Limits and Continuity 1 1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE 1.2 THE CONCEPT OF LIMIT 1.3 COMPUTATION OF LIMITS 1.4 CONTINUITY AND ITS CONSEQUENCES 1.5 LIMITS INVOLVING INFINITY; ASYMPTOTES 1.6 FORMAL DEFINITION OF THE LIMIT 1.7 LIMITS AND LOSS OF SIGNIFICANCE ERRORS © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 2

The Limit: Informal Idea 1.2 THE CONCEPT OF LIMIT The Limit: Informal Idea In this section, we develop the notion of limit using some common language and illustrate the idea with some simple examples. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 3

The Limit: Informal Idea 1.2 THE CONCEPT OF LIMIT The Limit: Informal Idea Suppose a function f is defined for all x in an open interval containing a, except possibly at x = a. If we can make f (x) arbitrarily close to some number L (i.e., as close as we’d like to make it) by making x sufficiently close to a (but not equal to a), then we say that L is the limit of f (x), as x approaches a, written © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 4

The Limit: Informal Idea 1.2 THE CONCEPT OF LIMIT The Limit: Informal Idea For instance, we have since as x gets closer and closer to 2, f (x) = x2 gets closer and closer to 4. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 5

1.2 THE CONCEPT OF LIMIT 2.1 Evaluating a Limit Slide 6 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 6

1.2 THE CONCEPT OF LIMIT 2.1 Evaluating a Limit Slide 7 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 7

1.2 THE CONCEPT OF LIMIT 2.1 Evaluating a Limit Slide 9 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 9

1.2 THE CONCEPT OF LIMIT 2.1 Evaluating a Limit Since the two one-sided limits of f (x) are the same, we summarize our results by saying that © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 11

1.2 THE CONCEPT OF LIMIT 2.1 Evaluating a Limit We can also determine the limit algebraically. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 12

1.2 THE CONCEPT OF LIMIT 2.2 A Limit that Does Not Exist Slide 15 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 15

1.2 THE CONCEPT OF LIMIT 2.2 A Limit that Does Not Exist Slide 16 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 16

1.2 THE CONCEPT OF LIMIT A limit exists if and only if both corresponding one-sided limits exist and are equal. That is, In other words, we say that if we can make f (x) as close as we might like to L, by making x sufficiently close to a (on either side of a), but not equal to a. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 13

1.2 THE CONCEPT OF LIMIT 2.3 Determining Limits Graphically Use the graph to determine © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 18

1.2 THE CONCEPT OF LIMIT 2.3 Determining Limits Graphically Slide 19 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 19

1.2 THE CONCEPT OF LIMIT 2.4 A Limit Where Two Factors Cancel Slide 16 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 16

1.2 THE CONCEPT OF LIMIT 2.4 A Limit Where Two Factors Cancel From the left: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 21

1.2 THE CONCEPT OF LIMIT 2.4 A Limit Where Two Factors Cancel From the right: Conjecture: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 22

1.2 THE CONCEPT OF LIMIT 2.4 A Limit Where Two Factors Cancel Slide 23 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 23

1.2 THE CONCEPT OF LIMIT 2.4 A Limit Where Two Factors Cancel Algebraic cancellation: Likewise: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 24

1.2 THE CONCEPT OF LIMIT 2.5 A Limit That Does Not Exist Slide 21 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 21

1.2 THE CONCEPT OF LIMIT 2.5 A Limit That Does Not Exist From the right: Conjecture: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 26

1.2 THE CONCEPT OF LIMIT 2.5 A Limit That Does Not Exist From the left: Conjecture: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 27

1.2 THE CONCEPT OF LIMIT 2.5 A Limit That Does Not Exist Conjecture: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 28

1.2 THE CONCEPT OF LIMIT 2.1 Computer or calculator computation of limits is unreliable. We use graphs and tables of values only as (strong) evidence pointing to what a plausible answer might be. To be certain, we need to obtain careful verification of our conjectures. We explore this in sections 1.3–1.7. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 25