# Limits at Infinity Limits at Infinity for Polynomials Limits at Infinity for the Exponential Function Function Dominance More on Asymptotes

Save this PDF as:

Size: px
Start display at page:

Download "Limits at Infinity Limits at Infinity for Polynomials Limits at Infinity for the Exponential Function Function Dominance More on Asymptotes"

## Transcription

1 Lecture 5 Limits at Infinity and Asymptotes Limits at Infinity Horizontal Asymptotes Limits at Infinity for Polynomials Limit of a Reciprocal Power The End Behavior of a Polynomial Evaluating the Limit at Infinity of a Polynomial Example 17 Limit of Rational Function Example 18 More Rational Functions Limits at Infinity for the Exponential Function Example 19 Limits of Exponential Functions Function Dominance Example 20 Exponential and Power Functions More on Asymptotes Vertical Asymptotes and Infinite Limits Horizontal Asymptotes and Limits at Infinity Example 21 Investigating Asymptotes Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 1/29

2 Limits at Infinity Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 2/29

3 Limits at Infinity Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition (Limits at Infinity) Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 2/29

4 Limits at Infinity Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition (Limits at Infinity) We say that f(x) = L x if f(x) can be made arbitrarily close to L by making x sufficiently large and positive Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 2/29

5 Limits at Infinity Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition (Limits at Infinity) We say that f(x) = L x if f(x) can be made arbitrarily close to L by making x sufficiently large and positive, and we say that f(x) = M x if f(x) can be made arbitrarily close to M by making x sufficiently large and negative. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 2/29

6 Horizontal Asymptotes Horizontal Asymptotes Horizontal Asymptotes Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 3/29

7 Horizontal Asymptotes Horizontal Asymptotes Horizontal Asymptotes If f(x) = L x then the graph of f has a horizontal asymptote along the line y = L on the right Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 3/29

8 Horizontal Asymptotes Horizontal Asymptotes Horizontal Asymptotes If f(x) = L x then the graph of f has a horizontal asymptote along the line y = L on the right, and if x f(x) = M then the graph of f has a horizontal asymptote along the line y = M on the left. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 3/29

9 Horizontal Asymptotes Horizontal Asymptotes Horizontal Asymptotes If f(x) = L x then the graph of f has a horizontal asymptote along the line y = L on the right, and if x f(x) = M then the graph of f has a horizontal asymptote along the line y = M on the left. If f has an unbounded domain, but f(x) does not approach a finite value as x goes to ±, there are two possibilities: Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 3/29

10 Horizontal Asymptotes Horizontal Asymptotes Horizontal Asymptotes If f(x) = L x then the graph of f has a horizontal asymptote along the line y = L on the right, and if x f(x) = M then the graph of f has a horizontal asymptote along the line y = M on the left. If f has an unbounded domain, but f(x) does not approach a finite value as x goes to ±, there are two possibilities: the it at infinity is infinite Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 3/29

11 Horizontal Asymptotes Horizontal Asymptotes Horizontal Asymptotes If f(x) = L x then the graph of f has a horizontal asymptote along the line y = L on the right, and if x f(x) = M then the graph of f has a horizontal asymptote along the line y = M on the left. If f has an unbounded domain, but f(x) does not approach a finite value as x goes to ±, there are two possibilities: the it at infinity is infinite the it at infinity does not exist Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 3/29

12 Horizontal Asymptotes Horizontal Asymptotes Horizontal Asymptotes If f(x) = L x then the graph of f has a horizontal asymptote along the line y = L on the right, and if x f(x) = M then the graph of f has a horizontal asymptote along the line y = M on the left. If f has an unbounded domain, but f(x) does not approach a finite value as x goes to ±, there are two possibilities: the it at infinity is infinite the it at infinity does not exist In either case, the graph of f does not have a horizontal asymptote. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 3/29

13 Limit of a Reciprocal Power Limit of a Reciprocal Power Limit of a Reciprocal Power Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 4/29

14 Limit of a Reciprocal Power Limit of a Reciprocal Power Limit of a Reciprocal Power If r > 0, then x 1 x r = 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 4/29

15 Limit of a Reciprocal Power Limit of a Reciprocal Power Limit of a Reciprocal Power If r > 0, then x 1 x r = 0 and x 1 x r = 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 4/29

16 Limit of a Reciprocal Power Limit of a Reciprocal Power Limit of a Reciprocal Power If r > 0, then and x x 1 x r = 0 1 x r = 0 if r is a positive integer or is a fraction that results is an even root. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 4/29

17 The End Behavior of a Polynomial The End Behavior of a Polynomial Limits at Infinity for Polynomials Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 5/29

18 The End Behavior of a Polynomial The End Behavior of a Polynomial Limits at Infinity for Polynomials For the n th degree polynomial function p(x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 the the end behavior is given by the it at infinity Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 5/29

19 The End Behavior of a Polynomial The End Behavior of a Polynomial Limits at Infinity for Polynomials For the n th degree polynomial function p(x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 the the end behavior is given by the it at infinity p(x) x ± Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 5/29

20 The End Behavior of a Polynomial The End Behavior of a Polynomial Limits at Infinity for Polynomials For the n th degree polynomial function p(x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 the the end behavior is given by the it at infinity p(x) x ± which is always infinite and whose sign depends only on the sign of a n and on whether n even or odd. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 5/29

21 Evaluating the Limit at Infinity of a Polynomial Evaluating the Limit at Infinity of a Polynomial The at infinity of a polynomial is ) p(x) = (a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 x ± x ± Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 6/29

22 Evaluating the Limit at Infinity of a Polynomial Evaluating the Limit at Infinity of a Polynomial The at infinity of a polynomial is ) p(x) = (a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 x ± x ± ( = x ± xn a n + a n a 2 x x n 2 + a 1 x n 1 + a ) 0 x n Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 6/29

