1. Finding Limits Graphically and Numerically An Introduction to Limits Limits that Fail to Exist A Formal Definition of a Limit

2. An Introduction to Limits Graph: What can we expect at x = 1? Approach x=1 from the left. Approach x=1 from the right. Are we approaching a specific value from both sides? What is that number? Do Now: evaluate f(1.1)

3. Numerically x0.750.900.990.99911.0011.011.101.25 f(x)? Fill in chart for all values of x:

4. Numerically x0.750.900.990.99911.0011.011.101.25 f(x)2.312.712.972.997?3.0033.03013.313.81

5. Notation The limit of f(x) as x approaches c is L.

6. Exploration x1.751.901.991.999 2 2.0012.012.102.25 f(x)

7. Exploration x1.751.901.991.999 2 2.0012.012.102.25 f(x).75.9.99.999 U n d. 1.0011.011.11.25

8. Example 1: Estimating a Limit Numerically Where is it undefined? What is the limit?

9. Example 1: Estimating a Limit Numerically Where is it undefined? 0 What is the limit? 2 x-.1-.01-.001 0.001.01.1 f(x)1.951.9951.9995 U n d. 2.000 5 2.0052.05

10. Estimating a Limit Numerically It is important to realize that the existence or nonexistence of f(x) at x = c has no bearing on the existence of the limit of f(x) as x approaches c. The value of f(c) may be the same as the limit as x approaches c, or it may not be.

11. Finding the limit by substitution Always try evaluating a function at c first: Examples:

12. Finding the limit by substitution Always try evaluating a function at c first: Simple and boring!

13. Substitution needing analytical approach: Factor and simplify:

14. Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Ex. Notice form Factor and cancel common factors Indeterminate Forms

15. Using Algebraic Methods When substitution renders an indeterminate value, try factoring and simplifying: Hint for # 3: use synthetic division to factor numerator (see if x+2 is a factor)

16. Using Algebraic Methods Now try to substitute in “c”

17. Using Algebraic Methods Substitution works for the simplified version.

18. More complicated algebraic methods Involving radicals:

19. Other Algebraic Methods: 1) Try simplifying a complex fraction 2) Try rationalizing (the numerator):

20. Other Algebraic Methods: Try simplifying a complex fraction or rationalizing (a numerator or denominator):

21. Do Now: graph the piecewise function:

22. Find -2-2 6 Note: f (-2) = 1 is not involved Using a graph to find the limit:

23. Ex 2: Finding the limit as x → 2 1.Numerical Approach – Construct a table of values. 2.Graphical Approach – Draw a graph by hand or using technology.

24. Ex 2: Finding the limit as x → 2 al Approach – Use algebra or calculus.

25. Use your calculator to evaluate the limits

26. Answer : 16 Answer : no limit 3) Use your calculator to evaluate the limits

27. Examples Do Now: Graph the function:

28. Limits that Fail to Exist-this one approaches a different value from the left and the right 1.Numerical Approach – Construct a table of values. 2.Graphical Approach – Draw a graph by hand or using technology. 3.Analytical Approach – Use algebra or calculus.

29. Ex 4: Unbounded Behavior 1.Numerical Approach – Construct a table of values. 2.Graphical Approach – Draw a graph by hand or using technology. 3.Analytical Approach – Use algebra or calculus.

30. Ex 5: Oscillating Behavior 1.Numerical Approach – Construct a table of values. 2.Graphical Approach – Draw a graph by hand or using technology. 3.Analytical Approach – Use algebra or calculus. x 1.5.1.01.001.0001As x approaches 0?

31. Ex 5: Oscillating Behavior 1.Numerical Approach – Construct a table of values. 2.Graphical Approach – Draw a graph by hand or using technology. 3.Analytical Approach – Use algebra or calculus. x 1.5.1.01.001.0001As x approaches 0?.84.91-.54-.51.827-.310? No! It doesn’t exist!

32. Common Types of Behavior Associated with the Nonexistence of a Limit 1.f(x) approaches a different number from the right side of c than it approaches from the left side. 2.f(x) increases or decreases without bound as x approaches c. 3.f(x) oscillates between two fixed values as x approaches c.

33. A Formal Definition of a Limit Lim x→c f(x) = L If for every number ε > 0 There is a number δ > 0 Such that |f(x) – L| < ε Whenever 0 < |x – c| < δ

34. Using the formal definition. Prove: lim x→3 (4x – 5) = 7 Lim x→c f(x) = L If for every number ε > 0 There is a number δ > 0 Such that |f(x) – L| < ε Whenever 0 < |x – c| < δ

35. The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. a L One-Sided Limit One-Sided Limits

36. The left-hand limit of f (x), as x approaches a, equals M written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. a M

37. 1. Given Find Examples Examples of One-Sided Limit

38. 1. Given Find Examples Examples of One-Sided Limit So and therefore, does not exist!

39. Find the limits: More Examples

40. Find the limits: More Examples

41. For the function Bu t This theorem is used to show a limit does not exist. A Theorem

42. Limits at infinity 3 cases: when the degree is: “top heavy”- goes to negative or positive infinity “bottom heavy”- goes to zero “equal” – put terms over each other and reduce. What does this mean?

43. Limits at Infinity For all n > 0, provided that is defined. Ex. Divide by

44. Limits at Infinity For all n > 0, provided that is defined. Ex. Divide by

45. More Examples

49. Limits at infinity When the numerator has a larger degree than the denominator…

50. Limits at infinity When the numerator has a larger degree than the denominator…

51. 51 Limits at infinity If n is a positive integer, the, where a is some constant. Property:

52. The denominator has a higher degree Find the limit

53. The denominator has a higher degree Find the limit

54. When the degrees are equal… Reduce the equal terms

55. When the degrees are equal… Reduce the equal terms

56. 56 Example Evaluate the limit

57. 57 Example Evaluate the limit

58. Continuity A function f is continuous at the point x = a if the following are true: This one fails iii ! a f(a)f(a)

59. A function f is continuous at the point x = a if the following are true: a f(a)f(a)

60. At which value(s) of x is the given function discontinuous? Continuous everywhere Continuous everywhere except at Examples

61. and Thus h is not cont. at x=1. h is continuous everywhere else and Thus F is not cont. at F is continuous everywhere else 0 o

62. Continuous Functions A polynomial function y = P(x) is continuous at every point x. A rational function is continuous at every point x in its domain. If f and g are continuous at x = a, then