1. Limits and Their Properties 11.2 Copyright © Cengage Learning. All rights reserved.

2. 2 Finding a limit graphically Warm-up :

3. Evaluating Limits Analytically Copyright © Cengage Learning. All rights reserved. 1.2

4. 4 Evaluate a limit using properties of limits. Develop and use a strategy for finding limits. Evaluate a limit using dividing out and rationalizing techniques. Evaluate a limit using the Squeeze Theorem. Objectives

5. 5 Properties of Limits

6. 6 The limit of f (x) as x approaches c does not depend on the value of f at x = c. It may happen, however, that the limit is precisely f (c). In such cases, the limit can be evaluated by direct substitution. That is, Such well-behaved functions are continuous at c. Properties of Limits

7. 7 Example 1 – Evaluating Basic Limits

8. 8 Example 2 – Evaluating Basic Limits

9. 9 The limit (as x → 2 ) of the polynomial function p(x) = 4x 2 + 3 is simply the value of p at x = 2. This direct substitution property is valid for all polynomial and rational functions with nonzero denominators. Properties of Limits

10. 10 Properties of Limits In other words, use direct substitution unless it results in division by zero.

11. 11 Find the limit: Solution: Because the denominator is not 0 when x = 1, you can apply Theorem 1.3 to obtain Example 3 – The Limit of a Rational Function

12. 12 Example 4(a) – The Limit of a Composite Function Because it follows that

13. 13 Example 4– The Limit of a Composite Function Find the limit:

14. 14 Example 5 – Limits of Trigonometric Functions

15. 15 A Strategy for Finding Limits

16. 16 A Strategy for Finding Limits

17. 17 Find the limit: Example 6 – Finding the Limit of a Function

18. 18 Example 6 – Solution So, for all x-values other than x = 1, the functions f and g agree, as shown in Figure 1.17 Figure 1.17 cont’d f and g agree at all but one point

19. 19 A Strategy for Finding Limits

20. 20 Dividing Out and Rationalizing Techniques

21. 21 Dividing Out and Rationalizing Techniques Two techniques for finding limits analytically are shown in Examples 7 and 8. The dividing out technique involves dividing out common factors, and the rationalizing technique involves rationalizing the numerator of a fractional expression.

22. 22 Example 7 – Dividing Out Technique Find the limit: Solution: Although you are taking the limit of a rational function, you cannot apply Theorem 1.3 because the limit of the denominator is 0.

23. 23 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 Using Theorem 1.7, it follows that Example 7 – Solution cont’d

24. 24 This result is shown graphically in Figure 1.18. 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). Example 7 – Solution Figure 1.18 cont’d

25. 25 Theorem You Try:

26. 26 You Try:

27. 27 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 evaluate a limit and encounter this form, remember that you must rewrite the fraction so that the new denominator does not have 0 as its limit. One way to do this is to divide out like factors, as shown in Example 7. A second way is to rationalize the numerator, as shown in Example 8. Dividing Out and Rationalizing Techniques

28. 28 Find the limit: Solution: By direct substitution, you obtain the indeterminate form 0/0. Example 8 – Rationalizing Technique

29. 29 In this case, you can rewrite the fraction by rationalizing the numerator. cont’d Example 8 – Solution

30. 30 Now, using Theorem 1.7, you can evaluate the limit as shown. cont’d Example 8 – Solution

31. 31 A table or a graph can reinforce your conclusion that the limit is. (See Figure 1.20.) Figure 1.20 Example 8 – Solution cont’d

32. 32 Example 8 – Solution cont’d

33. 33 You Try:

34. 34 You Try:

35. 35 Example: 11

36. 36

37. 37 You Try:

38. 38 You Try:

39. 39 You Try:

40. 40 Limit of a Difference Quotient:

41. 41 You Try:

42. 42 Limits at infinity  Find the following limits

43. 43 Homework Day 1 11.2 pg. 760 1-29 odd, 43-49 odd. Day 2: 11.2 pg. 760 2-24 even, 59-65 odd

44. 44 Mini Quiz 4/24 1) Find the limit analytically:

45. 45 Mini Quiz 4/24 2) Find the limit algebraically :

46. 46 Mini Quiz 4/24 2) Find the limit analytically :