23 Evaluating the Limit at Infinity of a Polynomial Evaluating the Limit at Infinity of a Polynomial The at infinity of a polynomial is ) p(x) = (a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 x ± x ± ( = x ± xn a n + a n a 2 x x n 2 + a 1 x n 1 + a ) 0 x n Using the the it at infinity of a reciprocal power, gives Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 6/29

24 Evaluating the Limit at Infinity of a Polynomial Evaluating the Limit at Infinity of a Polynomial The at infinity of a polynomial is ) p(x) = (a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 x ± x ± ( = x ± xn a n + a n a 2 x x n 2 + a 1 x n 1 + a ) 0 x n Using the the it at infinity of a reciprocal power, gives a n + a n a 2 x x n 2 + a 1 x n 1 p(x) = x ± x ± xn }{{} 0 }{{} 0 }{{} 0 + a 0 x n }{{} 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 6/29

25 Evaluating the Limit at Infinity of a Polynomial Evaluating the Limit at Infinity of a Polynomial The at infinity of a polynomial is ) p(x) = (a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 x ± x ± ( = x ± xn a n + a n a 2 x x n 2 + a 1 x n 1 + a ) 0 x n Using the the it at infinity of a reciprocal power, gives a n + a n a 2 x x n 2 + a 1 x n 1 p(x) = x ± x ± xn }{{} 0 All terms except the first go to zero. }{{} 0 }{{} 0 + a 0 x n }{{} 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 6/29

26 Evaluating the Limit at Infinity of a Polynomial Evaluating the Limit at Infinity of a Polynomial The at infinity of a polynomial is ) p(x) = (a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 x ± x ± ( = x ± xn a n + a n a 2 x x n 2 + a 1 x n 1 + a ) 0 x n Using the the it at infinity of a reciprocal power, gives a n + a n a 2 x x n 2 + a 1 x n 1 p(x) = x ± x ± xn }{{} 0 All terms except the first go to zero. Thus, p(x) = a n x ± x ± xn }{{} 0 }{{} 0 + a 0 x n }{{} 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 6/29

27 Evaluating the Limit at Infinity of a Polynomial Evaluating the Limit at Infinity of a Polynomial Now, if n > 0, then n even x ± xn = and n odd x ± xn = ± Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 7/29

28 Evaluating the Limit at Infinity of a Polynomial Evaluating the Limit at Infinity of a Polynomial Now, if n > 0, then Thus, n even x ± xn = and n odd x ± xn = ± a nx n + a n 1 x n a 2 x 2 + a 1 x + a 0 = x { + if an > 0 if a n < 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 7/29

29 Evaluating the Limit at Infinity of a Polynomial Evaluating the Limit at Infinity of a Polynomial Now, if n > 0, then Thus, and n even x ± xn = and n odd x ± xn = ± a nx n + a n 1 x n a 2 x 2 + a 1 x + a 0 = x { + if an > 0 if a n < 0 a nx n + a n 1 x n a 2 x 2 + a 1 x + a 0 = x { + if n even & an > 0 or n odd & a n < 0 if n even & a n < 0 or n odd & a n > 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 7/29

30 Example 17 Limit of Rational Function Example 17 Limit of Rational Function Evaluate the it x 4x 3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 8/29

31 Example 17 Limit of Rational Function Solution: Example 17 To make use of the its just discussed, the expression must involve terms with reciprocal powers. Thus divide the numerator and denominator by the highest power of x in the denominator, here x 3. This gives x 4x 3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 = x 4 3 x + 2 x 2 4 x x 4 x x 3 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 9/29

32 Example 17 Limit of Rational Function Solution: Example 17 To make use of the its just discussed, the expression must involve terms with reciprocal powers. Thus divide the numerator and denominator by the highest power of x in the denominator, here x 3. This gives x 4x 3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 = x 4 3 x + 2 x 2 4 x x 4 x x 3 = x }{{} x }{{} x }{{} x }{{} x x 3 }{{} x 3 }{{} 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 9/29

33 Example 17 Limit of Rational Function Solution: Example 17 To make use of the its just discussed, the expression must involve terms with reciprocal powers. Thus divide the numerator and denominator by the highest power of x in the denominator, here x 3. This gives x 4x 3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 = x 4 3 x + 2 x 2 4 x x 4 x x 3 = x = }{{} x }{{} x }{{} x }{{} x x 3 }{{} x 3 }{{} 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 9/29

34 Example 17 Limit of Rational Function Solution: Example 17 continue An identical argument for x shows that x 4x 3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 = 4 5 Hence the graph of the rational function r(x) = 4x3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 has a horizontal asymptote along the Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 10/29

35 Example 17 Limit of Rational Function Solution: Example 17 continue An identical argument for x shows that x 4x 3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 = 4 5 Hence the graph of the rational function r(x) = 4x3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 has a horizontal asymptote along the line y = 4 on both the right and 5 the left. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 10/29

36 Example 17 Limit of Rational Function Solution: Example 17 continue An identical argument for x shows that x 4x 3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 = 4 5 Hence the graph of the rational function r(x) = 4x3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 has a horizontal asymptote along the line y = 4 on both the right and 5 the left. Horizontal Asymptotes of a Rational Function Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 10/29

37 Example 17 Limit of Rational Function Solution: Example 17 continue An identical argument for x shows that x 4x 3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 = 4 5 Hence the graph of the rational function r(x) = 4x3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 has a horizontal asymptote along the line y = 4 on both the right and 5 the left. Horizontal Asymptotes of a Rational Function In general, for a rational function in which the degrees of polynomials in the numerator and denominator are equal, the it at ± equals the ratio of the coefficients of the highest powers, call it R, Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 10/29

38 Example 17 Limit of Rational Function Solution: Example 17 continue An identical argument for x shows that x 4x 3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 = 4 5 Hence the graph of the rational function r(x) = 4x3 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 has a horizontal asymptote along the line y = 4 on both the right and 5 the left. Horizontal Asymptotes of a Rational Function In general, for a rational function in which the degrees of polynomials in the numerator and denominator are equal, the it at ± equals the ratio of the coefficients of the highest powers, call it R, and the function has a horizontal asymptote along a line y = R on both sides. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 10/29

