1. Copyright © Cengage Learning. All rights reserved. 10 Introduction to the Derivative

2. Copyright © Cengage Learning. All rights reserved. 10.1 Limits: Numerical and Graphical Approaches

3. 33 Evaluating Limits Numerically

4. 44 Rates of change are calculated by derivatives, but an important part of the definition of the derivative is something called a limit. Start with a very simple example: Look at the function f (x) = 2 + x and ask: What happens to f (x) as x approaches 3? The following table shows the value of f (x) for values of x close to and on either side of 3:

5. 55 Evaluating Limits Numerically We have left the entry under 3 blank to emphasize that when calculating the limit of f (x) as x approaches 3, we are not interested in its value when x equals 3. Notice from the table that the closer x gets to 3 from either side, the closer f (x) gets to 5. We write this as The limit of f (x), as x approaches 3, equals 5.

6. 66 Example 1 – Estimating a Limit Numerically Use a table to estimate the following limits:

7. 77 Example 1(a) – Solution We cannot simply substitute x = 2, because the function is not defined at x = 2. Instead, we use a table of values, with x approaching 2 from both sides.

8. 88 Example 1(a) – Solution We notice that as x approaches 2 from either side, f (x) appears to be approaching 12. This suggests that the limit is 12, and we write cont’d

9. 99 Example 1(b) – Solution The function is not defined at x = 0 (nor can it even be simplified to one which is defined at x = 0). In the following table, we allow x to approach 0 from both sides: The table suggests that cont’d

10. 10 Evaluating Limits Numerically Definition of a Limit If f (x) approaches the number L as x approaches (but is not equal to) a from both sides, then we say that f (x) approaches L as x → a (“x approaches a”) or that the limit of f (x) as x → a is L. More precisely, we can make f (x) be as close to L as we like by choosing any x sufficiently close to (but not equal to) a on either side. We write or

11. 11 Evaluating Limits Numerically If f (x) fails to approach a single fixed number as x approaches a from both sides, then we say that f (x) has no limit as x → a, or does not exist. Quick Example: As x approaches –2, 3x approaches –6.

12. 12 Example 2 – Limits Do Not Always Exist Do the following limits exist?

13. 13 Example 2(a) – Solution Here is a table of values for with x approaching 0 from both sides. The table shows that as x gets closer to zero on either side, f (x) gets larger and larger without bound—that is, if you name any number, no matter how large, f (x) will be even larger than that if x is sufficiently close to 0.

14. 14 Example 2(a) – Solution Because f (x) is not approaching any real number, we conclude that does not exist. Because f (x) is becoming arbitrarily large, we also say that diverges to or just cont’d

15. 15 Example 2(b) – Solution Here is a table of values for with x approaching 0 from both sides. The table shows that f (x) does not approach the same limit as x approaches 0 from both sides. There appear to be two different limits: the limit as we approach 0 from the left and the limit as we approach from the right. cont’d

16. 16 Example 2(b) – Solution We write read as “the limit as x approaches 0 from the left (or from below) is –1” and read as “the limit as x approaches 0 from the right (or from above) is 1.” These are called the one-sided limits of f (x). In order for f to have a two-sided limit, the two one-sided limits must be equal. Because they are not, we conclude that lim x→0 f (x) does not exist. cont’d

17. 17 Example 2(c) – Solution Near x = 2, we have the following table of values for Because is approaching no (single) real number as x → 2, we see that does not exist. cont’d

18. 18 Example 2(c) – Solution Notice also that diverges to as x → 2 from the positive side (right half of the table) and to as x → 2 from the left (left half of the table). In other words, cont’d

19. 19 Estimating Limits Graphically

20. 20 Example 4 – Estimating Limits Graphically The graph of a function f is shown in Figure 1. (Recall that the solid dots indicate points on the graph, and the hollow dots indicate points not on the graph.) From the graph, analyze the following limits. Figure 1

