Worksheet: Comparing Rate of Growth of Functions Q1: For the functions π(π₯) = 2π₯ 2 and π(π₯) = 4π2π₯ , evaluate lim ( π₯ββ using lβHôpitalβs rule. A The limit does not exist. B 0 C β D 1 E 1 4 Q2: Evaluate lim ( π₯ββ 1 A β B 0 C 2 D ββ E 3 π₯2 ) using lβHôpitalβs rule. ππ₯ π(π₯) ) π(π₯) From the answer to the limit, what can you say about the growth rate of π₯ 2 compared to ππ₯ as π₯ β β? A The growth rate of π₯2 is greater than that of ππ₯ as π₯ β β. B The growth rate of π₯2 is smaller than that of ππ₯ as π₯ β β. C The growth rate of π₯2 is equal to that of ππ₯ as π₯ β β. D The growth rate of π₯ 2 becomes β, while the growth rate of ππ₯ becomes 0 as π₯ β β. E The growth rate of π₯2 becomes 0, while the growth rate of ππ₯ becomes β as π₯ β β. π(π₯) ) = 0, what do you notice about the growth rate of π(π₯) π₯ββ π(π₯) compared to π(π₯) as π₯ β β? Q3: If lim ( A The growth rate of π(π₯) is equal to that of π(π₯) as π₯ β β. B The growth rate of π(π₯) becomes 0, while the growth rate of π(π₯) becomes β as π₯ β β. 2 C The growth rate of π(π₯) is greater than that of π(π₯) as π₯ β β. D The growth rate of π(π₯) becomes β, while the growth rate of π(π₯) becomes 0 as π₯ β β. E The growth rate of π(π₯) is smaller than that of π(π₯) as π₯ β β. π(π₯) ) = β, what do you notice about the growth rate of π(π₯) π₯ββ π(π₯) compared to π(π₯) as π₯ β β? Q4: If lim ( A The growth rate of π(π₯) becomes β, while the growth rate of π(π₯) becomes 0 as π₯ β β. B The growth rate of π(π₯) is equal to that of π(π₯) as π₯ β β. C The growth rate of π(π₯) is smaller than that of π(π₯) as π₯ β β. D The growth rate of π(π₯) is greater than that of π(π₯) as π₯ β β. E The growth rate of π(π₯) becomes 0, while the growth rate of π(π₯) becomes β as π₯ β β. Q5: Compare the growth rate of the two functions π(π₯) = (π₯ + 4)2 and π(π₯) = ln π₯ using limits as π₯ β β. A The growth rate of π(π₯) is equal to the growth rate of π(π₯). B The growth rate of π(π₯) becomes β, while the growth rate of π(π₯) becomes 0 as π₯ β β. C The growth rate of π(π₯) is greater than the growth rate of π(π₯). D The growth rate of π(π₯) becomes 0, while the growth rate of π(π₯) becomes β as π₯ β β. E 3 The growth rate of π(π₯) is smaller than the growth rate of π(π₯). Q6: For the functions π(π₯) = ln (2π₯) and π(π₯) = 4π₯ 3 , evaluate lim ( π₯ββ π(π₯) ) π(π₯) using lβHôpitalβs rule. A β B 1 12 C 1 6 D 0 E The limit does not exist. Q7: Compare the growth rate of the two functions π(π₯) = π₯ 10 and π(π₯) = 10π₯ using limits as π₯ β β. A The growth rate of π(π₯) becomes 0, while the growth rate of π(π₯) becomes β as π₯ β β. B The growth rate of π(π₯) is greater than the growth rate of π(π₯). C The growth rate of π(π₯) is greater than the growth rate of π(π₯). D The growth rate of π(π₯) is equal to the growth rate of π(π₯). E The growth rate of π(π₯) becomes 0, while the growth rate of π(π₯) becomes β as π₯ β β. 4 Q8: Compare the growth rate of the two functions π(π₯) = ππ₯ and π(π₯) = ln π₯ using limits as π₯ β β. A The growth rate of ln π₯ becomes 0, while the growth rate of ππ₯ becomes β as π₯ β β. B The growth rate of ln π₯ is greater than the growth rate of ππ₯ . C The growth rate of ππ₯ is greater than the growth rate of ln π₯. D The growth rate of ππ₯ becomes 0, while the growth rate of ln π₯ becomes β as π₯ β β. E The growth rate of ππ₯ is equal to the growth rate of ln π₯. Q9: Compare the growth rate of the two functions π(π₯) = π₯ 2 and π(π₯) = using limits as π₯ β β. A The growth rate of π(π₯) becomes 0, while the growth rate of π(π₯) becomes β as π₯ β β. B The growth rate of π(π₯) is equal to the growth rate of π(π₯). C The growth rate of π(π₯) is smaller than the growth rate of π(π₯). D The growth rate of π(π₯) is greater than the growth rate of π(π₯). E The growth rate of π(π₯) becomes β, while the growth rate of π(π₯) becomes 0 as π₯ β β. Q10: Consider the functions π(π₯) = π₯ 50 and π(π₯) = 2π . 5 1 π₯ Evaluate lim ( π₯ββ A The limit does not exist. B 1 C 1 2 D 0 E β Evaluate lim ( π₯ββ 6 π(π₯) ) using lβHôpitalβs rule. π(π₯) π(π₯) ) using lβHôpitalβs rule. π(π₯) A The limit does not exist. B 1 C 2 D β E 0 What do you get from the two results about the growth rates of π₯ 50 and 2π₯ as π₯ β β? 7 A The growth rate of π₯ 50 becomes 0, while the growth rate of 2π₯ becomes β as π₯ β β. B The growth rate is the same for both as π₯ β β. C The growth rate of 2π₯ becomes 0, while the growth rate of π₯ 50 becomes β as π₯ β β. D The growth rate of 2π₯ is greater than that of π₯ 50 as π₯ β β. E The growth rate of π₯50 is greater than that of 2π₯ as π₯ β β.