39 Example 18 More Rational Functions Example 18 More Rational Functions Evaluate the its (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 (b) x ± 4x 4 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 11/29

40 Example 18 More Rational Functions Example 18 More Rational Functions Evaluate the its (a) x ± Solution: 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 (b) x ± 4x 4 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 11/29

41 Example 18 More Rational Functions Example 18 More Rational Functions Evaluate the its (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 (b) x ± 4x 4 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 Solution: (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 = x ± 4 x 3 x x 3 4 x x 2 4 x x 4 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 11/29

42 Example 18 More Rational Functions Example 18 More Rational Functions Evaluate the its (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 (b) x ± 4x 4 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 Solution: (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 = x ± 4 x 3 x x 3 4 x x 2 4 x x 4 = 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 11/29

43 Example 18 More Rational Functions Example 18 More Rational Functions Evaluate the its (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 (b) x ± 4x 4 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 Solution: (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 = x ± 4 x 3 x x 3 4 x x 2 4 x x 4 = 0 (b) x ± 4x 4 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 11/29

44 Example 18 More Rational Functions Example 18 More Rational Functions Evaluate the its (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 (b) x ± 4x 4 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 Solution: (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 = x ± 4 x 3 x x 3 4 x x 2 4 x x 4 = 0 (b) x ± 4x 4 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 = x ± 4x 3 x + 2 x 2 4 x x 4 x x 3 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 11/29

45 Example 18 More Rational Functions Example 18 More Rational Functions Evaluate the its (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 (b) x ± 4x 4 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 Solution: (a) x ± 4x 3 3x 2 + 2x 4 5x 4 + 2x 2 4x + 3 = x ± 4 x 3 x x 3 4 x x 2 4 x x 4 = 0 (b) x ± 4x 4 3x 2 + 2x 4 5x 3 + 2x 2 4x + 3 = x ± 4x 3 x + 2 x 2 4 x x 4 x x 3 = ± Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 11/29

46 Limits at Infinity for the Exponential Function The graph of the natural exponential function f(x) = e x looks like this. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 12/29

47 Limits at Infinity for the Exponential Function The graph of the natural exponential function f(x) = e x looks like this. f(x) = e x Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 12/29

48 Limits at Infinity for the Exponential Function The graph of the natural exponential function f(x) = e x looks like this. The graph of the function g(x) = e x is obtained by reflecting across the y-axis. f(x) = e x Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 12/29

49 Limits at Infinity for the Exponential Function The graph of the natural exponential function f(x) = e x looks like this. The graph of the function g(x) = e x is obtained by reflecting across the y-axis. It looks like this. f(x) = e x g(x) = e x Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 12/29

50 Limits at Infinity for the Exponential Function Limits at Infinity for the Exponential Function Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 13/29

51 Limits at Infinity for the Exponential Function Limits at Infinity for the Exponential Function From the graphs just shown x ex = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 13/29

52 Limits at Infinity for the Exponential Function Limits at Infinity for the Exponential Function From the graphs just shown x ex = and x ex = 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 13/29

53 Limits at Infinity for the Exponential Function Limits at Infinity for the Exponential Function From the graphs just shown and x ex = and x ex = 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 13/29

54 Limits at Infinity for the Exponential Function Limits at Infinity for the Exponential Function From the graphs just shown and x ex = and x ex = 0 x e x Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 13/29

55 Limits at Infinity for the Exponential Function Limits at Infinity for the Exponential Function From the graphs just shown and x ex = and x ex = 0 x e x = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 13/29

56 Limits at Infinity for the Exponential Function Limits at Infinity for the Exponential Function From the graphs just shown and x ex = and x ex = 0 x e x = and e x = 0 x Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 13/29

57 Example 19 Limits of Exponential Functions Example 19 Limits of Exponential Functions Evaluate the its (a) x 2e x + 1 e x + 2 (b) 2e x + 1 x e x + 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 14/29

58 Example 19 Limits of Exponential Functions Example 19 Limits of Exponential Functions Evaluate the its (a) x 2e x + 1 e x + 2 Solution: (a) Since x ex = 0 we have (b) 2e x + 1 x e x + 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 14/29

59 Example 19 Limits of Exponential Functions Example 19 Limits of Exponential Functions Evaluate the its (a) x 2e x + 1 e x + 2 Solution: (a) Since x ex = 0 we have x 2e x + 1 e x + 2 (b) 2e x + 1 x e x + 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 14/29

60 Example 19 Limits of Exponential Functions Example 19 Limits of Exponential Functions Evaluate the its (a) x 2e x + 1 e x + 2 Solution: (a) Since x ex = 0 we have (b) 2e x + 1 x e x + 2 x 2e x + 1 e x + 2 = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 14/29

61 Example 19 Limits of Exponential Functions Example 19 Limits of Exponential Functions Evaluate the its (a) x 2e x + 1 e x + 2 Solution: (a) Since x ex = 0 we have (b) 2e x + 1 x e x + 2 x 2e x + 1 e x + 2 = = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 14/29

62 Example 19 Limits of Exponential Functions Example 19 Limits of Exponential Functions Evaluate the its (a) x 2e x + 1 e x + 2 Solution: (a) Since x ex = 0 we have (b) 2e x + 1 x e x + 2 x 2e x + 1 e x + 2 = = (b) Now we use x e x = 0. To do the we first multiply numerator and denominator by e x to give Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 14/29

63 Example 19 Limits of Exponential Functions Example 19 Limits of Exponential Functions Evaluate the its (a) x 2e x + 1 e x + 2 Solution: (a) Since x ex = 0 we have (b) 2e x + 1 x e x + 2 x 2e x + 1 e x + 2 = = (b) Now we use x e x = 0. To do the we first multiply numerator and denominator by e x to give x 2e x + 1 e x + 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 14/29

