Algebra of Limits

1. Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:

2. Calculating LimitsFinding the limit of a function f a point x = a. Distinguishing the following cases: • The case when f is continuous a x = a. • The case 0/0. • The case ∞/ ∞ • The case of an infinite limit • The case of the function f defined by a formula involving absolute values. • The case c/∞, where c is a real number. • Other cases: the case ∞- ∞ • The case, when it is possible to use the squeeze theorem.

3. 1. The case when f is continuous at x = a If f is continues at x=a, then: Notice:1. Polynomial functions and the cubic root function ( & all functions of its two families) are everywhere continuous.2. Rational, trigonometric and root functions are continuous at every point of their domains.3. If f and g are continuous a x=a, then so are cf, f+g, f-g, fg and f/g (provided that he limit of f at x=a is not zero)

4. Examples for the case when f is continuous at x = a

5. Examples for the case when f is continuous at x = a

6. Examples for the case when f is continuous at x= a

7. 2. The case 0/0 Suppose we want to find: For the case when: Then this is called the case 0/0. Caution: The limit is not equal 0/0. This is just a name that classifies the type of limits having such property.

8. Examples for the case 0/0

9. Examples for the case 0/0

10. 3. The case ∞/ ∞ Suppose we want to find: For the case when the limits of both functions f and g are infinite Then this is called the case ∞/ ∞. Caution: The limit is not equal ∞/∞. This is just a name that classifies the type of limits having such property.

11. Examples for the case ∞/∞ • See the examples involving rational functions in the file on the limit a infinity. • Examples involving roots: See the following slides

12. Limits at infinity A function y=f(x) may approach a real number b as x increases or decreases with no bound. When this happens, we say that f has a limit at infinity, and that the line y=b is a horizontal asymptote for f.

13. 1. Limit at infinity: The Case of Rational Functions A rational function r(x) = p(x)/q(x) has a limit at infinity if the degree of p(x) is equal or less than the degree of q(x). A rational function r(x) = p(x)/q(x)does not have a limit at infinity (but has rather infinite right and left limits) if the degree of p(x) is greater than the degree of q(x).

14. Example (1) Let Find Solution: Since the degree of the polynomial in the numerator, which is 9, is equal to the degree of the polynomial in the denominator, then

15. To show that, we follow the following steps:

16. Example (2) Let Find Solution: Since the degree of the polynomial in the numerator, which is 9, is less than the degree of the polynomial in the denominator, which is 12, then

17. To show that, we follow the following steps:

18. Example (3) Let Find Solution: Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 9, then

19. They are infinite limits. To show that, we follow the following steps:

20. Example (4) Let Find Solution: Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 9, then

21. They are infinite limits. To show that, we follow the following steps:

22. Example (5) Let Find Solution: Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 8, then

23. They are infinite limits. To show that, we follow the following steps:

24. Example (6) Let Find Solution: Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 8, then

25. They are infinite limits. To show that, we follow the following steps:

26. Limits @ Infinity2. Problems Involving Roots

27. Introduction We know that: √x2 = |x|, which is equal x is x non-negative and equal to – x if x is negative For if x = 2, then √(2)2 = √4 = |2|=2 & if x = - 2, then √(-2)2= √4 = |-2|=-(-2) = 2

28. Example

29. Example (1)

32. Example (2)

35. Homework: Problems:Example (4) & (5) –Section 3.4. Page: 228 Exercises 3.4 Page: 235 :Problems: 9, 10, 11, 15, 23, 25,17 & 40

36. 4. The case of infinite limit See the examples in the file on the infinite limits and also the examples of infinite limits in the file on limits at infinity.

37. Infinite Limits A function f may increases or decreases with no bound near certain values c for the independent variable x. When this happens, we say that f has an infinite limit, and that f has a vertical asymptoteat x = c The line x=c is called a vertical asymptote for f.

38. Infinite Limits- The Case of Rational Functions A rational function has an infinite limit if the limit of the denominator is zero and the limit of the numerator is not zero. The sign of the infinite limit is determined by the sign of both the numerator and the denominator at values close to the considered point x=c approached by the variable x.

39. Example (1) Let Find Solution: First x=0 is a zero of the denominator which is not a zero of the numerator.

40. a. As x approaches 0 from the right, the numerator is always positive ( it is equal to 1) and the denominator approaches 0 while keeping positive; hence, the function increases with no bound. Thus: The function has a vertical asymptote at x = 0, which is the line x = 0 (see the graph in the file on basic algebraic functions). b. As x approaches 0 from the left, the numerator is always positive ( it is equal to 1) and the denominator approaches 0 while keeping negative; hence, the function decreases with no bound.

41. Example (2) Let Find Solution: First x=1 is a zero of the denominator which is not a zero of the numerator.

42. a. As x approaches 1 from the right, the numerator approaches 6 (thus keeping positive), and the denominator approaches 0 while keeping positive; hence, the function increases with no bound. Thus, The function has a vertical asymptote at x = 1, which is the line x = 1 b. As x approaches 1 from the left, the numerator approaches 6 (thus keeping positive), and the denominator approaches 0 while keeping negative; hence, the function decreases with no bound. The function has a vertical asymptote at x=1, which is the line x = 1. Thus:

43. Example (3) Let Find Solution:First, rewrite: x=0 is a zero of the denominator which is not a zero of the numerator.

44. a. As x approaches 0 from the right, the numerator approaches -1 (thus keeping negative), and the denominator approaches 0 while keeping positive; hence, the function decreases with no bound. Thus: The function has a vertical asymptote at x = 0, which is the line x = 0 b. As x approaches 0 from the left, the numerator approaches -1 (thus keeping negative), and the denominator approaches 0 while keeping negative; hence, the function increases with no bound. Thus:

45. Same Type Problems from the Homework Exercises 1.5 Pages 59-61Problems: 29, 31, 33, 37

46. 5. The case of discontinuous function f defined by a formula involving absolute values

47. 5. The case of discontinuous function f defined by a formula involving absolute values

48. Same Type Problems from the Homework Exercises 1.6 Pages 69-71Problems: 43, 45, 42, 44

49. 6. The case constant/∞ Suppose we want to find: For the case when: In this case, no mater what the formulas of g and h are, we will always have: Then this is called the case c/∞. Caution: The limit is not equal c/∞. This is just a name that classifies the type of limits having such property. This limit is always equal zero

50. Example on the case constant/∞