1. Asymptotes, End Behavior, and Infinite Limits
Keeper 12
Honors Calculus

2. End Behavior of Polynomial Functions

3. End Behavior of Polynomial Functions

4. Example
Describe the End Behavior − 𝑥 7 +2 𝑥 5 −4 𝑥 2 +2𝑥−4

5. Example
Describe the End Behavior 𝑥 8 +2 𝑥 5 −4 𝑥 2 +2𝑥−4

6. Example
Describe the End Behavior − 𝑥 7 +2 𝑥 5 −4 𝑥 10 +2𝑥−4

7. Example
Describe the End Behavior − 𝑥 7 +2 𝑥 5 −4 𝑥 2 +2 𝑥 9 −4

8. Asymptotes Of rational Functions
Vertical Asymptotes – Set the denominator equal to 0 and solve Horizontal Asymptotes – Focus on the degree of the numerator and denominator - degree of numerator = degree of denominator 𝑦= 𝑙𝑒𝑎𝑑 𝑐𝑜 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑒𝑎𝑑 𝑐𝑜 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 - degree of numerator < degree of denominator 𝑦=0 - degree of numerator > degree of denominator – No horizontal asymptote Oblique/Slant Asymptote – degree of numerator = degree of denominator +1 - use long division to find equation of oblique asymptote ***Watch out for holes!!! - factor in the numeration = factor in denominator

9. Find all Asymptotes
𝑓 𝑥 = 4𝑥+5 8𝑥−3

10. Find all Asymptotes
𝑓 𝑥 = 6 𝑥 2 −𝑥 3 𝑥 3 +1

11. Find all Asymptotes
𝑓 𝑥 = 5 𝑥 4 9 𝑥 3 +2𝑥

12. Find all Asymptotes
𝑓 𝑥 = 6 𝑥 2 −3𝑥+4 2 𝑥 2 −8

13. Find all Asymptotes
𝑓 𝑥 = 𝑥 2 −9 𝑥+10

14. Find all Asymptotes
𝑓 𝑥 = 𝑥−12 2 𝑥 2 +5𝑥−3

15. Find the asymptotes, sketch the graph and discuss the end behavior
𝑓 𝑥 = 5 𝑥 3 𝑥 2 −4𝑥+2

16. Find the asymptotes, sketch the graph and discuss the end behavior
𝑓 𝑥 = 7𝑥−2 𝑥+3

17. Find the asymptotes, sketch the graph and discuss the end behavior
𝑓 𝑥 = 3 𝑥 2 −𝑥+12 2 𝑥 2 −6𝑥+7

18. Find the asymptotes, sketch the graph and discuss the end behavior
𝑓 𝑥 = 4𝑥+7 6 𝑥 2 −5

19. Find the asymptotes, sketch the graph and discuss the end behavior
𝑓 𝑥 = 8 𝑥 2 −5𝑥+1 4 𝑥 2 −3

20. Limits of Power Functions at Infinity

21. Limits of Polynomial Functions at Infinity

22. Example
lim 𝑥→−∞ ( 𝑥 3 −2 𝑥 2 +5𝑥−1)

23. Example
lim 𝑥→∞ (4+3𝑥− 𝑥 2 )

24. Example
lim 𝑥→−∞ (5 𝑥 4 −3𝑥)

25. Limits of Rational Functions

26. Algebraic Rules
If the degree of the numerator is greater than the degree of the denominator then the lim 𝑥→∞ 𝑓 𝑥 =±∞
If the degree of the numerator is equal to the degree of the denominator then the lim 𝑥→∞ 𝑓(𝑥) = 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠.
If the degree of the numerator is less than the degree of the denominator then the lim 𝑥→∞ 𝑓(𝑥) =0

27. Example
lim 𝑥→∞ 4𝑥+5 8𝑥−3

28. Example
lim 𝑥→−∞ 6 𝑥 2 −𝑥 3 𝑥 3 +1

29. Example
lim 𝑥→∞ 5 𝑥 4 9 𝑥 3 +2𝑥

30. Example
lim 𝑥→∞ 15 𝑥 2 −2𝑥+3 5 𝑥 2 −7

31. Example
lim 𝑥→∞ 𝑥 𝑥−3

32. Example
lim 𝑥→−∞ 𝑥 𝑥−3

33. Example
lim 𝑥→∞ 4−3 𝑥 3 2 𝑥 3 +3𝑥−1

34. Example
lim 𝑥→∞ 10−3𝑥 2𝑥+1 3

35. Example
lim 𝑥→∞ tan 𝑥