64 Example 19 Limits of Exponential Functions Example 19 Limits of Exponential Functions Evaluate the its (a) x 2e x + 1 e x + 2 Solution: (a) Since x ex = 0 we have (b) 2e x + 1 x e x + 2 x 2e x + 1 e x + 2 = = (b) Now we use x e x = 0. To do the we first multiply numerator and denominator by e x to give x 2e x + 1 e x + 2 = x ( 2e x + 1 e x + 2 ) ( e x ) e x Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 14/29

65 Example 19 Limits of Exponential Functions Example 19 Limits of Exponential Functions Evaluate the its (a) x 2e x + 1 e x + 2 Solution: (a) Since x ex = 0 we have (b) 2e x + 1 x e x + 2 x 2e x + 1 e x + 2 = = (b) Now we use x e x = 0. To do the we first multiply numerator and denominator by e x to give x 2e x + 1 e x + 2 = x ( 2e x + 1 e x + 2 ) ( e x ) e x = x 2 + e x 1 + 2e x Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 14/29

66 Example 19 Limits of Exponential Functions Example 19 Limits of Exponential Functions Evaluate the its (a) x 2e x + 1 e x + 2 Solution: (a) Since x ex = 0 we have (b) 2e x + 1 x e x + 2 x 2e x + 1 e x + 2 = = (b) Now we use x e x = 0. To do the we first multiply numerator and denominator by e x to give x 2e x + 1 e x + 2 = x ( 2e x + 1 e x + 2 ) ( e x ) e x = x 2 + e x 1 + 2e x = 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 14/29

67 Example 19 Limits of Exponential Functions Solution: Example 19 continued The graph of the function f(x) = 2ex + 1 e x has a horizontal asymptotes + 2 along y = 1 2 on the left and y = 2 on the right. Its graph looks like this. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 15/29

68 Example 19 Limits of Exponential Functions Solution: Example 19 continued The graph of the function f(x) = 2ex + 1 e x has a horizontal asymptotes + 2 along y = 1 2 on the left and y = 2 on the right. Its graph looks like this. 2 f(x) = 2ex + 1 e x Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 15/29

69 Example 19 Limits of Exponential Functions Horizontal Asymptotes for Functions Involving the Exponential Function Horizontal Asymptotes for Exponential Functions Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 16/29

70 Example 19 Limits of Exponential Functions Horizontal Asymptotes for Functions Involving the Exponential Function Horizontal Asymptotes for Exponential Functions For a function involving an exponential function the its at + and can be different. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 16/29

71 Example 19 Limits of Exponential Functions Horizontal Asymptotes for Functions Involving the Exponential Function Horizontal Asymptotes for Exponential Functions For a function involving an exponential function the its at + and can be different. This means that such functions can have a different horizontal asymptote on the right (x ) and on the left (x ). Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 16/29

72 Function Dominance When comparing functions, an important question is: which function grows faster as x gets large? This question can be answered using its at infinity. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 17/29

73 Function Dominance When comparing functions, an important question is: which function grows faster as x gets large? This question can be answered using its at infinity. Function Dominance Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 17/29

74 Function Dominance When comparing functions, an important question is: which function grows faster as x gets large? This question can be answered using its at infinity. Function Dominance For two function f and g, if x f(x) g(x) = 0 we say that g dominates f as x gets large and positive. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 17/29

75 Function Dominance When comparing functions, an important question is: which function grows faster as x gets large? This question can be answered using its at infinity. Function Dominance For two function f and g, if x f(x) g(x) = 0 we say that g dominates f as x gets large and positive. Equivalently, we have g(x) x f(x) = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 17/29

76 Example 20 Exponential and Power Functions Example 20 Exponential and Power Functions Consider the family of functions f(x) = x n e x = x e x for x 0 where the parameter n is a positive integer. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 18/29

77 Example 20 Exponential and Power Functions Example 20 Exponential and Power Functions Consider the family of functions f(x) = x n e x = x e x for x 0 where the parameter n is a positive integer. The graph of the single member of the family with n = 1 looks like this. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 18/29

78 Example 20 Exponential and Power Functions Example 20 Exponential and Power Functions Consider the family of functions f(x) = x n e x = x e x for x 0 where the parameter n is a positive integer. The graph of the two members of the family with n = 1 and n = 2 look like this. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 18/29

79 Example 20 Exponential and Power Functions Example 20 Exponential and Power Functions Consider the family of functions f(x) = x n e x = x e x for x 0 where the parameter n is a positive integer. The graph of the three members of the family with n = 1, n = 2, and n = 3 look like this. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 18/29

80 Example 20 Exponential and Power Functions Example 20 Exponential and Power Functions Consider the family of functions f(x) = x n e x = x e x for x 0 where the parameter n is a positive integer. The graph of the four members of the family with n = 1, n = 2, n = 3, and n = 4 look like this. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 18/29

81 Example 20 Exponential and Power Functions Example 20 Exponential and Power Functions Consider the family of functions f(x) = x n e x = x e x for x 0 where the parameter n is a positive integer. The graph of the four members of the family with n = 1, n = 2, n = 3, and n = 4 look like this. Based on these graphs what can you conclude about the dominance relationship between the exponential function e x and the power function x n? Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 18/29

82 Example 20 Exponential and Power Functions Solution: Example 20 From the graphs it appears that for any integer n 1 we have x x n e x = 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 19/29

83 Example 20 Exponential and Power Functions Solution: Example 20 From the graphs it appears that for any integer n 1 we have x x n e x = 0 since for each of the graphs the x-axis, y = 0, is a horizontal asymptote. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 19/29

84 Example 20 Exponential and Power Functions Solution: Example 20 From the graphs it appears that for any integer n 1 we have x x n e x = 0 since for each of the graphs the x-axis, y = 0, is a horizontal asymptote. Thus, we conclude that the exponential function e x dominates x n as x. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 19/29

85 Example 20 Exponential and Power Functions Solution: Example 20 From the graphs it appears that for any integer n 1 we have x x n e x = 0 since for each of the graphs the x-axis, y = 0, is a horizontal asymptote. Thus, we conclude that the exponential function e x dominates x n as x. In other words, the exponential function grows more rapidly than the power function as x gets large and positive. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 19/29

