U2+Quiz+Review+1.7+-+1.11+KEY.pdf
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Review A: (Topics 1.7 – 1.11) Rational Functions Name: Solutions Directions: For each of the following rational functions, write limit statements to describe the left and right end behaviors. 1. 2....
Review A: (Topics 1.7 – 1.11) Rational Functions Name: Solutions Directions: For each of the following rational functions, write limit statements to describe the left and right end behaviors. 1. 2. 𝐆𝐫𝐚𝐩𝐡 𝐨𝐟 𝒇(𝒙) 𝐆𝐫𝐚𝐩𝐡 𝐨𝐟 𝒈(𝒙) Left: lim 𝑓(𝑥) = 1 Left: lim 𝑔(𝑥) = 2 !→#$ !→#$ Right: lim 𝑓(𝑥) = 1 Right: lim 𝑔(𝑥) = 2 !→$ !→$ 2𝑥 % − 2𝑥 + 1 2𝑥 & 2 2𝑥(𝑥 − 3) (2𝑥)(𝑥) 2𝑥 % 2 3. ℎ(𝑥) = 𝑦→ = 4. 𝑘(𝑥) = 𝑘(𝑥) → = & = 3𝑥 % + 5𝑥 + 7 3𝑥 & 3 (𝑥 + 2)% (𝑥 − 1) ( 𝑥 )% ( 𝑥 ) 𝑥 𝑥 2𝑥 & 2 Left: lim = 2 !→#$ 3𝑥 & 3 Left: lim =0 !→#$ 𝑥 2𝑥 & 2 2 Right: lim = Right: lim =0 !→$ 3𝑥 & 3 !→$ 𝑥 −2𝑥 ' + 3𝑥 % + 𝑥 − 1 −2𝑥 ' 2 % 3(𝑥 − 1)% (𝑥 + 5) 5. 𝑟(𝑥) = 𝑟(𝑥) → = − 𝑥 6. 𝑚(𝑥) = 5𝑥 % + 2𝑥 + 3 5𝑥 % 5 (2𝑥 + 3)% 3(𝑥)% (𝑥) 3𝑥 & 3 𝑚(𝑥) → = %= 𝑥 2 (2𝑥 )% 4𝑥 4 Left: lim − 𝑥 % = −∞ !→#$ 5 3 Left: lim 𝑥 = −∞ !→#$ 4 2 Right: lim − 𝑥 % = −∞ !→$ 5 3 Right: lim 𝑥=∞ !→$ 4 𝐑𝐞𝐯𝐢𝐞𝐰 𝐀: Topics 1.7 − 1.11 Rational Functions Created by Bryan Passwater Solutions by Ted Gott [email protected] Directions: Write a limit statement describing the output values for the following graphs and verbal descriptions of the input values. 7. The input values decrease without bound 8. The input values increase without bound 𝐆𝐫𝐚𝐩𝐡 𝐨𝐟 𝒇(𝒙) 𝐆𝐫𝐚𝐩𝐡 𝐨𝐟 𝒈(𝒙) 7. Limit Statement: lim 𝑓(𝑥) = −∞ 8. Limit Statement: lim 𝑔(𝑥) = 1 !→#$ !→$ Directions: The graphs of the functions ℎ and 𝑘 are given below. Use the graphs to find the following limits. 𝐆𝐫𝐚𝐩𝐡 𝐨𝐟 𝒉(𝒙) 𝐆𝐫𝐚𝐩𝐡 𝐨𝐟 𝒌(𝒙) 9. lim! ℎ(𝑥) = −∞ 10. lim" ℎ(𝑥) = ∞ 13. lim! 𝑘(𝑥) = −2 14. lim" 𝑘(𝑥) = ∞ !→% !→% !→( !→& 11. lim! ℎ(𝑥) = 3 12. lim" ℎ(𝑥) = 3 15. lim 𝑘(𝑥) = ∞ 16. lim 𝑘(𝑥) = −∞ !→& !→& !→#$ !