Limits and Their Properties. Copyright Cengage Learning. All rights reserved.

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1 1 Limits and Their Properties Copyright Cengage Learning. All rights reserved.

2 1.1 A Preview of Calculus Copyright Cengage Learning. All rights reserved.

3 What Is Calculus? 3

4 Calculus Calculus is the mathematics of change. For instance, calculus is the mathematics of velocities, accelerations, tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, and a variety of other concepts that have enabled scientists, engineers, and economists to model real-life situations. Although precalculus mathematics also deals with velocities, accelerations, tangent lines, slopes, and so on, there is a fundamental difference between precalculus mathematics and calculus. Precalculus mathematics is more static, whereas calculus is more dynamic. 4

5 Precalculus concepts 5

6 Precalculus concepts cont d 6

7 Precalculus concepts cont d 7

8 Precalculus concepts cont d 8

9 1.2 Finding Limits Graphically and Numerically Copyright Cengage Learning. All rights reserved. 9

10 Objectives Estimate a limit using a numerical or graphical approach. (GNAW on Calculus) Learn different ways that a limit can fail to exist. 10

11 An Introduction to Limits What is a limit? (Class example) 11

12 An Introduction to Limits Suppose you are asked to sketch the graph of the function f given by For all values other than x = 1, you can use standard curve-sketching techniques. However, at x = 1, it is not clear what to expect. 12

13 An Introduction to Limits To get an idea of the behavior of the graph of f near x = 1, you can use two sets of x- values one set that approaches 1 from the left and one set that approaches 1 from the right, as shown in the table. x f(x) 13

14 An Introduction to Limits To get an idea of the behavior of the graph of f near x = 1, you can use two sets of x-values one set that approaches 1 from the left and one set that approaches 1 from the right, as shown in the table. 14

15 An Introduction to Limits The graph of f is a parabola that has a gap at the point (1, 3), as shown in the Figure 1.5. Although x can not equal 1, you can move arbitrarily close to 1, and as a result f(x) moves arbitrarily close to 3. Using limit notation, you can write Figure 1.5 This is read as the limit of f(x) as x approaches 1 is 3. 15

16 An Introduction to Limits This discussion leads to an informal definition of limit. If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c, is L. This limit is written as 16

17 Limits That Fail to Exist (See figure 1.10 on page 51) 17

18 Example 3 Behavior That Differs from the Right and from the Left Show that the limit does not exist. Solution: Consider the graph of the function Figure 1.8 and the definition of absolute value. From you can see that Figure

19 Limits That Fail to Exist What about f(x) increasing or decreasing without bound as x approaches c? 19

20 1.3 Evaluating Limits Analytically Copyright Cengage Learning. All rights reserved. 20

21 Objectives Develop and use a strategy for finding limits. Evaluate a limit using dividing out (factor and cancel) and rationalizing techniques. 21

22 An Introduction to Limits Remember this function? Could we have figured out the limit as x approaches 1 without graphing or doing a table? 22

23 Let s do some algebra An expression such as 0/0 is called an indeterminate form because you cannot (from the form alone) determine the limit. (When you try to plug in x = 1, you get the 0/0.) 23

24 Example 6 Solution cont d Because exists, you can apply Theorem 1.7 to conclude that f and g have the same limit at x = 1. 24

25 Example 6 Solution cont d So, for all x-values other than x = 1, the functions f and g agree, as shown in Figure 1.17 f and g agree at all but one point Figure

26 Limits So But what about: = 1 or = 3 26

27 Limits The first thing we do when finding limits is to try plugging in the x to see what y value we get. If you can t plug in the x, then try doing some algebra and then see if you can plug in the x, (factor & cancel, or rationalize). If that doesn t work, use a graph or table to determine the limit. 27

28 Example 1 Evaluating Basic Limits 28

29 Example 3 The Limit of a Rational Function Find the limit: 29

30 Example 4(a) The Limit of a Composite Function 30

31 Example 4(b) The Limit of a Composite Function 31

32 Example 5 Limits of Trigonometric Functions 32

33 Example 7 Dividing Out (factor & cancel) Find the limit: What happens to the numerator and the denominator when you plug in the -3? It is an indeterminate form because you cannot (from the form alone) determine the limit. (When you plug in x = -3, you get the fraction 0/0 which is undefined.) 33

34 Example 7 Solution cont d Because the limit of the numerator is also 0, the numerator and denominator have a common factor of (x + 3). So, for all x 3, you can divide out this factor to obtain It follows that: 34

35 Example 7 Solution cont d This result is shown graphically in Figure Note that the graph of the function f coincides with the graph of the function g(x) = x 2, except that the graph of f has a gap at the point ( 3, 5). Figure

36 Rationalizing Technique (another way to simplify) 36

37 Example 8 Rationalizing Technique Find the limit: (By direct substitution, you obtain the indeterminate form 0/0.) Solution: 37

38 Example 8 Solution cont d In this case, you can rewrite the fraction by rationalizing the numerator. (Continued on next page) 38

39 Example 8 Solution cont d Now, using Theorem 1.7, you can evaluate the limit as shown. 39

40 Example 8 Solution cont d A table or a graph can reinforce your conclusion that the limit is. (See Figure 1.20.) Figure