1. Evaluating Limits Analytically
Lesson 1.3

2. 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.

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

4. Examples:

5. 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….

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

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

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

9. Examples of Direct Substitution - EASY

10. Examples

11. 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…

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

13. 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…

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

15. 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.

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

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

20. Two Special Trigonometric Limits
In your text, read about the Squeeze Theorem on page 65. Following the Squeeze Theorem are the proofs of two special trig limits…I will not expect you to be able to prove the two limits, so you’ll just want to memorize them. The next slide will give them to you and then we’ll use them in a few examples.

21. Trig Limits (A star will indicate the need to memorize!!!)
Think of this as the limit as “something” approaches 0 of the sine of “something” over the same “something” is equal to 1.
(A star will indicate the need to memorize!!!)

22. Example- Using Trig Limits
equals 1
Before you decide to even use a special trig limit, make sure that direct substitution won’t work. In this case, direct substitution really won’t work, so let’s try to get this to look like one of those special trig limits.
This 5 is a constant and can be pulled out in front of the limit.
Example 10, page 66, in your text is another example.
Now, the 5x is like the heart. You will need the bottom to also be 5x in order to use the trig limit. So, multiply the top and bottom by 5. You won’t have changed the fraction. Watch how to do it.

23. Example
Example:
Direct substitution won’t work. We can use the sine trig limit, but first we’ll have to use some algebra since we need the bottom to be a 3x.
To create a 3x on the bottom, we’ll multiply the bottom by 3/2. To be “fair”, we’ll have to multiply the top by 3/2 as well. Watch how I would do it.
equals 1

24. Example
Looking at the unit circle, this value is 1
Example:
This example was thrown in to keep you on your toes. Direct substitution works at this point since the bottom of the fraction will not be 0 when you use π/2.

25. Example Example: Use the Pythagorean Trig Identity
This is the 2nd special trig limit and you should know that this limit is 0. Let’s prove it by using the 1st trig limit.
Again, in this case, it’s best not to distribute on the bottom. You’ll see why it helps to leave it in factored form for now.
Be creative
Special Trig Limit

26. General Strategies

27. Example- Factoring Example:
Direct substitution at this point will give you 0 in the denominator. Using a bit of algebra, we can try to find the limit.
Now direct substitution will work
Graph on your calculator and use the table to check your result

28. Example- Factoring Example:
Direct substitution results in 0 in the denominator. Try factoring.
Now direct substitution will work
Use your calculator to reinforce your result

29. Example Example: The limit DNE. Verify the result on your calculator.
Sum of cubes
Not factorable
Example:
Direct substitution results in 0 in the denominator. Try factoring.
None of the factors can be divided out, so direct substitution still won’t work.
The limit DNE. Verify the result on your calculator.
The limits from the right and left do not equal each other, thus the limit DNE.
Observe how the right limit goes to off to positive infinity and the left limit goes to negative infinity.

30. Example- Rationalizing Technique
Direct substitution results in 0 in the denominator. I see a radical in the numerator. Let’s try rationalizing the numerator.
Multiply the top and bottom by the conjugate of the numerator.
Note: It was convenient NOT to distribute on the bottom, but you did need to FOIL on the top
Now direct substitution will work
Go ahead and graph to verify.

31. Three Special Limits
Try it out!

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

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

35. Find each limit, if it exists.

36. 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.

37. Find each limit, if it exists.

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

39. Special Trig Limits

40. Special Trig Limits
Trig limit
D.S.

41. Evaluate in any way you chose.

42. Evaluate in any way you chose.

43. Evaluate in any way you chose.

44. Evaluate in any way you chose.

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

51. Evaluate:

52. Evaluate:

53. Evaluate:

54. Evaluate:

55. Evaluate:

56. Evaluate:

57. Evaluate:

58. Evaluate:

59. Evaluate:

60. Evaluate:

61. Evaluate:

62. Evaluate:

63. Evaluate:

64. Evaluate:

65. Evaluate:

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

67. Three Special Limits
Try it out!