86 Example 20 Exponential and Power Functions Solution: Example 20 From the graphs it appears that for any integer n 1 we have x x n e x = 0 since for each of the graphs the x-axis, y = 0, is a horizontal asymptote. Thus, we conclude that the exponential function e x dominates x n as x. In other words, the exponential function grows more rapidly than the power function as x gets large and positive. Further, from the graphs it appears that as n gets larger, we must go to larger x values for the dominance to take effect. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 19/29

87 Vertical Asymptotes and Infinite Limits Vertical Asymptotes and Infinite Limits Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 20/29

88 Vertical Asymptotes and Infinite Limits Vertical Asymptotes and Infinite Limits Where Are Vertical Asymptotes? Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 20/29

89 Vertical Asymptotes and Infinite Limits Vertical Asymptotes and Infinite Limits Where Are Vertical Asymptotes? Where the function has an infinite it. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 20/29

90 Vertical Asymptotes and Infinite Limits Vertical Asymptotes and Infinite Limits Where Are Vertical Asymptotes? Where the function has an infinite it. There may be no vertical asymptotes, or many. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 20/29

91 Vertical Asymptotes and Infinite Limits Vertical Asymptotes and Infinite Limits Where Are Vertical Asymptotes? Where the function has an infinite it. There may be no vertical asymptotes, or many. A rational function has a vertical asymptote for any x value where the denominator is zero but the numerator is non-zero. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 20/29

92 Vertical Asymptotes and Infinite Limits Vertical Asymptotes and Infinite Limits Where Are Vertical Asymptotes? Where the function has an infinite it. There may be no vertical asymptotes, or many. A rational function has a vertical asymptote for any x value where the denominator is zero but the numerator is non-zero. Some functions have vertical asymptotes without having an obvious denominator to make equal to zero. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 20/29

93 Vertical Asymptotes and Infinite Limits Vertical Asymptotes and Infinite Limits Where Are Vertical Asymptotes? Where the function has an infinite it. There may be no vertical asymptotes, or many. A rational function has a vertical asymptote for any x value where the denominator is zero but the numerator is non-zero. Some functions have vertical asymptotes without having an obvious denominator to make equal to zero. For example, tan x has vertical asymptotes at any odd multiple of π 2. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 20/29

94 Horizontal Asymptotes and Limits at Infinity Horizontal Asymptotes and Limits at Infinity Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 21/29

95 Horizontal Asymptotes and Limits at Infinity Horizontal Asymptotes and Limits at Infinity What Are the Horizontal Asymptotes? Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 21/29

96 Horizontal Asymptotes and Limits at Infinity Horizontal Asymptotes and Limits at Infinity What Are the Horizontal Asymptotes? To identify any horizontal asymptotes for a function determine the its at infinity. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 21/29

97 Horizontal Asymptotes and Limits at Infinity Horizontal Asymptotes and Limits at Infinity What Are the Horizontal Asymptotes? To identify any horizontal asymptotes for a function determine the its at infinity. There can be at most two different horizontal asymptotes: on the right (x ) and on the left (x ). Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 21/29

98 Horizontal Asymptotes and Limits at Infinity Horizontal Asymptotes and Limits at Infinity What Are the Horizontal Asymptotes? To identify any horizontal asymptotes for a function determine the its at infinity. There can be at most two different horizontal asymptotes: on the right (x ) and on the left (x ). A rational function can have only one horizontal asymptote. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 21/29

99 Horizontal Asymptotes and Limits at Infinity Horizontal Asymptotes and Limits at Infinity What Are the Horizontal Asymptotes? To identify any horizontal asymptotes for a function determine the its at infinity. There can be at most two different horizontal asymptotes: on the right (x ) and on the left (x ). A rational function can have only one horizontal asymptote. If the degree of the denominator and numerator are equal, there is a non-zero horizontal asymptote. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 21/29

100 Horizontal Asymptotes and Limits at Infinity Horizontal Asymptotes and Limits at Infinity What Are the Horizontal Asymptotes? To identify any horizontal asymptotes for a function determine the its at infinity. There can be at most two different horizontal asymptotes: on the right (x ) and on the left (x ). A rational function can have only one horizontal asymptote. If the degree of the denominator and numerator are equal, there is a non-zero horizontal asymptote. If the degree of the denominator is greater than the degree of the numerator, there is a horizontal asymptote at y = 0. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 21/29

101 Horizontal Asymptotes and Limits at Infinity Horizontal Asymptotes and Limits at Infinity What Are the Horizontal Asymptotes? To identify any horizontal asymptotes for a function determine the its at infinity. There can be at most two different horizontal asymptotes: on the right (x ) and on the left (x ). A rational function can have only one horizontal asymptote. If the degree of the denominator and numerator are equal, there is a non-zero horizontal asymptote. If the degree of the denominator is greater than the degree of the numerator, there is a horizontal asymptote at y = 0. If the degree of the denominator is less than the degree of the numerator, there is no horizontal asymptote. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 21/29

102 Horizontal Asymptotes and Limits at Infinity Horizontal Asymptotes and Limits at Infinity What Are the Horizontal Asymptotes? To identify any horizontal asymptotes for a function determine the its at infinity. There can be at most two different horizontal asymptotes: on the right (x ) and on the left (x ). A rational function can have only one horizontal asymptote. If the degree of the denominator and numerator are equal, there is a non-zero horizontal asymptote. If the degree of the denominator is greater than the degree of the numerator, there is a horizontal asymptote at y = 0. If the degree of the denominator is less than the degree of the numerator, there is no horizontal asymptote. Functions involving exponential functions may have different horizontal asymptotes on the left and the right. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 21/29

