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

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT Formal Definition of the Limit According to the theorems we’ve seen thus far, So, we should be able to make (3x + 4) as close as we like to 10, just by making x sufficiently close to 2. But can we actually do this? For instance, can we force (3x + 4) to be within distance 1 of 10? © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 3

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT Formal Definition of the Limit To see what values of x will guarantee this, write an inequality that says that (3x + 4) is within 1 unit of 10: or Solve for x: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 4

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT Formal Definition of the Limit If x is between 2 – 1/3 and 2 + 1/3, 3x + 4 is guaranteed to be between 9 and 11. While we can make (3x + 4) within 1 unit of 10, we wish to make it arbitrarily close, as close as anyone would ever demand. We accomplish this by repeating the arguments in this example , this time for an unspecified distance, call it ε (epsilon, where ε > 0). © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 5

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.2 Verifying a Limit Show that we can make (3x + 4) within any specified distance ε > 0 of 10 (no matter how small ε is), just by making x sufficiently close to 2. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 6

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.2 Verifying a Limit © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 7

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.2 Verifying a Limit No matter how close we are asked to make (3x + 4) to 10, we can accomplish this simply by taking x to be in the specified interval. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 8

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.3 Proving That a Limit Is Correct © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 9

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.3 Proving That a Limit Is Correct © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 10

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.3 Proving That a Limit Is Correct Since x = 1, © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 11

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.3 Proving That a Limit Is Correct Taking δ = ε/2 and working backward, we see that requiring x to satisfy will guarantee that © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 12

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.1 (Precise Definition of Limit) For a function f defined in some open interval containing a (but not necessarily at a itself), we say if given any number ε > 0, there is another number δ > 0, such that 0 < |x − a| < δ guarantees that | f (x) − L| < ε. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 13

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.1 It is rather difficult to explicitly find δ as a function of ε, for all but a few simple examples. Despite this, learning how to work through the definition, even for a small number of problems, will shed considerable light on a deep concept. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 14

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.4 Using the Precise Definition of Limit © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 15

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.4 Using the Precise Definition of Limit If this limit is correct, then given any ε > 0, there must be a δ > 0 for which 0 < |x − 2| < δ guarantees that Notice that Assume that x lies in the interval [1, 3]; thus: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 16

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.4 Using the Precise Definition of Limit Finally, if we require that © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 17

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.4 Using the Precise Definition of Limit We now have two restrictions: So, choose: Working backward: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 18

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT Exploring the Definition of Limit Graphically Finding a δ for a given ε is not always easily accomplished. However, we can explore the definition graphically for any function. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 19

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.5 Exploring the Precise Definition of Limit Graphically © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 20

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.5 Exploring the Precise Definition of Limit Graphically In example 6.4, we saw that , © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 20

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.5 Exploring the Precise Definition of Limit Graphically Confirmed. We can draw virtually the same picture for any given value of ε, since we have an explicit formula for finding δ given ε. We are not proving that the above limit is correct. To prove this requires symbolically finding a δ for every ε > 0. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 22

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.7 Exploring the Definition of Limit Where the Limit Does Not Exist © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 23

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.7 Exploring the Definition of Limit Where the Limit Does Not Exist Choose Find a δ > 0 for which 0 < |x| < δ guarantees that © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 24

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.7 Exploring the Definition of Limit Where the Limit Does Not Exist Try δ = 0.1. Setting the x-range to the interval [−0.1, 0.1] and the y-range to the interval [0.5, 1.5] and redrawing gives: No points are plotted in the window for any x < 0. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 25

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.7 Exploring the Definition of Limit Where the Limit Does Not Exist In fact, there is no choice of δ that makes the defining inequality true for ε = 1/2. Thus, the conjectured limit of 1 is incorrect. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 26

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT Limits Involving Infinity Recall that we write whenever the function increases without bound as x → a. That is, we can make f (x) as large as we like, simply by making x sufficiently close to a. So, given any large positive number, M, we must be able to make f (x) > M, for x sufficiently close to a. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 27

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.2 For a function f defined in some open interval containing a (but not necessarily at a itself), we say if given any number M > 0, there is another number δ > 0, such that 0 < |x − a| < δ guarantees that f (x) > M. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 28

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.3 For a function f defined in some open interval containing a (but not necessarily at a itself), we say if given any number N < 0, there is another number δ > 0, such that 0 < |x − a| < δ guarantees that f (x) < N. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 29

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.8 Using the Definition of Limit Where the Limit Is Infinite © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 30

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.8 Using the Definition of Limit Where the Limit Is Infinite Given any (large) number M > 0, we need to find a distance δ > 0 such that if x is within δ of 0 (but not equal to 0) then Since both M and x2 are positive, this is equivalent to © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 31

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.8 Using the Definition of Limit Where the Limit Is Infinite So, for any M > 0, if we take and work backward, we have that 0 < |x − 0| < δ guarantees that © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 32

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT Limits Involving Infinity If we write we mean that as x increases without bound, f (x) gets closer and closer to L. That is, we can make f (x) as close to L as we like, by choosing x sufficiently large. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 33

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.4 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 34

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.5 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 35

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.9 Using the Definition of Limit Where x Tends to −∞ © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 36

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.9 Using the Definition of Limit Where x Tends to −∞ Since x < 0, |x| = −x. Thus: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 37

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.9 Using the Definition of Limit Where x Tends to −∞ Dividing both sides by ε and multiplying by x (remember that x < 0 and ε > 0, so that this will change the direction of the inequality), we get So, if we take and work backward, we have satisfied the definition and thereby proved that the limit is correct. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 38

FORMAL DEFINITION OF THE LIMIT 1.6 FORMAL DEFINITION OF THE LIMIT 6.1 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 39