21. 21 Example 4 – Solution Since we are given only a graph of f, we must analyze these limits graphically. (a)Imagine that Figure 1 was drawn on a graphing calculator equipped with a trace feature that allows us to move a cursor along the graph and see the coordinates as we go. To simulate this, place a pencil point on the graph to the left of x = –2, and move it along the curve so that the x-coordinate approaches –2. (See Figure 2.) Figure 2

22. 22 Example 4 – Solution We evaluate the limit numerically by noting the behavior of the y-coordinates. However, we can see directly from the graph that the y-coordinate approaches 2. Similarly, if we place our pencil point to the right of x = –2 and move it to the left, the y coordinate will approach 2 from that side as well (Figure 3). Therefore, as x approaches –2 from either side, f (x) approaches 2, so Figure 3 cont’d

23. 23 Example 4 – Solution (b)This time we move our pencil point toward x = 0. Referring to Figure 4, if we start from the left of x = 0 and approach 0 (by moving right), the y-coordinate approaches –1. However, if we start from the right of x = 0 and approach 0 (by moving left), the y-coordinate approaches 3. Thus and cont’d Figure 4

24. 24 Example 4 – Solution Because these limits are not equal, we conclude that does not exist. In this case there is a “break” in the graph at x = 0, and we say that the function is discontinuous at x = 0. (c)Once more we think about a pencil point moving along the graph with the x-coordinate this time approaching x = 1 from the left and from the right (Figure 5). cont’d Figure 5

25. 25 Example 4 – Solution As the x-coordinate of the point approaches 1 from either side, the y-coordinate approaches 1 also. Therefore, (d)For this limit, x is supposed to approach infinity. We think about a pencil point moving along the graph further and further to the right as shown in Figure 6. cont’d Figure 6

26. 26 Example 4 – Solution As the x-coordinate gets larger, the y-coordinate also gets larger and larger without bound. Thus, f (x) diverges to Similarly, cont’d

27. 27 Estimating Limits Graphically Evaluating Limits Graphically To decide whether lim x→a f (x) exists and to find its value if it does: 1. Draw the graph of f (x) by hand or with graphing technology. 2. Position your pencil point (or the Trace cursor) on a point of the graph to the right of x = a. 3. Move the point along the graph toward x = a from the right and read the y-coordinate as you go. The value the y-coordinate approaches (if any) is the limit lim x→a + f (x).

28. 28 Estimating Limits Graphically 4. Repeat Steps 2 and 3, this time starting from a point on the graph to the left of x = a, and approaching x = a along the graph from the left. The value the y-coordinate approaches (if any) is lim x→a – f (x). 5. If the left and right limits both exist and have the same value L, then lim x→a f (x) = L. Otherwise, the limit does not exist. The value f (a) has no relevance whatsoever.

29. 29 Estimating Limits Graphically 6. To evaluate lim x→+ f (x), move the pencil point toward the far right of the graph and estimate the value the y-coordinate approaches (if any). For lim x→– f (x), move the pencil point toward the far left. 7. If x = a happens to be an endpoint of the domain of f, then only a one-sided limit is possible at x = a. For instance, if the domain is (–, 4], then lim x→4 – f (x) can be computed, but not lim x→4 + f (x). In this case, we have said that lim x→4 f (x) is just the one-sided limit lim x→4 – f (x).

30. 30 Application

31. 31 Example 6 – Broadband The percentage of U.S. Internet-connected households that have broadband connections can be modeled by where t is time in years since 2000. a. Estimate and interpret the answer. b. Estimate and interpret the answer.

32. 32 Example 6(a) – Solution Figure 8 shows a plot of P(t) for 0 ≤ t ≤ 20. Using either the numerical or the graphical approach, we find Thus, in the long term (as t gets larger and larger), the percentage of U.S. Internet connected households that have broadband is expected to approach 100%. Figure 8

33. 33 Example 6(b) – Solution The limit here is (Notice that in this case, we can simply put t = 0 to evaluate this limit.) Thus, the closer t gets to 0 (representing 2000) the closer P(t) gets to 8.475%, meaning that, in 2000, about 8.5% of Internet-connected households had broadband. cont’d