2 LIMITS

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GROUP WORK 4, SECTION 2.4The Significance of the ‘‘For Every’’ (Part 2)Try the three limits again, this time for ε =100 1 . Make sure that when you are trying to verify the condition|x − x 0 | < δ, you check values of x 0 > x and x 0 < x. Do you wish to change your answer to Problem 3 fromPart 1?68
2.5 CONTINUITYSUGGESTED TIME AND EMPHASIS1–1 1 2 classes (1 class for an overview, 1 1 2for a more detailed treatment)POINTS TO STRESS1. The graphical and mathematical definitions of continuity, and the basic principles.2. Examples of discontinuity.3. The Intermediate Value Theorem: mathematical statement, graphical examples, and applied examples.QUIZ QUESTIONS• Text Question: The text says that y = tan x is discontinuous at x = π 2. This would seem to contradictTheorem 7, which states that trigonometric functions are continuous at every number in their domains.Does it? Why or why not?Answer: It does not; tan x is indeed continuous at every point in its domain, but x = π 2is not in itsdomain.• Drill Question: Assume that f (1) =−5, and f (3) = 5. Doestherehavetobeavalueofx, between 1and3,suchthat f (x) = 0?Answer: No, there does not. Only if the function is continuous does the IVT indicate that there must besuch a value.MATERIALS FOR LECTURE• Discuss the idea of continuity at a point, continuity on an interval, and the basic types of discontinuities.• Note that the statement “ f is continuous at x = a” is implicitly saying three things:1. f (a) exists.2. lim f (x) exists.x→a3. The two quantities are equal.To show that all three statements are important to continuity, have the students come up with exampleswhere the first holds and the second does not, the second holds and the first does not, and where the firsttwo hold and the third does not. Examples are sketched below.yyyxxx• Some students tend to believe that all piecewise functions are discontinuous at the border points. Examine⎧2⎪⎨ π x if x < π 2the function f (x) = sin x if π 2 ⎪⎩≤ x ≤ 2π at the points x = π 2and x = 2π. This would be a good2π if x ≥ 2πtime to point out that the function |x| is continuous everywhere, including at x = 0.69
Page 1 and 2: 2 LIMITS2.1 THE TANGENT AND VELOCIT
Page 3: SECTION 2.1THE TANGENT AND VELOCITY
Page 8 and 9: 2.2 THE LIMIT OF A FUNCTIONSUGGESTE
Page 10 and 11: CHAPTER 2LIMITSGROUP WORK 2: The Sh
Page 12 and 13: CHAPTER 2LIMITS3. Answers will vary
Page 14 and 15: GROUP WORK 2, SECTION 2.2The Shape
Page 16 and 17: GROUP WORK 2, SECTION 2.2The Shape
Page 18 and 19: 2.3 CALCULATING LIMITS USING THE LI
Page 20 and 21: CHAPTER 2LIMITSGROUP WORK 3: The Sq
Page 22 and 23: GROUP WORK 2, SECTION 2.3Fixing a H
Page 24 and 25: 2.4 THE PRECISE DEFINITION OF A LIM
Page 26 and 27: CHAPTER 2LIMITSGROUP WORK 2: The Di
Page 28 and 29: GROUP WORK 1, SECTION 2.4A ‘‘Ji
Page 30 and 31: GROUP WORK 2, SECTION 2.4The Dire W
Page 32 and 33: Consider the following functions:GR
Page 36 and 37: CHAPTER 2LIMITS• Start by stating
Page 38 and 39: CHAPTER 2LIMITSGROUP WORK 3: The Tw
Page 40 and 41: Exploring Continuity3. Consider the
Page 42 and 43: The following are graphs of y = f (
Page 44 and 45: GROUP WORK 4, SECTION 2.5Swimming t
Page 46 and 47: CHAPTER 2LIMITS3. Find the vertical
Page 48 and 49: CHAPTER 2LIMITS7. Let⎧ √⎨ 3
Page 50: CHAPTER 2LIMITS6. (a) L = 0√0.01(

2 LIMITS

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