→$ 𝐑𝐞𝐯𝐢𝐞𝐰 𝐀: Topics 1.7 − 1.11 Rational Functions Created by Bryan Passwater Solutions by Ted Gott [email protected] Directions: For each of the following, write the left and right limit statements for 𝑓(𝑥) as 𝑥 approaches 1. (𝑥 − 1)(𝑥 + 5) (𝑥 − 2)(𝑥 − 4) −2(𝑥 + 3)(𝑥 + 1) 17. 𝑓(𝑥) = 18. 𝑓(𝑥) = 19. 𝑓(𝑥) = (𝑥 − 1)(𝑥 + 2) (𝑥 − 1)(𝑥 + 2) (𝑥 − 1)% 1+5 6 (1 − 2)(1 − 4) −2(1 + 3)(1 + 1) Left: lim! 𝑓(𝑥) = = =2 Left: lim! 𝑓(𝑥) = # Left: lim! 𝑓(𝑥) = !→) 1+2 3 !→) (1 − 1)(1 + 2) !→) (1# − 1)% (−1)(−3) 3 −2(4)(2) −16 1+5 6 → → # → −∞ → → → −∞ Right: lim" 𝑓(𝑥) = = =2 (0 )(3) # 0 (0# )% 0 !→) 1+2 3 (1 − 2)(1 − 4) −2(4)(2) Right: lim" 𝑓(𝑥) = Right: lim" 𝑓(𝑥) = !→) (1* − 1)(1 + 2) !→) (0* )% (−1)(−3) 3 −16 → → *→∞ → * → −∞ (0 )(3) * 0 0 Directions: For each of the following rational functions, determine and label any values of x where the graph has a hole or vertical asymptote. (𝑥 + 3)(𝑥 − 2) (𝑥 + 7)(𝑥 + 2)& 𝑥& − 𝑥% 20. 𝑦 = 21. 𝑘(𝑥) = 22. 𝑟(𝑥) = (𝑥 + 3)% (𝑥 − 2) (𝑥 + 1)(𝑥 + 2)% 𝑥 % + 2𝑥 + 1 1 1 (𝑥 + 7)(𝑥 + 2) 𝑥 % (𝑥 − 1) lim 𝑦 = lim = lim 𝑘(𝑥) = lim = !→% !→% 𝑥 + 3 5 !→#% !→#% 𝑥+1 (𝑥 + 1)% hole at 𝑥 = 2 (5)(0) No hole because no common factors = =0 1 1 −1 in numerator and denominator. lim 𝑦 = lim → ∓ → ∓∞ !→#& !→#& 𝑥 + 3 0 hole at 𝑥 = −2 𝑥 % (𝑥 − 1) ⇒ vertical asymptote at 𝑥 = −3 (𝑥 + 7)(𝑥 + 2) lim 𝑟(𝑥) = lim !→#) !→#) (𝑥 + 1)% lim 𝑘(𝑥) = lim !→#) !→#) 𝑥+1 (1)(−2) (6)(1) → → −∞ → → ∓∞ 0 0∓ ⇒ vertical asymptote at 𝑥 = −1 ⇒ vertical asymptote at 𝑥 = −1 Directions: Solve the following inequalities. Write your answers using interal notation. 𝑥−3 (𝑥 − 1)% (𝑥 + 2) 23. ≤0 (−2,3] 24. >0 (−∞, −2) ∪ (−1,1) ∪ (1, ∞) 𝑥+2 (𝑥 + 1) (#) (#) (*) (*)(#) (*)(*) (*)(*) (*)(*) (#) (*) (*) (#) (*) (*) * # * (#) * # * * #% & #% #) ) 𝑥 % − 𝑥 − 12 (𝑥 − 4)(𝑥 + 3) −2𝑥(𝑥 − 3)% 25. ≥0⇒ ≥0 26. #%. & ' #&. ' 𝐑𝐞𝐯𝐢𝐞𝐰 𝐀: Topics 1.7 − 1.11 Rational Functions Created by Bryan Passwater Solutions by Ted Gott [email protected]
