1. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Does not exist
number
number

2. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Example:
Consider the function:
Study the behavior of the function around x=0. 10
0.1
100
0.01
1000
0.001
10000
0.0001
Vertical Asymptote

3. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Example: 10
1.1
100
1.01
1000
1.001
10000
1.0001
-10
0.9
-1000
0.999
-10000
Vertical Asymptote

4. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

5. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

6. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

7. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Infinite Limits

8. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

9. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Vertical Asymptote &

10. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Find all vertical asymptotes

11. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
EXAMPLE

12. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Does not exist
number
number

13. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Example:
Example: 1
0.1 10
0.01
100
0.001
1000
------
-----
1,000,000 1
0.01 10
0.0001
100
1000
------
-----
10^(-12)
1,000,000

14. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

15. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Example:
Example:
Example:
Example:

16. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Limits at Infinity of Rational Functions
To evaluate the limit at infinity of any rational function, we first divide both the
numerator and denominator by the highest power of that occurs in the denominator.
Example:
Example:

17. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Remark:
To evaluate the limit at infinity of any rational function, we first divide both the
numerator and denominator by the highest power of that occurs in the denominator.
Example:
Example:
Example:
Remark:

18. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Notes:
Notes:

19. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Notes:
Notes:

20. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Example: (ex80p116)
Multiply by conjugate radical.
Example:
Example:

21. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

22. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Example:
Example:
Example:

23. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Vertical Asymptote &
Horizontal Asymptote &
The line
Is a horizontal asymptote

24. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Example:
The line
Is a horizontal asymptote

25. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Example:
The line
Is a horizontal asymptote

26. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Example:

27. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
EXAM-1 TERM-121

28. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

29. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
To evaluate the limit at infinity of any rational function, we first divide both the
numerator and denominator by the highest power of that occurs in the denominator.
Multiply by conjugate radical.
Factor then take the limit

30. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Vertical Asymptote
Horizontal Asymptote
Oblique Asymptote
If the degree of the numerator of a rational function is 1 greater than the degree of the denominator, the graph has an oblique or slant line asymptote.
We find an equation for the asymptote by dividing numerator by denominator
Remark:

31. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Who is going faster to infinity
Example:

32. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Example: Sketch the graph of

33. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

34. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Exam1-Term101

35. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphsb

36. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Limit Laws

37. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

38. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

39. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

40. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Do you have the graph
Find
From the graph study the limit from right and left
Does f contain greatest integer or absolute value
Study the limit from right and left
Substitute and find the limit
Can we use
Direct substitution
Use: 1)factor then cancel
2)Multiply by conjugate
3)Make common denominator
Write f as:
Use squeeze theorem

41. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Factor then take the limit
Sandwich thm
high pwr in denomi
Rational func
Containing radicals
Remove the | |
Containing absolute value
Multiply by conjugate radical.
Containing noninteger
Use
Use graph
Use

42. Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs
Reminder:
After sec2.5 continuity