Section 3.5 – Limits at Infinity

1. Section 3.5 – Limits at Infinity

2. Vertical Asymptotes and Limits When we investigated infinite limits and vertical asymptotes, we let x approach a number. The result was that the values of y became arbitrarily large (positive or negative).

3. White Board Challenge Analytically find the vertical asymptote(s) of:

4. Horizontal Asymptotes and Limits When we investigate infinite limits and horizontal asymptotes, we will let x become arbitrarily large (positive or negative) and see what happens to y. This will be referred to as the end behavior.

5. End Behavior Let f be a function defined on some interval (a,∞). Then means that the values of f(x) get closer to L as x increases.

6. End Behavior Let f be a function defined on some interval (-∞, a). Then means that the values of f(x) get closer to L as x decreases.

7. End Behavior Let f be a function defined on some interval (a,∞). Then means that the values of f(x) become large (positive or negative) as x increases.

8. End Behavior Let f be a function defined on some interval (-∞, a). Then means that the values of f(x) become large (positive or negative) as x decreases.

9. White Board Challenge Sketch a graph of a function with the following characteristics: The function is continuous for all reals except 5.

10. Calculating Limits at Infinity Our book focuses on three ways: • Numerical Approach – Construct a table of values • Graphical Approach – Draw a graph • Analytic Approach – Use Algebra or calculus First Second

11. Example 1 Use the graph and complete the table to find the limit (if it exists). 0.999998 -1 0 0.923 0.980 0.9992 0.9998 As x increases, the value of the function approaches 1.

12. Example 2 Use the graph and complete the table to find the limit (if it exists). -3003 -303 -153 -32.778 -17.5 UND -5 As x decreases, the value of the function decreases.

13. Example 3 Use the graph and complete the table to find the limit (if it exists). 0.000005 0.000000005 UND 5 0.04 0.005 0.00004 As x increases, the value of the function approaches 0.

14. Example 4 Use the graph and complete the table to find the limit (if it exists). 0.001 0.002 0.01 0.1 1 10 UND As x decreases, the value of the function approaches 0.

15. “Special Property” of Limits to Infinity If A is any real number and r is a positive rational number then, Furthermore, if r is such that xr is defined for x < 0, then

16. White Board Challenge Use a table or graph to find the limit:

17. Two Procedures for Analytically Determining Infinite Limits If the function is a rational function or a radical/rational function: • Divide each term in the numerator and denominator by the highest power of x that occurs in the denominator. • Use basic limit laws and the “Special Property” of Infinite Limits to evaluate the limit. OR Use L’Hôpital’s Rule to evaluate the limit (Only if L’Hôpital’s Rule applies.)

18. Reminder

19. Example 1 (Procedure 1) Analytically evaluate . In order to use previous results, divide both the numerator and denominator by the highest power of x appearing in the fraction Use “Direct Substitution” and previous results.

20. ***Aside*** Analytically evaluate . For this example, the limit’s value does not change if x approaches negative infinity.

21. Example 1 (Procedure 2) Analytically evaluate L’Hôpital’s Rule applies since this is an indeterminate form. In order to use L’Hôpital’s Rule direct substitution must result in 0/0 or ∞/∞. Differentiate the numerator and the denominator. Find the limit of the quotient of the derivatives. This is still an indeterminate form, apply L’Hôpital’s Rule again to the new limit. Differentiate the new numerator and the denominator. Find the limit of the quotient of the second derivatives. Since the result is finite or infinite, the result is valid.

22. Example 2 (Procedure 1) Analytically evaluate . In order to use previous results, divide both the numerator and denominator by the highest power of x appearing in the fraction Use “Direct Substitution” and previous results.

23. ***Aside*** Analytically evaluate . For this example, the limit’s value does not change if x approaches negative infinity.

