1. OUR GOAL Find: Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
From sec 2.1 :
OUR GOAL
Find:

2. OUR GOAL Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
From sec 2.1 :
OUR GOAL

3. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

4. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

5. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

6. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
For the function whose graph is given below
Determine the following limits

7. Exam-I Term-122 Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Exam-I
Term-122

8. 1 2 Find: Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Find: 1
Use the Graph to find the limit 2
Create a table: watch the behavior

9. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Example:
Find
We say that the limit of f(x) as x approaches 2 from the left is equal to 4
We say that the limit of f(x) as x approaches 2 from the right is equal to 4
The symbol x  2- means that we consider only values of x < 2
The symbol x  2+ means that we consider only values of x > 2

10. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Example:
Find

11. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Example:
Example:
Example:
Example:

12. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Can we use Direct Substitution Prop.?? -

13. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Can we use Direct Substitution Prop.??

14. 1 2 3 Find: Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Find: 1
Use the Graph to find the limit 2
Create a table: watch the behavior 3
Given the function f(x)

15. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

16. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
What is the difference ??

17. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Example:
case
1)Common Factors
2)Conjugate radical
Exam-1 Term 122
Exam-1 Term 121
Example

18. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
case
1)Common Factors
2)Conjugate radical
3)Open brackets

19. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

20. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Absolute Value Functions
NOTE:
1) Study
2) In each side limit, use the definition of the absolute value

21. piecewise functions Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
piecewise functions
NOTE:
1) Study
2) In each side limit, use the definition of the piecewise function

22. 3 4 -1 1 4 Example: Example: Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
The Greatest Integer functions
The largest integer that is less than or equal to x 3 4 -1 1 4
Example:
Example:
NOTE:
1) Study
2) In each side limit, substitute by a number

23. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

24. Squeeze theorem = Sandwich theorem
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Squeeze theorem = Sandwich theorem

25. Example: Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Example:

26. Exam1-091 Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Exam1-091

27. Postpone to sec2.6: Limits at Infinity:
Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Postpone to sec2.6: Limits at Infinity:
Limit from one side does not exist
Find an example for a function such that the limit from right does not exist

28. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

29. INFINITE LIMITS NOTE THAT: Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
INFINITE LIMITS
NOTE THAT:
The line x = 0 is a vertical asymptote

30. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
NOTE THAT:
INFINITE LIMITS
the line
All red lines are ???????
Can we conclude? VA

31. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Example:
Example: x y
3.1 10
3.001
1000
100000 x y
2.9 10
2.999
1000
100000
Note:
Examples:

32. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Rational function
(substitute by a)
1)Common Factors
2)Open brackets
3)……..
Direct substitution

33. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
Graph: y = ln x
Graph: y = tan x
is a vertical asymptote
is a vertical asymptote

34. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
NOTE THAT:
1) Find the domain of f(x)
2) Study the limit
3) If the limit is + or - infinity

35. Exam-1 Term 152 Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS
NOTE THAT:
1) Find the domain of f(x)
2) Study the limit
3) If the limit is + or - infinity
Exam-1 Term 152

36. Sec 2.2: Limit of Function and Limit Laws
Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS