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

2. Evaluating Limits Analytically Copyright © Cengage Learning. All rights reserved. 1.3

3. 3 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

4. 4 Properties of Limits

5. 5 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

6. 6 Book uses more formal proof, but we can just look graphically (see next slide).

7. 7 Example 1 – Evaluating Basic Limits

8. 8 Properties of Limits

9. 9 Example 2 – The Limit of a Polynomial

10. 10 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 Another example:

11. 11 Properties of Limits

12. 12 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 Another example: Example of power of function:

13. 13 Properties of Limits We will provide example of this property shortly once we talk about the limit of a composite function

14. 14 Properties of Limits

15. 15 Example 4(a) – The Limit of a Composite Function Because it follows that

16. 16 Example 4(b) – The Limit of a Composite Function Because it follows that

17. 17 Properties of Limits

18. 18 Example 5 – Limits of Trigonometric Functions Another example:

19. 19 A Strategy for Finding Limits Several limits before could be evaluated by direct substitutions because functions were continuous. Now, consider more complicated cases, when function has a discontinuity at the point there we want to compute the limit.

20. 20 A Strategy for Finding Limits

21. 21

22. 22 My example Long division can be used to see this analytically

23. 23 A Strategy for Finding Limits

24. 24 Dividing Out and Rationalizing Techniques

25. 25 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.

26. 26 Example 7 – Dividing Out Technique Find the limit: Solution: Although you are taking the limit of a rational function, you can NOT apply Theorem 1.3 because the limit of the denominator is 0.

27. 27 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

28. 28 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

29. 29 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

30. 30 Find the limit: Solution: By direct substitution, you obtain the indeterminate form 0/0. Example 8 – Rationalizing Technique

31. 31 In this case, you can rewrite the fraction by rationalizing the numerator. cont’d Example 8 – Solution

32. 32 Now, using Theorem 1.7, you can evaluate the limit as shown. cont’d Example 8 – Solution

33. 33 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

34. 34 Example 8 – Solution cont’d

35. 35 More examples:

36. 36 The Squeeze Theorem

37. 37 The next theorem concerns the limit of a function that is squeezed between two other functions, each of which has the same limit at a given x-value, as shown in Figure 1.21 The Squeeze Theorem Figure 1.21

38. 38 Squeeze Theorem is also called the Sandwich Theorem or the Pinching Theorem. The Squeeze Theorem

39. 39

40. 40 use squeeze theorem.

41. 41 Find the limit: Solution: Direct substitution yields the indeterminate form 0/0. To solve this problem, you can write tan x as (sin x)/(cos x) and obtain Example 9 – A Limit Involving a Trigonometric Function

42. 42 Example 9 – Solution cont’d Now, because you can obtain

43. 43 (See Figure 1.23.) Figure 1.23 Example 9 – Solution cont’d

44. 44 My example