1. Finding Limits Graphically and Numerically 2015 Limits Introduction Copyright © Cengage Learning. All rights reserved. 1.2

2. Estimate a limit using a numerical or graphical approach. Learn different ways that a limit can fail to exist. Study and use the informal definition of limit. Objectives

3. Formal definition of a 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. “The limit of f of x as x approaches c is L.” This limit is written as

4. Limits can be found in various ways: a)Graphically b)Numerically c)Algebraically Ex: Find the following limit:

5. An Introduction to Limits Ex: Find the following limit:

6. Looks like y=1

7. An Introduction to Limits Ex: Find the following limit:

8. Start by sketching a graph of the function 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. We can find this limit numerically: An Introduction to Limits

9. 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. An Introduction to Limits

10. 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 An Introduction to Limits This is read as “the limit of f(x) as x approaches 1 is 3.” Figure 1.5

11. This discussion leads to an informal definition of a limit: A limit is the value (meaning y value) a function approaches as x approaches a particular value from the left and from the right.

12. Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

13. The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1

14. 1234 1 2 At x=1:left hand limit right hand limit value of the function does not exist because the left and right hand limits do not match! DNE

15. At x=2:left hand limit right hand limit value of the function because the left and right hand limits match. 1234 1 2

16. At x=3:left hand limit right hand limit value of the function because the left and right hand limits match. 1234 1 2

17. You Try– Estimating a Limit Numerically Use the table feature of your graphing calculator to evaluate the function at several points near x = 0 and use the results to estimate the limit:

18. Example 1 – Solution The table lists the values of f(x) for several x-values near 0.

19. Example 1 – Solution From the results shown in the table, you can estimate the limit to be 2. This limit is reinforced by the graph of f (see Figure 1.6.) cont’d Figure 1.6

20. Use a graphing utility to estimate the limit:

21. Find

22. Limits That Fail to Exist

23. Show that does not exist Non-existance Because the behavior differs from the right and from the left of zero, the limit DNE.

24. Discuss the existence of the limit: Solution: Using a graphical representation, you can see that x does not approach any number. Therefore, the limit does not exist.

25. Example 5: xx→0 sin 1/x11 1 DNE Make a table approaching 0 The graph oscillates, so the limit does not exist.

26. Fig. 1.10, p. 51

27. Limits That Fail to Exist - 3 Reasons

28. Properties of Limits

29. Limits Basics Examples

30. Properties of Limits

31. Using Properties of Limits

32. Find the following limits:

33. Properties of Limits

34. Compute the following limits

35. Let’s take a look at the last one What happened when we plugged in 1 for x? When we get we have what’s called an indeterminate form Let’s see how we can solve it

36. Let’s look at the graph of Is the function continuous at x = 1?

37. You Try: Find the limit:

38. You Try:

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

40. In this case, you can rewrite the fraction by rationalizing the numerator. cont’d Solution

41. Now, using Theorem 1.7, you can evaluate the limit as shown. cont’d Solution

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

43. Solution cont’d

44. Group Work : Sketch the graph of f. Identify the values of c for which exists.

45. Homework p.54 8-25 all