1. 1.3 Evaluating Limits Analytically

2. After this lesson, you will be able to:
Evaluate a limit using the properties of limits
Develop and use a strategy for finding limits
Evaluate a limit using dividing out and rationalizing techniques
Evaluate a limit using Squeeze Theorem

3. Limits Analytically
In the previous lesson, you learned how to find limits numerically and graphically. In this lesson you will be shown how to find them analytically…using algebra or calculus.

4. Theorem 1.1 Some Basic Limits
Let b and c be real numbers and let n be a positive integer.
Examples 1:
Think of it graphically…
Let
Let
Let
(y scale was adjusted to fit)
As x approaches 3, f(x) approaches 4
As x approaches 2, f(x) approaches 2
As x approaches 5, f(x) approaches 125

5. Direct Substitution
Some limits can be evaluated by direct substituting for x.
Direct substitution works on continuous functions.
Continuous functions do not have any holes, breaks or gaps.
Note: Direct substitution is valid for all polynomial functions and rational functions whose denominators are not zero (or not approaches to zero) as the x approaches to a certain value.
Example
However

6. Theorem 1.2 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 limits:
and
Scalar multiple:
Sum or difference:
Product:
Quotient:
Power:

7. Theorem 1.2 Properties of Limits
Note: The following assumptions are necessary to those properties of limits. You need provide all the counterexamples to some of the properties if the assumptions are not provided.
and
Scalar multiple:
Sum or difference:
Product:
Quotient:
Power:

8. Limit of a Polynomial Function
Example 2:
Since a polynomial function is a continuous function, then we know the limit from the right and left of any number will be the same. Thus, we may use direct substitution.
Ans: The limit is 5

9. Theorem 1.3 Limits of Polynomial and Rational Function
If p is a polynomial function and c is a real number, then
If p(x) and q(x) are polynomial functions and r(x) = p(x)/q(x) and c is a real number such that q(c) ≠ 0, then

10. Limit of a Rational Function
Make sure the denominator doesn’t = 0 !
Example 3:
Ans: 1
If the denominator had been 0, we would NOT have been able to use direct substitution.

11. Theorem 1.4 The Limit of a Function Involving a Radical
Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is even.

12. Theorem 1.5 The Limit of a Composite Function
If f and g are functions such that lim g(x) = L and lim f (x) = f (L), then

13. Limit of a Composite Function-part a
Example 4: Given
a) Find
Direct substitution works here

14. Limit of a Composite Function -part b
Example 4: Given
b) Find
Direct substitution works here

15. Limit of a Composite Function -part c
Example 4: Given
c) Find
From part a, we know that the limit of f(x) as x approaches 4 is 21

16. Theorem 1.6 The Limits of Trig Functions
Let c be a real number in the domain of the given trigonometric function 1. 2. 3. 4. 5. 6.

17. Limits of Trig Functions
Examples 5:

18. A Strategy for Finding Limits
Try Direct Substitution
If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to divide out common factors or to rationalize the numerator or the denominator so that direct substitution works.
Use a graph or table to reinforce your result.

19. Theorem 1.7 Functions That Agree at All But One Point
Let c be a real number and let f (x) = g (x) for all x ≠ c in an open interval containing c. If exists, then
also exists and

20. Example- Factoring Example 6:
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

21. Example- Factoring Example 7:
Direct substitution results in 0 in the denominator. Try factoring.
Now direct substitution will work.
Use your calculator to reinforce your result.

22. Example
Sum of cubes
Not factorable
Example 8:
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.

23. 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.

24. Theorem 1.8 The Squeeze Theorem
If h(x) ≤ f (x) ≤ g(x) for all x in an open interval containing c, except possibly at c itself, and if
then

25. 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.

26. 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!!!)

27. Example- Using Trig Limits=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.

28. Example
Example 12
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.
= 1

29. Example Looking at the unit circle, this value is 1 Example 13
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.

30. Example
Example 14
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.
Use the Pythagorean Trig Identity
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