103 Example 21 Investigating Asymptotes Example 21 Investigating Asymptotes For each function below, determine any vertical and horizontal asymptotes and sketch the graph of the function. For the vertical asymptote(s) determine the behavior of the function on either side of the asymptote by calculating the appropriate one-sided infinite its. (a) f(x) = 1 x + 1 (b) g(x) = x x (c) h(x) = (x + 1) 2 x (d) F(x) = (x + 1) 2 x 2 (e) G(x) = (x + 1) 2 x 3 (f) H(x) = (x + 1) 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 22/29

104 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

105 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where x + 1 = 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

106 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where x + 1 = 0 x = 1 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

107 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where x + 1 = 0 x = 1 The behavior on either side of the vertical asymptote at x = 1 is given by 1 x 1 x + 1 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

108 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where x + 1 = 0 x = 1 The behavior on either side of the vertical asymptote at x = 1 is given by 1 x 1 x + 1 = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

109 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where x + 1 = 0 x = 1 The behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 x + 1 x 1 + x + 1 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

110 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where x + 1 = 0 x = 1 The behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 x + 1 x 1 + x + 1 = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

111 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where x + 1 = 0 x = 1 The behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 x + 1 x 1 + x + 1 = Further, x ± 1 x + 1 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

112 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where x + 1 = 0 x = 1 The behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 x + 1 x 1 + x + 1 = Further, x ± 1 x + 1 = 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

113 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where x + 1 = 0 x = 1 The behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 x + 1 x 1 + x + 1 = Further, x ± 1 x + 1 = 0 So that, the x-axis, y = 0, is the horizontal asymptote for the graph of the function. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

114 Example 21 Investigating Asymptotes Solution: Example 21(a) For f(x) = 1 x + 1 there is a vertical asymptote where x + 1 = 0 x = 1 x = 1 The behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 x + 1 x 1 + x + 1 = Further, x ± 1 x + 1 = 0 So that, the x-axis, y = 0, is the horizontal asymptote for the graph of the function. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 23/29

115 Example 21 Investigating Asymptotes Solution: Example 21(b) For g(x) = x x + 1 again there is a vertical asymptote at Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 24/29

116 Example 21 Investigating Asymptotes Solution: Example 21(b) For g(x) = x x + 1 again there is a vertical asymptote at x = 1. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 24/29

117 Example 21 Investigating Asymptotes Solution: Example 21(b) For g(x) = x x + 1 again there is a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x x + 1 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 24/29

118 Example 21 Investigating Asymptotes Solution: Example 21(b) For g(x) = x x + 1 again there is a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x x + 1 = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 24/29

119 Example 21 Investigating Asymptotes Solution: Example 21(b) For g(x) = x x + 1 again there is a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x = and x + 1 x 1 + x x + 1 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 24/29

120 Example 21 Investigating Asymptotes Solution: Example 21(b) For g(x) = x x + 1 again there is a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x = and x + 1 x 1 + x x + 1 = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 24/29

121 Example 21 Investigating Asymptotes Solution: Example 21(b) For g(x) = x x + 1 again there is a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x = and x + 1 x 1 + x x + 1 = Further, x ± x x + 1 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 24/29

122 Example 21 Investigating Asymptotes Solution: Example 21(b) For g(x) = x x + 1 again there is a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x = and x + 1 x 1 + x x + 1 = Further, x ± x x + 1 = 1 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 24/29

123 Example 21 Investigating Asymptotes Solution: Example 21(b) For g(x) = x x + 1 again there is a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x = and x + 1 x 1 + x x + 1 = Further, x ± x x + 1 = 1 So that y = 1 is the horizontal asymptote for the graph of the function. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 24/29

124 Example 21 Investigating Asymptotes Solution: Example 21(b) For g(x) = x x + 1 again there is a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x = and x + 1 x 1 + x x + 1 = x = 1 y = 1 Further, x ± x x + 1 = 1 So that y = 1 is the horizontal asymptote for the graph of the function. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 24/29

125 Example 21 Investigating Asymptotes Solution: Example 21(c) For 1 h(x) = (x + 1) 2 there is still a vertical asymptote at Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 25/29

126 Example 21 Investigating Asymptotes Solution: Example 21(c) For 1 h(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 25/29

127 Example 21 Investigating Asymptotes Solution: Example 21(c) For 1 h(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by 1 x 1 (x + 1) 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 25/29

128 Example 21 Investigating Asymptotes Solution: Example 21(c) For 1 h(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by 1 x 1 (x + 1) 2 = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 25/29

129 Example 21 Investigating Asymptotes Solution: Example 21(c) For 1 h(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 2 (x + 1) x 1 + (x + 1) 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 25/29

130 Example 21 Investigating Asymptotes Solution: Example 21(c) For 1 h(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 2 (x + 1) x 1 + (x + 1) 2 = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 25/29

131 Example 21 Investigating Asymptotes Solution: Example 21(c) For 1 h(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 2 (x + 1) x 1 + (x + 1) 2 = Further, x ± 1 (x + 1) 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 25/29

132 Example 21 Investigating Asymptotes Solution: Example 21(c) For 1 h(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 2 (x + 1) x 1 + (x + 1) 2 = Further, x ± 1 (x + 1) 2 = 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 25/29

133 Example 21 Investigating Asymptotes Solution: Example 21(c) For 1 h(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by 1 1 = and x 1 2 (x + 1) x 1 + (x + 1) 2 = Further, x ± 1 (x + 1) 2 = 0 So that y = 0 is the horizontal asymptote for the graph of the function. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 25/29

134 Example 21 Investigating Asymptotes Solution: Example 21(c) For 1 h(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x = = and x 1 2 (x + 1) x 1 + (x + 1) 2 = Further, x ± 1 (x + 1) 2 = 0 So that y = 0 is the horizontal asymptote for the graph of the function. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 25/29

135 Example 21 Investigating Asymptotes Solution: Example 21(d) For x F(x) = (x + 1) 2 there is still a vertical asymptote at Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 26/29

136 Example 21 Investigating Asymptotes Solution: Example 21(d) For x F(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 26/29

