Worksheet: Comparing Rate of Growth of Functions

In this worksheet, we will practice using limits to compare the relative magnitudes of functions and their rates of change.

Q1:

For the functions 𝑓(π‘₯)=2π‘₯ and 𝑔(π‘₯)=4π‘’οŠ¨ο—, evaluate limο—β†’βˆžο€½π‘“(π‘₯)𝑔(π‘₯) using l’HΓ΄pital’s rule.

Q2:

Evaluate limο—β†’βˆžοŠ¨ο—ο€Ύπ‘₯π‘’οŠ using l’HΓ΄pital’s rule.

From the answer to the limit, what can you say about the growth rate of π‘₯ compared to 𝑒 as π‘₯β†’βˆž?

  • AThe growth rate of π‘₯ is smaller than that of 𝑒 as π‘₯β†’βˆž.
  • BThe growth rate of π‘₯ becomes ∞, while the growth rate of 𝑒 becomes 0 as π‘₯β†’βˆž.
  • CThe growth rate of π‘₯ is equal to that of 𝑒 as π‘₯β†’βˆž.
  • DThe growth rate of π‘₯ is greater than that of 𝑒 as π‘₯β†’βˆž.
  • EThe growth rate of π‘₯ becomes 0, while the growth rate of 𝑒 becomes ∞ as π‘₯β†’βˆž.

Q3:

If limο—β†’βˆžο€½π‘“(π‘₯)𝑔(π‘₯)=0, what do you notice about the growth rate of 𝑓(π‘₯) compared to 𝑔(π‘₯) as π‘₯β†’βˆž?

  • AThe growth rate of 𝑓(π‘₯) is greater than that of 𝑔(π‘₯) as π‘₯β†’βˆž.
  • BThe growth rate of 𝑓(π‘₯) is smaller than that of 𝑔(π‘₯) as π‘₯β†’βˆž.
  • CThe growth rate of 𝑓(π‘₯) becomes 0, while the growth rate of 𝑔(π‘₯) becomes ∞ as π‘₯β†’βˆž.
  • DThe growth rate of 𝑓(π‘₯) becomes ∞, while the growth rate of 𝑔(π‘₯) becomes 0 as π‘₯β†’βˆž.
  • EThe growth rate of 𝑓(π‘₯) is equal to that of 𝑔(π‘₯) as π‘₯β†’βˆž.

Q4:

If limο—β†’βˆžο€½π‘“(π‘₯)𝑔(π‘₯)=∞, what do you notice about the growth rate of 𝑓(π‘₯) compared to 𝑔(π‘₯) as π‘₯β†’βˆž?

  • AThe growth rate of 𝑓(π‘₯) becomes 0, while the growth rate of 𝑔(π‘₯) becomes ∞ as π‘₯β†’βˆž.
  • BThe growth rate of 𝑓(π‘₯) is greater than that of 𝑔(π‘₯) as π‘₯β†’βˆž.
  • CThe growth rate of 𝑓(π‘₯) is smaller than that of 𝑔(π‘₯) as π‘₯β†’βˆž.
  • DThe growth rate of 𝑓(π‘₯) is equal to that of 𝑔(π‘₯) as π‘₯β†’βˆž.
  • EThe growth rate of 𝑓(π‘₯) becomes ∞, while the growth rate of 𝑔(π‘₯) becomes 0 as π‘₯β†’βˆž.

Q5:

Compare the growth rate of the two functions 𝑓(π‘₯)=(π‘₯+4) and 𝑔(π‘₯)=π‘₯ln using limits as π‘₯β†’βˆž.

  • AThe growth rate of 𝑓(π‘₯) is equal to the growth rate of 𝑔(π‘₯).
  • BThe growth rate of 𝑓(π‘₯) is greater than the growth rate of 𝑔(π‘₯).
  • CThe growth rate of 𝑓(π‘₯) becomes ∞, while the growth rate of 𝑔(π‘₯) becomes 0 as π‘₯β†’βˆž.
  • DThe growth rate of 𝑓(π‘₯) is smaller than the growth rate of 𝑔(π‘₯).
  • EThe growth rate of 𝑓(π‘₯) becomes 0, while the growth rate of 𝑔(π‘₯) becomes ∞ as π‘₯β†’βˆž.

Q6:

For the functions 𝑓(π‘₯)=(2π‘₯)ln and 𝑔(π‘₯)=4π‘₯, evaluate limο—β†’βˆžο€½π‘“(π‘₯)𝑔(π‘₯) using l’HΓ΄pital’s rule.

Q7:

Compare the growth rate of the two functions 𝑓(π‘₯)=π‘₯ and 𝑔(π‘₯)=10 using limits as π‘₯β†’βˆž.

  • AThe growth rate of 𝑔(π‘₯) is greater than the growth rate of 𝑓(π‘₯).
  • BThe growth rate of 𝑓(π‘₯) is greater than the growth rate of 𝑔(π‘₯).
  • CThe growth rate of 𝑓(π‘₯) is equal to the growth rate of 𝑔(π‘₯).
  • DThe growth rate of 𝑓(π‘₯) becomes 0, while the growth rate of 𝑔(π‘₯) becomes ∞ as π‘₯β†’βˆž.
  • EThe growth rate of 𝑔(π‘₯) becomes 0, while the growth rate of 𝑓(π‘₯) becomes ∞ as π‘₯β†’βˆž.

