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Comparing Rates of Growth of Functions

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Worksheet: Comparing Rate of Growth
of Functions
Q1: For the functions 𝑓(π‘₯) = 2π‘₯ 2 and 𝑔(π‘₯) = 4𝑒2π‘₯ , evaluate lim (
π‘₯β†’βˆž
using l’Hôpital’s rule.
A
The limit does not exist.
B
0
C
∞
D
1
E
1
4
Q2:
Evaluate lim (
π‘₯β†’βˆž
1
A
∞
B
0
C
2
D
βˆ’βˆž
E
3
π‘₯2
) using l’Hôpital’s rule.
𝑒π‘₯
𝑓(π‘₯)
)
𝑔(π‘₯)
From the answer to the limit, what can you say about the growth rate of π‘₯ 2
compared to 𝑒π‘₯ as π‘₯ β†’ ∞?
A
The growth rate of π‘₯2 is greater than that of 𝑒π‘₯ as π‘₯ β†’ ∞.
B
The growth rate of π‘₯2 is smaller than that of 𝑒π‘₯ as π‘₯ β†’ ∞.
C
The growth rate of π‘₯2 is equal to that of 𝑒π‘₯ as π‘₯ β†’ ∞.
D
The growth rate of π‘₯ 2 becomes ∞, while the growth rate of 𝑒π‘₯ becomes
0 as π‘₯ β†’ ∞.
E
The growth rate of π‘₯2 becomes 0, while the growth rate of 𝑒π‘₯ becomes
∞ as π‘₯ β†’ ∞.
𝑓(π‘₯)
) = 0, what do you notice about the growth rate of 𝑓(π‘₯)
π‘₯β†’βˆž 𝑔(π‘₯)
compared to 𝑔(π‘₯) as π‘₯ β†’ ∞?
Q3: If lim (
A
The growth rate of 𝑓(π‘₯) is equal to that of 𝑔(π‘₯) as π‘₯ β†’ ∞.
B
The growth rate of 𝑓(π‘₯) becomes 0, while the growth rate of 𝑔(π‘₯)
becomes ∞ as π‘₯ β†’ ∞.
2
C
The growth rate of 𝑓(π‘₯) is greater than that of 𝑔(π‘₯) as π‘₯ β†’ ∞.
D
The growth rate of 𝑓(π‘₯) becomes ∞, while the growth rate of 𝑔(π‘₯)
becomes 0 as π‘₯ β†’ ∞.
E
The growth rate of 𝑓(π‘₯) is smaller than that of 𝑔(π‘₯) as π‘₯ β†’ ∞.
𝑓(π‘₯)
) = ∞, what do you notice about the growth rate of 𝑓(π‘₯)
π‘₯β†’βˆž 𝑔(π‘₯)
compared to 𝑔(π‘₯) as π‘₯ β†’ ∞?
Q4: If lim (
A
The growth rate of 𝑓(π‘₯) becomes ∞, while the growth rate of 𝑔(π‘₯)
becomes 0 as π‘₯ β†’ ∞.
B
The growth rate of 𝑓(π‘₯) is equal to that of 𝑔(π‘₯) as π‘₯ β†’ ∞.
C
The growth rate of 𝑓(π‘₯) is smaller than that of 𝑔(π‘₯) as π‘₯ β†’ ∞.
D
The growth rate of 𝑓(π‘₯) is greater than that of 𝑔(π‘₯) as π‘₯ β†’ ∞.
E
The growth rate of 𝑓(π‘₯) becomes 0, while the growth rate of 𝑔(π‘₯)
becomes ∞ as π‘₯ β†’ ∞.
Q5: Compare the growth rate of the two functions 𝑓(π‘₯) = (π‘₯ + 4)2 and
𝑔(π‘₯) = ln π‘₯ using limits as π‘₯ β†’ ∞.
A
The growth rate of 𝑓(π‘₯) is equal to the growth rate of 𝑔(π‘₯).
B
The growth rate of 𝑓(π‘₯) becomes ∞, while the growth rate of 𝑔(π‘₯)
becomes 0 as π‘₯ β†’ ∞.
C
The growth rate of 𝑓(π‘₯) is greater than the growth rate of 𝑔(π‘₯).
D
The growth rate of 𝑓(π‘₯) becomes 0, while the growth rate of 𝑔(π‘₯)
becomes ∞ as π‘₯ β†’ ∞.
