebraic Answers Chapter 2 Limits and Continuity Exercise 2.1 1E Chapter 2 Limits and Continuity Exercise 2.1 1QR Chapter 2 Limits and Continuity Exercise 2.1 2E Chapter 2 Limits and Continuity Exercise 2.1 2QR Chapter 2 Limits and […] 0000019289 00000 n x��YKo�F��W�9��.�AMm�"��J�#R��;�r)R6[i��D��;��fv5�HM�&��d��L'��L��I4��H�~�z��ʮ�|Nc%��xJyVl��2�l{i^�6�iP��/j|��^Da� 0000023356 00000 n Ê $ 94. Let’s call the function that pairs the distance traveled d in time t function f. So in function notation we have d f(t). 48 Chapter 1 Limits and Their Properties 4. 2 We already know how to nd the average velocity over an interval of time. If 1 x fx x − = and gx x() 1= − Then fgx( ( )) = (A) 1 x x − − (B) 1 1−x (C) 1 1 x x − − (D) 1 x (E) 1 x −x _____ 2. As x approaches 1 from the right, g(x) approaches 0. /Type/Font /Filter[/FlateDecode] chapter 2 limits and continuity 2.1 rates of change and tangents to curves 1. 14.2 – Multivariable Limits 14.2 Limits and Continuity In this section, we will learn about: Limits and continuity of various types of functions. 0000003244 00000 n 0000036943 00000 n 0000046870 00000 n << Limits and Continuity 2.5 Continuity Note. endobj Chapter 2 Limits and Continuity 2.1 Limits The limits is the fundamental notion of calculus. CHAPTER 2: Limits and Continuity 2.1: An Introduction to Limits 2.2: Properties of Limits 2.3: Limits and Infinity I: Horizontal Asymptotes (HAs) 2.4: Limits and Infinity II: Vertical Asymptotes (VAs) 2.5: The Indeterminate Forms 0/0 and / 2.6: The Squeeze (Sandwich) Theorem 2.7: Precise Definitions of Limits 2.8: Continuity We saw in Section 2.2 that, by Dr. Bob’s Anthropomorphic Deﬁnition of Limit, a function f deﬁned on an open interval about c, except possibly at c itself, has a limit L, lim x→c f(x) = L, if the graph of … 694.5 295.1] Chapter 2 Limits and Continuity 2.1 Limits The limits is the fundamental notion of calculus. 0000045167 00000 n 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 /Type/Font Chapter 2 Limits and Continuity Page 2 of 14 The graph below is a plot of distance vs. time for your trip. 0000019635 00000 n /FontDescriptor 14 0 R 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 << 0000003775 00000 n %PDF-1.4 %�� 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 Afunction fis a set of ordered pairs of numbers (x;y) satisfying the property that if (x;y. A t Q (5 3.9) 2.1 gal/day" ßœ 3.9 6 54 Q (4.5 4.8) 2.4 gal/day# ßœ 4.8 6 4.5 4 Q (4.1 5.7) 3 gal/day$ ßœ 5.7 6 As x approaches 1 from the right, g(x) approaches 0. LIMITS AND CONTINUITY Algebra reveals much about many functions. endobj That … In particular the left and right hand limits do not coincide. Chapter 1: Limits and Continuity Spring 2018 Department of Mathematics Hong Kong Baptist University 1/75. ��]�[{��f��$" ��6��n�, ��A1Ԇ�>�2��e��~��LǸ� I"f9�(��9��D��׍��y73 1= y. 0000014549 00000 n Mathematic I Prepared by Dr. Sahar abdul Hadi 1Page LIMITS AND CONTINUITY CHAPTER TWO LIMITS AND CONTINUITY LIMITS OF FUNCTION: Y-axis Definitions: L y=f(x) ܔܑܕ࢞→ࢇࢌ(࢞) = ࡸ ܯ݁ܽ݊ ݐℎܽݐ ݓℎ݁݊ ܽ … Falls approximately s(t)=16t2 feet in t seconds. 0000029193 00000 n limits_1_the_limit_and_computing_the_limit_completed_v2.pdf: File Size: 5596 kb: File Type: pdf << /BaseFont/ABYQLR+CMSY10 What value should be assigned For this limit it is completely irrelevant that f(3) = 2, We only care about what happens to f(x) for x … 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 Chapter 2 Limits and Continuity Exercise 2.2 29E. 0000028463 00000 n 0000013968 00000 n 0000002666 00000 n 0000037430 00000 n 0000038587 00000 n In later chapters you’ll see that the derivative, the integral and inﬁnite series—all of calculus depends on it, so your understanding of the topic is critical. Chapter 3: Derivatives. Chapter 1 DACS 1222 Lok 2004/05 1 CHAPTER 1 LIMITS AND CONTINUITY 1.1 LIMITS (page 2) Limits are used to explain changes that arise for a particular function when the value of an independent variable approaches a certain value. Falls approximately s(t)=16t2 feet in t seconds. 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 Take x to be a point which approaches y without being equal to y.If there exists a number L that the values f(x) approach as x … 2.5 Continuity LIMITS AND DERIVATIVES In this section, we will: See that the mathematical definition of continuity corresponds closely with the meaning of the word continuity in everyday language. 0000035789 00000 n 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 For x ≠4, the function h(x) is equal to 2 20 4 xx x +− −. 0000031502 00000 n As x approaches 1 from the left, g(x) approaches 1. /Subtype/Type1 In this chapter, we develop the most fundamental idea behind calculus, that of a limit. Chapter 2: Functions, Limits and Continuity 2. so that yis called the image of xunder the function f; xis a pre-image of yunder f. We also say that \yis the function value of xunder f." In the function f : X !Y, the set X containing all of the rst Chapter 2 The Derivative Business Calculus 80 (c) When x is close to 3 (or as x approaches the value 3), the values of f(x) are close to 1 (or approach the value 1), so lim x→3 f(x) = 1. 