24. Example 2 (Procedure 2) L’Hôpital’s Rule applies since this is an indeterminate form. Analytically evaluate . In order to use L’Hôpital’s Rule direct substitution must result in 0/0 or ∞/∞. Differentiate the numerator and the denominator. Find the limit of the quotient of the derivatives. This is still an indeterminate form, apply L’Hôpital’s Rule again to the new limit. Differentiate the new numerator and the denominator. Find the limit of the quotient of the second derivatives. This is still an indeterminate form, apply L’Hôpital’s Rule again to the new limit. Differentiate the new numerator and the denominator. Find the limit of the quotient of the third derivatives. Since the result is finite or infinite, the result is valid.

25. Example 3 (Procedure 1) In order to use the previous result, divide both the numerator and denominator by the highest power of x appearing in the fraction Analytically evaluate . But, in order to simplify the numerator, you must rewrite 1/x Use “Direct Substitution” and previous results

26. ***Aside*** Analytically evaluate . For this example, the limit’s value does change if x approaches positive infinity.

27. Example 3 (Procedure 2) Analytically evaluate . L’Hôpital’s Rule applies since this is an indeterminate form. In order to use L’Hôpital’s Rule direct substitution must result in 0/0 or ∞/∞. L’Hôpital’s Rule has failed to find a limit. This final result is almost identical to the original. The first procedure is more applicable. Differentiate the numerator and the denominator. Find the limit of the quotient of the derivatives. This is still an indeterminate form, apply L’Hôpital’s Rule again to the new limit. Differentiate the new numerator and the denominator. Find the limit of the quotient of the second derivatives.

28. Example 4 (Procedure 1) Strategy: Rewrite one factor so its numerator is 1. Analytically evaluate the following limit: Rewrite the expression as a ratio in order to use the first procedure. Now evaluate the limit: Since the denominator is not a polynomial, we can not use the first procedure. We need to try something new.

29. Example 4 (Procedure 1) Strategy: Rewrite one factor so its numerator is 1. Analytically evaluate the following limit: Rewrite the expression as a ratio in order to use L’Hôpital’s Rule. L’Hôpital’s Rule applies since this is an indeterminate form. In order to use L’Hôpital’s Rule direct substitution must result in 0/0 or ∞/∞. Differentiate the numerator and the denominator. Since the result is finite or infinite, the result is valid. Find the limit of the quotient of the derivatives.

30. ***Aside*** Analytically evaluate the following limit: For this example, the limit’s value does not change if x approaches negative infinity.

31. Day 45: November 10th Objective: Determine (finite) limits at infinity, horizontal asymptotes of a graph if they exist, and infinite limits at infinity Homework Questions Notes: Section 3.5 Conclusion Homework: Read pgs. 198-204 and complete 3.5

32. White Board Challenge Analytically evaluate each limit below:

33. Horizontal Asymptotes and Limits The line y = L is called a horizontal asymptote of the curve y = f(x) if Lis finite and either Since: Then y = 1 is a horizontal asymptote.

34. Procedure for Finding Horizontal Asymptotes For a function f : • Find the limit of the function as x goes to positive infinity. • Find the limit of the function as x goes to negative infinity. • If either of the above limits is finite, then they represent a horizontal asymptote(s) (remember to write the result as y =)

35. Examples Continued For our previous examples:

36. Whiteboard Challenge On a calculator, graph What is a characteristic of this graph that we have not discussed?

37. Whiteboard Challenge Slant/Oblique Asymptotes.

38. Oblique/Slant Asymptote For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division. Degree = 2 Degree = 1

39. Procedure for Finding Oblique/Slant Asymptotes of a Rational Function In a rational function f , if the degree of the numerator is one more than the degree of the denominator: • Perform Polynomial division. • Ignoring the remainder, the result is the oblique/slant asymptote. (remember to write the result as y =)

40. Example Analytically find the slant asymptote of x 2 Rm Perform Polynomial Division. x x2 2x 4 - 3 -3x -6 x2 – x – 2 Ignore the remainder Thus: This means y = x + 2 is a slant asymptote because:

41. Asymptotes Summary The following asymptotes exists if… Vertical: When there is a non-removable discontinuity (a value for x that makes the denominator 0 and the numerator non-zero) Horizontal: When the limit as x approaches infinity (positive or negative), the value for y approaches a real number. Slant: For a rational function, the degree of the numerator is one more than the degree of the denominator.