E
3
The growth rate of 𝑓(π‘₯) is smaller than the growth rate of 𝑔(π‘₯).
Q6: For the functions 𝑓(π‘₯) = ln (2π‘₯) and 𝑔(π‘₯) = 4π‘₯ 3 , evaluate lim (
π‘₯β†’βˆž
𝑓(π‘₯)
)
𝑔(π‘₯)
using l’Hôpital’s rule.
A
∞
B
1
12
C
1
6
D
0
E
The limit does not exist.
Q7: Compare the growth rate of the two functions 𝑓(π‘₯) = π‘₯ 10 and 𝑔(π‘₯) =
10π‘₯ using limits as π‘₯ β†’ ∞.
A
The growth rate of 𝑔(π‘₯) becomes 0, while the growth rate of 𝑓(π‘₯)
becomes ∞ as π‘₯ β†’ ∞.
B
The growth rate of 𝑔(π‘₯) is greater than the growth rate of 𝑓(π‘₯).
C
The growth rate of 𝑓(π‘₯) is greater than the growth rate of 𝑔(π‘₯).
D
The growth rate of 𝑓(π‘₯) is equal to the growth rate of 𝑔(π‘₯).
E
The growth rate of 𝑓(π‘₯) becomes 0, while the growth rate of 𝑔(π‘₯)
becomes ∞ as π‘₯ β†’ ∞.
4
Q8: Compare the growth rate of the two functions 𝑓(π‘₯) = 𝑒π‘₯ and 𝑔(π‘₯) = ln π‘₯
using limits as π‘₯ β†’ ∞.
A
The growth rate of ln π‘₯ becomes 0, while the growth rate of 𝑒π‘₯ becomes
∞ as π‘₯ β†’ ∞.
B
The growth rate of ln π‘₯ is greater than the growth rate of 𝑒π‘₯ .
C
The growth rate of 𝑒π‘₯ is greater than the growth rate of ln π‘₯.
D
The growth rate of 𝑒π‘₯ becomes 0, while the growth rate of ln π‘₯ becomes
∞ as π‘₯ β†’ ∞.
E
The growth rate of 𝑒π‘₯ is equal to the growth rate of ln π‘₯.
Q9: Compare the growth rate of the two functions 𝑓(π‘₯) = π‘₯ 2 and 𝑔(π‘₯) =
using limits as π‘₯ β†’ ∞.
A
The growth rate of 𝑓(π‘₯) becomes 0, while the growth rate of 𝑔(π‘₯)
becomes ∞ as π‘₯ β†’ ∞.
B
The growth rate of 𝑓(π‘₯) is equal to the growth rate of 𝑔(π‘₯).
C
The growth rate of 𝑓(π‘₯) is smaller than the growth rate of 𝑔(π‘₯).
D
The growth rate of 𝑓(π‘₯) is greater than the growth rate of 𝑔(π‘₯).
E
The growth rate of 𝑓(π‘₯) becomes ∞, while the growth rate of 𝑔(π‘₯)
becomes 0 as π‘₯ β†’ ∞.
Q10: Consider the functions 𝑓(π‘₯) = π‘₯ 50 and 𝑔(π‘₯) = 2𝑋 .
5
1
π‘₯
Evaluate lim (
π‘₯β†’βˆž
A
The limit does not exist.
B
1
C
1
2
D
0
E
∞
Evaluate lim (
π‘₯β†’βˆž
6
𝑓(π‘₯)
) using l’Hôpital’s rule.
𝑔(π‘₯)
𝑔(π‘₯)
) using l’Hôpital’s rule.
𝑓(π‘₯)
A
The limit does not exist.
B
1
C
2
D
∞
E
0
What do you get from the two results about the growth rates of π‘₯ 50 and 2π‘₯
as π‘₯ β†’ ∞?
7
A
The growth rate of π‘₯ 50 becomes 0, while the growth rate of 2π‘₯ becomes
∞ as π‘₯ β†’ ∞.
B
The growth rate is the same for both as π‘₯ β†’ ∞.
C
The growth rate of 2π‘₯ becomes 0, while the growth rate of π‘₯ 50 becomes
∞ as π‘₯ β†’ ∞.
D
The growth rate of 2π‘₯ is greater than that of π‘₯ 50 as π‘₯ β†’ ∞.
E
The growth rate of π‘₯50 is greater than that of 2π‘₯ as π‘₯ β†’ ∞.