137 Example 21 Investigating Asymptotes Solution: Example 21(d) For x F(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x (x + 1) 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 26/29

138 Example 21 Investigating Asymptotes Solution: Example 21(d) For x F(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x (x + 1) 2 = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 26/29

139 Example 21 Investigating Asymptotes Solution: Example 21(d) For x F(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x (x + 1) 2 = Further, x ± x (x + 1) 2 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 26/29

140 Example 21 Investigating Asymptotes Solution: Example 21(d) For x F(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x (x + 1) 2 = Further, x ± x (x + 1) 2 = 0 Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 26/29

141 Example 21 Investigating Asymptotes Solution: Example 21(d) For x F(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x (x + 1) 2 = Further, x ± x (x + 1) 2 = 0 So that y = 0 is the horizontal asymptote for the graph of the function. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 26/29

142 Example 21 Investigating Asymptotes Solution: Example 21(d) For x F(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x (x + 1) 2 = x = 1 Further, x ± x (x + 1) 2 = 0 So that y = 0 is the horizontal asymptote for the graph of the function. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 26/29

143 Example 21 Investigating Asymptotes Solution: Example 21(e) For x G(x) = 2 (x + 1) 2 there is still a vertical asymptote at Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 27/29

144 Example 21 Investigating Asymptotes Solution: Example 21(e) For x G(x) = 2 (x + 1) 2 there is still a vertical asymptote at x = 1. Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 27/29

145 Example 21 Investigating Asymptotes Solution: Example 21(e) For x 2 G(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x 2 (x + 1) 2 = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 27/29

146 Example 21 Investigating Asymptotes Solution: Example 21(e) For x 2 G(x) = (x + 1) 2 there is still a vertical asymptote at x = 1. Now the behavior on either side of the vertical asymptote at x = 1 is given by x 1 x 2 (x + 1) 2 = Clint Lee Math 112 Lecture 5: Limits at Infinity and Asymptotes 27/29

### Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.

Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote

### Asymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if

Section 2.1: Vertical and Horizontal Asymptotes Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if, lim x a f(x) =, lim x a x a x a f(x) =, or. + + Definition.

### MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity

MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x

### WARM UP EXERCSE. 2-1 Polynomials and Rational Functions

WARM UP EXERCSE Roots, zeros, and x-intercepts. x 2! 25 x 2 + 25 x 3! 25x polynomial, f (a) = 0! (x - a)g(x) 1 2-1 Polynomials and Rational Functions Students will learn about: Polynomial functions Behavior

### Review for Calculus Rational Functions, Logarithms & Exponentials

Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

### 55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

### Solutions to Self-Test for Chapter 4 c4sts - p1

Solutions to Self-Test for Chapter 4 c4sts - p1 1. Graph a polynomial function. Label all intercepts and describe the end behavior. a. P(x) = x 4 2x 3 15x 2. (1) Domain = R, of course (since this is a

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### Section 3.7 Rational Functions

Section 3.7 Rational Functions A rational function is a function of the form where P and Q are polynomials. r(x) = P(x) Q(x) Rational Functions and Asymptotes The domain of a rational function consists

### MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial

### Situation: Dividing Linear Expressions

Situation: Dividing Linear Expressions Date last revised: June 4, 203 Michael Ferra, Nicolina Scarpelli, Mary Ellen Graves, and Sydney Roberts Prompt: An Algebra II class has been examining the product

### Graphing Rational Functions

Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could

### Rational functions are defined for all values of x except those for which the denominator hx ( ) is equal to zero. 1 Function 5 Function

Section 4.6 Rational Functions and Their Graphs Definition Rational Function A rational function is a function of the form that h 0. f g h where g and h are polynomial functions such Objective : Finding

PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

### 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

### Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

### Pre Cal 2 1 Lesson with notes 1st.notebook. January 22, Operations with Complex Numbers

0 2 Operations with Complex Numbers Objectives: To perform operations with pure imaginary numbers and complex numbers To use complex conjugates to write quotients of complex numbers in standard form Complex

### Rational Functions, Limits, and Asymptotic Behavior

Unit 2 Rational Functions, Limits, and Asymptotic Behavior Introduction An intuitive approach to the concept of a limit is often considered appropriate for students at the precalculus level. In this unit,

### LIMITS AND CONTINUITY

LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

### Name: where Nx ( ) and Dx ( ) are the numerator and

Oblique and Non-linear Asymptote Activity Name: Prior Learning Reminder: Rational Functions In the past we discussed vertical and horizontal asymptotes of the graph of a rational function of the form m

### Math 21A Brian Osserman Practice Exam 1 Solutions

Math 2A Brian Osserman Practice Exam Solutions These solutions are intended to indicate roughly how much you would be expected to write. Comments in [square brackets] are additional and would not be required.

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Rational Functions 5.2 & 5.3

Math Precalculus Algebra Name Date Rational Function Rational Functions 5. & 5.3 g( ) A function is a rational function if f ( ), where g( ) and h( ) are polynomials. h( ) Vertical asymptotes occur at

### Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply?

Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply? Commentary In our foci, we are assuming that we have a

### Horizontal and Vertical Asymptotes of Graphs of Rational Functions

PRECALCULUS AND ADVANCED OPICS Horizontal and Vertical Asymptotes of Graphs of Rational Functions Student Outcomes Students identify vertical and horizontal asymptotes of rational functions. Lesson Notes

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### APPLICATIONS OF DIFFERENTIATION

4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

### Polynomial and Rational Functions

Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### 8 Polynomials Worksheet

8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph

### Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships

Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships These materials are for nonprofit educational purposes

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Math Rational Functions

Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:

Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goes

### Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions.

Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions. Basic Functions In several sections you will be applying shifts

### Polynomial & Rational Functions

4 Polynomial & Rational Functions 45 Rational Functions A function f is a rational function if there exist polynomial functions p and q, with q not the zero function, such that p(x) q(x) for all x for

### In this lesson you will learn to find zeros of polynomial functions that are not factorable.

2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Calculus Card Matching

Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### Chapter 8. Investigating Rational Equations

Chapter 8 Investigating Rational Equations Suppose that you started a business selling picnic tables and wanted to determine the cost of each table you made. Part of your cost is for materials and is proportional

### Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

### Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Chapter 11 - Curve Sketching. Lecture 17. MATH10070 - Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson.

