1. EVALUATING LIMITS ANALYTICALLY
Section 1.3

2. When you are done with your homework, you should be able to…
Evaluate a Limit Using Properties of Limits
Develop and Use a Strategy for Finding Limits
Evaluate a Limit Using Dividing Out and Rationalizing Techniques
Evaluate a Limit Using the Squeeze Theorem

3. SOME BASIC LIMITS
Let and b and c be real numbers and let n be a positive integer.

4. Evaluate 5 -3
Does not exist

5. Evaluate 1 -1 5
Does not exist

6. Properties of Limits
Let and b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:
1. Scalar multiple
Sum or difference
Product
Quotient
5. Power

7. Evaluate -4 -2 1
Does not exist

8. Limits of Polynomial and Rational Functions
If p is a polynomial function and c is a real number, then
If r is a rational function given by
and c is a real number
such that then

9. Evaluate 5 -5
Does not exist

10. Evaluate the function at x = 21
DNE

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

12. The Limit of a Composite Function
Let f and g be functions with the following limits:
Then

13. Limits of Trigonometric Functions
Let c be a real number in the domain of the given trigonometric function.

14. STRATEGIES FOR FINDING LIMITS
Functions That Agree at All But One Point
Let c be a real number and let for all x in an open interval containing c. If the limit of g as x approaches c exists, then the limit of f also exists and

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.
Apply .
Use a graph or table to reinforce your
conclusion.

16. Dividing Out Techniques
Example:

17. Evaluate.
Does not exist
1/16

18. Rationalizing Techniques
Example:

19. Evaluate the exact limit.
.25
0.0

20. TWO SPECIAL TRIGONOMETRIC LIMITS

21. Evaluate the exact answer.
5.0
0.0