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 Example 2 Describe the behavior of the function f(x) = chapter 2 limits and continuity 2.1 rates of change and tangents to curves 1. Chapter 2: Limits and Continuity 2.1 Rates of change and limits ____ Determine limits as x approaches a constant using graphs and expressions p 67: 49; p 97: 5, 8 ____ Find one sided limits, and understand their relation to general limits P 66: 41 ____ Use the properties of limits when finding limits Pg 67: 55 2.2 Limits involving infinity 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 0000033149 00000 n /Type/Font /FirstChar 33 CAS Exercise Examples for Chapter 2: Limits and Continuity à Section 2.1 Rates of Change and Limits Graphical Estimates of Limits In Exercises 41 - 46, you are asked to estimate the value of lim xØx 0 fHxL by plotting the graph of y = f HxL near x = x 0 and to then evaluate the limit symbolically. 0000014957 00000 n 21. That … chapter 2 limits and continuity 2.1 rates of change and tangents to curves 1. 0000037499 00000 n /BaseFont/UGWCBZ+CMSY8 There is no single number L that all the values g(x) get arbitrarily close to as x 1.Ä (b) 1 (c) 0 2. (a) (3) (2) 28 9 32 1 ff f 19 x 0000039161 00000 n In Section 2.1 we introduce the notion of limit, we discuss some of the elementary properties, especially in connection with the order and the algebraic operations in ℝ, and finally, we prove that monotone functions always have limits. 761.6 272 489.6] /BaseFont/KNUNVN+CMR12 De nition 1. SBS Chapter 2: Limits & continuity (SBS 2.1) Limit of a function Consider a free falling body with no air resistance. Chapter 2 Limits and Continuity Exercise 2.2 25E. Chapter 2 Limits and Continuity Page 2 of 14 The graph below is a plot of distance vs. time for your trip. 0 0 0 0 722.2 555.6 777.8 666.7 444.4 666.7 777.8 777.8 777.8 777.8 222.2 388.9 777.8 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 1062.5 826.4] /FirstChar 33 As x approach 0 from the left, the value of the function is getting closer to 1, so lim ( ) 1 0 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 c. Looking at the graph, describe this trip in words. The following are the daily homework assignments for Chapter 2 – Limits and Continuity Section Pages Topics Assignment 2.1 Day 1 9/15 p.63-72 Limits of function values, limits by tables and graphs, definition of a limit, two-sided limits p.72-75: # 5 – 21 odd Worksheet 2.1-1 2.1 Day 2 9/16 p.63-72 Rules of Calculating Limits~ Part 1 0000034162 00000 n 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 0000044321 00000 n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 There is no single number L that all the values g(x) get arbitrarily close to as x 1.Ä (b) 1 (c) 0 2. As x approaches 1 from the right, g(x) approaches 0. /Subtype/Type1 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 611.1 777.8 777.8 388.9 500 777.8 666.7 944.4 722.2 777.8 611.1 777.8 722.2 555.6 SBS Chapter 2: Limits & continuity (SBS 2.1) Limit of a function Consider a free falling body with no air resistance. Calculus: Graphical, Numerical, Algebraic, 3rd Edition Answers Ch 2 Limits and Continuity Ex 2.1 Calculus: Graphical, Numerical, Algebraic Answers Chapter 2 Limits and Continuity Exercise 2.1 1E Chapter 2 Limits and Continuity Exercise 2.1 1QR Chapter 2 Limits and Continuity Exercise 2.1 2E Chapter 2 Limits and Continuity Exercise 2.1 2QR Chapter 2 Limits and […] Chapter 2 Limits and continuity 2.1 The deﬂnition of a limit Deﬂnition 2.1 ("-– deﬂnition).Let f be a function and y 2 Ra ﬂxed number. 0000037141 00000 n Let’s call the function that pairs the distance traveled d in time t function f. So in function notation we have d f(t). Thomas Calculus Early Transcendentals 14th Edition Hass SOLUTIONS MANUAL CHAPTER 2 LIMITS AND CONTINUITY 2.1 RATES OF CHANGE AND TANGENTS TO CURVES /LastChar 196 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 /Subtype/Type1 Chapter 2: Functions, Limits and Continuity 1. As x approaches 1 from the left, g(x) approaches 1. x1.1 Examples where limits arise Calculus has two basic procedures: di erentiation and integration. (a) Does not exist. (a) 0000004283 00000 n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 826.4 295.1 826.4 531.3 826.4 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 So the left limit is -2 while the right limit is 2. 2) 2f then y. /BaseFont/DJKQJZ+MSBM10 (a) Does not exist. 0000005952 00000 n You already probably have an intuitive idea of what it means for a function to be continuous. /Subtype/Type1 12 0 obj 0000023713 00000 n >> endobj Functions. /FirstChar 0 Section 11.3 Limits and Continuity 1063 Limits and Continuity Figure 11.12 shows three graphs that cannot be drawn without lifting a pencil from the paper.In each case,there appears to be an interruption of the graph of at f x = a. 21 0 obj 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 /Name/F2 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4
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