Q8:

Compare the growth rate of the two functions 𝑓(π‘₯)=𝑒 and 𝑔(π‘₯)=π‘₯ln using limits as π‘₯β†’βˆž.

  • AThe growth rate of 𝑒 becomes 0, while the growth rate of lnπ‘₯ becomes ∞ as π‘₯β†’βˆž.
  • BThe growth rate of 𝑒 is greater than the growth rate of lnπ‘₯.
  • CThe growth rate of lnπ‘₯ becomes 0, while the growth rate of 𝑒 becomes ∞ as π‘₯β†’βˆž.
  • DThe growth rate of 𝑒 is equal to the growth rate of lnπ‘₯.
  • EThe growth rate of lnπ‘₯ is greater than the growth rate of 𝑒.

Q9:

Compare the growth rate of the two functions 𝑓(π‘₯)=π‘₯ and 𝑔(π‘₯)=1π‘₯ using limits as π‘₯β†’βˆž.

  • AThe growth rate of 𝑓(π‘₯) becomes ∞, while the growth rate of 𝑔(π‘₯) becomes 0 as π‘₯β†’βˆž.
  • BThe growth rate of 𝑓(π‘₯) is greater than the growth rate of 𝑔(π‘₯).
  • CThe growth rate of 𝑓(π‘₯) is smaller than the growth rate of 𝑔(π‘₯).
  • DThe growth rate of 𝑓(π‘₯) is equal to the growth rate of 𝑔(π‘₯).
  • EThe growth rate of 𝑓(π‘₯) becomes 0, while the growth rate of 𝑔(π‘₯) becomes ∞ as π‘₯β†’βˆž.

Q10:

Consider the functions 𝑓(π‘₯)=π‘₯ and 𝑔(π‘₯)=2.

Evaluate limο—β†’βˆžο€½π‘“(π‘₯)𝑔(π‘₯) using l’HΓ΄pital’s rule.

Evaluate limο—β†’βˆžο€½π‘”(π‘₯)𝑓(π‘₯) using l’HΓ΄pital’s rule.

  • A∞
  • B2
  • C0
  • DThe limit does not exist.
  • E1

What do you get from the two results about the growth rates of π‘₯ and 2 as π‘₯β†’βˆž?

  • AThe growth rate of 2 becomes 0, while the growth rate of π‘₯ becomes ∞ as π‘₯β†’βˆž.
  • BThe growth rate of π‘₯ is greater than that of 2 as π‘₯β†’βˆž.
  • CThe growth rate is the same for both as π‘₯β†’βˆž.
  • DThe growth rate of π‘₯ becomes 0, while the growth rate of 2 becomes ∞ as π‘₯β†’βˆž.
  • EThe growth rate of 2 is greater than that of π‘₯ as π‘₯β†’βˆž.

Q11:

Given that 𝑓(π‘₯)=√5π‘₯ and 𝑔(π‘₯)=5π‘₯log, use limο—β†’βˆžπ‘“(π‘₯)𝑔(π‘₯) to determine whether 𝑓(π‘₯) or 𝑔(π‘₯) is dominant.

  • A𝑔(π‘₯) is the dominant function.
  • BNeither 𝑓(π‘₯) nor 𝑔(π‘₯) is the dominant function.
  • C𝑓(π‘₯) is the dominant function.

Q12:

Given that 𝑓(π‘₯)=π‘₯ and 𝑔(π‘₯)=4π‘’οŠ©ο—, find whether 𝑓(π‘₯) or 𝑔(π‘₯) is the dominant function.

  • A𝑓(π‘₯) is the dominant function.
  • B𝑔(π‘₯) is the dominant function.
  • CNeither 𝑓(π‘₯) nor 𝑔(π‘₯) is the dominant function.

Q13:

Compare the growth rates of the functions 𝑓(π‘₯)=2+5 and 𝑔(π‘₯)=π‘₯+3 taking the limits as π‘₯β†’βˆž.

  • AThe growth rate of 𝑔(π‘₯) is greater than the growth rate of 𝑓(π‘₯).
  • BThe growth rate of 𝑓(π‘₯) is greater than the growth rate of 𝑔(π‘₯).
  • CThe growth rate of 𝑓(π‘₯) is equal to the growth rate of 𝑔(π‘₯).

Q14:

Given that 𝑓(π‘₯)=3π‘₯+7π‘₯ and 𝑔(π‘₯)=5π‘₯+4, use limο—β†’βˆžπ‘“(π‘₯)𝑔(π‘₯) to determine whether 𝑓(π‘₯) or 𝑔(π‘₯) is dominant.

  • ANeithernoristhedominantfunction𝑓(π‘₯)𝑔(π‘₯).
  • B𝑓(π‘₯).isthedominantfunction
  • C𝑔(π‘₯).isthedominantfunction

Q15:

Given that 𝑓(π‘₯)=π‘₯ln and 𝑔(π‘₯)=5π‘₯+3, use limο—β†’βˆžπ‘“(π‘₯)𝑔(π‘₯) to determine whether 𝑓(π‘₯) or 𝑔(π‘₯) is dominant.

  • ANeither 𝑓(π‘₯) nor 𝑔(π‘₯) is the dominant function.
  • B𝑔(π‘₯) is the dominant function.
  • C𝑓(π‘₯) is the dominant function.

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