1. Welcome to Class
Arrival Instructions: Factor each of the following functions.

2. Evaluating Limits Algebraically
Day 3
Evaluating Limits Algebraically

3. Algebraic Techniques Direct substitution Factoring Rationalizing roots
Multiply by 1
Synthetic Division
A few to memorize
Sometimes the limit does not exist

4. Based on yesterday’s notes:
Is often independent of f(c)
However, for continuous functions
We will refer to this as DIRECT SUBSTITUTION to find a limit.

5. Example

6. DIRECT SUBSTITUTION can be used if. . .
The function is continuous at c.
Which means (informally):
No holes
No jumps
No vertical asymptotes

7. f(-4) D.N.E., however if we factor we can find
Dealing with “holes”
f(-4) D.N.E., however if we factor we can find

8. Practice:

9. Dealing with zero Undefined and indeterminate are NOT the same thing!
Undefined is an answer and means it does not exist.
Indeterminate means we do not know the answer…yet.

10. “Indeterminate Form”: when direct substitution produces 0/0
Remember Conjugates?!?...

11. You try:

12. Back in Time Fractions LCDs (Least Common Denominators)
To get rid of the denominators, multiply by . . .

13. Multiply by 1 in a “convenient form” (The common denominator)

14. Synthetic Division

15. KNOW THESE!!

16. Together Let’s Try

17. Summary Direct substitution Factoring Rationalizing roots
Multiply by 1
Synthetic Division
A few to memorize
Sometimes the limit does not exist

18. One last way to find a limit…
Squeeze Thm (aka Sandwich Thm): If
and
then

19. Sandwich Thm Example