Lecture 17 MATH10070 - Introduction to Calculus maths.ucd.ie/modules/math10070 Kevin Hutchinson 28th October 2010 Z Chain Rule (I): If y = f (u) and u = g(x) dy dx = dy du du dx Z Chain rule (II): d dx

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### 3.4 Limits at Infinity - Asymptotes

3.4 Limits at Infinity - Asymptotes Definition 3.3. If f is a function defined on some interval (a, ), then f(x) = L means that values of f(x) are very close to L (keep getting closer to L) as x. The line

### List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated

MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible

### Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

### Math 234 February 28. I.Find all vertical and horizontal asymptotes of the graph of the given function.

Math 234 February 28 I.Find all vertical and horizontal asymptotes of the graph of the given function.. f(x) = /(x 3) x 3 = 0 when x = 3 Vertical Asymptotes: x = 3 H.A.: /(x 3) = 0 /(x 3) = 0 Horizontal

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is one-to-one, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

### 1 Shapes of Cubic Functions

MA 1165 - Lecture 05 1 1/26/09 1 Shapes of Cubic Functions A cubic function (a.k.a. a third-degree polynomial function) is one that can be written in the form f(x) = ax 3 + bx 2 + cx + d. (1) Quadratic

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1

Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)

### 106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM 5.1.1 Fermat s Theorem f is differentiable at a, then f (a) = 0.

5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function

### Main page. Given f ( x, y) = c we differentiate with respect to x so that

Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching - asymptotes Curve sketching the

### Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions

Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions A) Vertical Asymptotes A rational function, in lowest terms, will have vertical asymptotes at the real zeros of the denominator

### GRAPH OF A RATIONAL FUNCTION

GRAPH OF A RATIONAL FUNCTION Find vertical asmptotes and draw them. Look for common factors first. Vertical asmptotes occur where the denominator becomes zero as long as there are no common factors. Find

### Principles of Math 12 - Transformations Practice Exam 1

Principles of Math 2 - Transformations Practice Exam www.math2.com Transformations Practice Exam Use this sheet to record your answers. NR. 2. 3. NR 2. 4. 5. 6. 7. 8. 9. 0.. 2. NR 3. 3. 4. 5. 6. 7. NR

### M 1310 4.1 Polynomial Functions 1

M 1310 4.1 Polynomial Functions 1 Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a, a,..., a, a, a n n1 2 1 0, be real numbers, with a

### Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

### Practice Problems for Exam 1 Math 140A, Summer 2014, July 2

Practice Problems for Exam 1 Math 140A, Summer 2014, July 2 Name: INSTRUCTIONS: These problems are for PRACTICE. For the practice exam, you may use your book, consult your classmates, and use any other

### Algebra II Unit Number 4

Title Polynomial Functions, Expressions, and Equations Big Ideas/Enduring Understandings Applying the processes of solving equations and simplifying expressions to problems with variables of varying degrees.

### Exploring Asymptotes. Student Worksheet. Name. Class. Rational Functions

Exploring Asymptotes Name Student Worksheet Class Rational Functions 1. On page 1.3 of the CollegeAlg_Asymptotes.tns fi le is a graph of the function 1 f(x) x 2 9. a. Is the function defi ned over all

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### 5.3 Improper Integrals Involving Rational and Exponential Functions

Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a

### CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

### MTH 3005 - Calculus I Week 8: Limits at Infinity and Curve Sketching

MTH 35 - Calculus I Week 8: Limits at Infinity and Curve Sketching Adam Gilbert Northeastern University January 2, 24 Objectives. After reviewing these notes the successful student will be prepared to

### Discrete Mathematics: Homework 7 solution. Due: 2011.6.03

EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

### f(x) = g(x), if x A h(x), if x B.

1. Piecewise Functions By Bryan Carrillo, University of California, Riverside We can create more complicated functions by considering Piece-wise functions. Definition: Piecewise-function. A piecewise-function

### HFCC Math Lab Arithmetic - 4. Addition, Subtraction, Multiplication and Division of Mixed Numbers

HFCC Math Lab Arithmetic - Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers.

### Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

### Limits and Continuity

Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### i is a root of the quadratic equation.

13 14 SEMESTER EXAMS 1. This question assesses the student s understanding of a quadratic function written in vertex form. y a x h k where the vertex has the coordinates V h, k a) The leading coefficient

### Some Lecture Notes and In-Class Examples for Pre-Calculus:

Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax

### Procedure for Graphing Polynomial Functions

Procedure for Graphing Polynomial Functions P(x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine

### ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals

ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

### Introduction to Calculus

Introduction to Calculus Contents 1 Introduction to Calculus 3 11 Introduction 3 111 Origin of Calculus 3 112 The Two Branches of Calculus 4 12 Secant and Tangent Lines 5 13 Limits 10 14 The Derivative

### Section 3.2 Polynomial Functions and Their Graphs

Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P(x) = 3, Q(x) = 4x 7, R(x) = x 2 +x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 +2x+4 (b)

### Unit 3 - Lesson 3. MM3A2 - Logarithmic Functions and Inverses of exponential functions

Math Instructional Framework Time Frame Unit Name Learning Task/Topics/ Themes Standards and Elements Lesson Essential Questions Activator Unit 3 - Lesson 3 MM3A2 - Logarithmic Functions and Inverses of

### Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

### Larson, R. and Boswell, L. (2016). Big Ideas Math, Algebra 2. Erie, PA: Big Ideas Learning, LLC. ISBN

ALG B Algebra II, Second Semester #PR-0, BK-04 (v.4.0) To the Student: After your registration is complete and your proctor has been approved, you may take the Credit by Examination for ALG B. WHAT TO