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

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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.

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