1. 10.3 – More on Limits Take a few minutes to review your last test!!
We will spend 15 min. on most frequently missed questions

2. Vocabulary Limit at a (informal definition) Average Rate of Change
Limit at Infinity (informal definition)
Properties of Limits
Limits at Infinity (informal definition)

3. Algebra Review (8 min. – No Talking!!)
Simplify x3 – 8
x - 2
Evaluate to find:
f(-5) =
f(-2) =
f(0) =
f(2) =

4. 10.3 More on Limits In this section, you will …
Informally define a limit
Examine properties of limits
Determine limits of continuous functions
Determine one-sided and two-sided limits
Determine limits involving infinity

5. Use the table and the stated increment to determine the limit
Start at 2.99 and  = 0.01
Start at and  = 0.001
Start at 100 and  = 100
Start at 10 and  = 10

6. Use the table and the stated increment to determine the limit
These limits are relatively easy to support using a graphing calculator!!
Type them in and verify your answers!!

7. Informal Definition of a Limit
When we write “ ” we mean that
f(x) gets arbitrarily close to L as x gets arbitrarily close (but NOT equal) to a.

8. Exploration “What’s the Limit?”
Complete the exploration on p. 816 in your textbook
6 min. – no talking (individual work)
4 min. – discuss your responses with a neighbor
5 min. – share results with the class

9. Finding a Limit Solve Graphically Solve Numerically (Use a table)
Solve Algebraically
(Factor & evaluate)

10. Finding a Limit Solve Graphically Solve Numerically (Use a table)
Solve Algebraically
(Factor & evaluate)

11. Properties of Limits If and both exist, then 1. Sum Rule
2. Difference Rule
3. Product Rule
4. Constant Multiple Rule

12. Properties of Limits (cont.)
If and both exist, then
5. Quotient Rule
6. Power Rule
7. Root Rule

13. Example
Use the properties of limits to evaluate each of the following. Note:

14. Limits of Continuous Functions
Recall a function is continuous at a if
The properties of limits apply to all continuous functions!!

15. Using Substitution
Find the limits

16. More examples
Evaluate each limit or state that it does not exist

17. One-sided vs Two-sided limits
One-sided limits are determined from either the left-hand side (below) or the right-hand side (above)
Note: The limit from one-side may exist whereas the limit from the other may NOT!!

18. One-sided limits
Left-hand: The limit of f as x approaches c from the left
Right-hand: The limit of f as x approaches c from the right
The limit is sometimes call the two- sided limit of f at c to distinguish it from the one-sided left-hand and right-hand limits of f at c.

19. Theorem One-sided and Two-sided limits
The function f(x) has a limit as x approaches c if and only if the left-hand and right-hand limits at c EXIST AND are EQUAL. That is,

20. Try these
Evaluate each limit or state that it does not exist

21. Infinite Limits Vs Limits at Infinity
Limits at Infinity: f(x) gets arbitrarily
close to L as x gets arbitrarily large. f has a limit as x approaches 
Infinite Limits: The function values of f
as x approaches c approaches  or Does Not Exist (DNE)

22. Examples

23. Your Turn

24. Exploration Time!!  Complete the Exploration 2 on p. 823
2 min – No Talking (Individual work)
1 min – Discuss answers with a neighbor
2 min – Share results with the class

25. Limits at x = 0 Use a table of values to verify
Use a graph to determine

26. Your Turn
Find