1. Evaluating Limits Analytically
Lesson 1.3

2. What Is the Squeeze Theorem?
Today we look at various properties of limits, including the Squeeze Theorem

3. How do we evaluate limits?
Numerically
Construct a table of values.
Graphically
Draw a graph by hand or use TI’s.
Analytically
Use algebra or calculus.

4. Properties of Limits The Fundamentals
Basic Limits:
Let b and c be real numbers and
let n be a positive integer:

5. Examples:

6. Properties of Limits Algebraic Properties
Algebraic Properties of Limits:
Let b and c be real numbers, let n be a positive integer, and let f and g be functions
with the following properties:
Too many to fit on this page….

7. Properties of Limits Algebraic Properties
Let:
and
Scalar Multiple:
Sum or Difference:
Product:

8. Properties of Limits Algebraic Properties
Let:
and
Quotient:
Power:

9. Evaluate by using the properties of limits
Evaluate by using the properties of limits. Show each step and which property was used.

10. Examples of Direct Substitution - EASY

11. Examples

12. Properties of Limits nth roots
Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for all c > 0 if n is even…

13. Properties of Limits Composite Functions
If f and g are functions such that…
and
then…

14. Example:
By now you should have already arrived at the conclusion that many algebraic functions can be evaluated by direct substitution. The six basic trig functions also exhibit this desirable characteristic…

15. Properties of Limits Six Basic Trig Function
Let c be a real number in the domain of the
given trig function.

16. A Strategy For Finding Limits
Learn to recognize which limits can be evaluated by direct substitution.
If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to find a function g that agrees with f for all x other than x = c.
Use a graph or table to find, check or reinforce your answer.

17. The Squeeze Theorem
FACT: If
for all x on
and
then,

18. Example:
GI-NORMOUS PROBLEMS!!!
Use Squeeze Theorem!

21. Example:
Use the squeeze theorem to find:

22. Properties of Limits Two Special Trig Function

23. General Strategies

24. Some Examples Consider Strategy: simplify the algebraic fraction
Why is this difficult?
Strategy: simplify the algebraic fraction

25. Reinforce Your Conclusion
Graph the Function
Trace value close to specified point
Use a table to evaluate close to the point in question

26. Find each limit, if it exists.

27. Find each limit, if it exists.
Don’t forget, limits can never be undefined!
Direct Substitution doesn’t work!
Factor, cancel, and try again!
D.S.

28. Find each limit, if it exists.

29. Find each limit, if it exists.
Direct Substitution doesn’t work.
Rationalize the numerator.
D.S.

30. Special Trig Limits

31. Special Trig Limits
Trig limit
D.S.

32. Evaluate in any way you chose.

33. Evaluate in any way you chose.

34. Evaluate in any way you chose.

35. Evaluate in any way you chose.

36. Evaluate by using a graph. Is there a better way?

42. Evaluate:

43. Evaluate:

44. Evaluate:

45. Evaluate:

46. Evaluate:

47. Evaluate:

48. Evaluate:

49. Evaluate:

50. Evaluate:

51. Evaluate:

52. Evaluate:

53. Evaluate:

54. Evaluate:

55. Evaluate:

56. Evaluate:

57. Note possibilities for piecewise defined functions
Note possibilities for piecewise defined functions. Does the limit exist?

58. Three Special Limits
Try it out!

60. Squeeze Rule
Given g(x) ≤ f(x) ≤ h(x) on an open interval containing c And …
Then

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

62. Gap in graph Asymptote
Oscillates c c c