# Comparing Rates of Growth of Functions

```Worksheet: Comparing Rate of Growth
of Functions
Q1: For the functions π(π₯) = 2π₯ 2 and π(π₯) = 4π2π₯ , evaluate lim (
π₯ββ
using lβHôpitalβs rule.
A
The limit does not exist.
B
0
C
β
D
1
E
1
4
Q2:
Evaluate lim (
π₯ββ
1
A
β
B
0
C
2
D
ββ
E
3
π₯2
) using lβHôpitalβs rule.
ππ₯
π(π₯)
)
π(π₯)
From the answer to the limit, what can you say about the growth rate of π₯ 2
compared to ππ₯ as π₯ β β?
A
The growth rate of π₯2 is greater than that of ππ₯ as π₯ β β.
B
The growth rate of π₯2 is smaller than that of ππ₯ as π₯ β β.
C
The growth rate of π₯2 is equal to that of ππ₯ as π₯ β β.
D
The growth rate of π₯ 2 becomes β, while the growth rate of ππ₯ becomes
0 as π₯ β β.
E
The growth rate of π₯2 becomes 0, while the growth rate of ππ₯ becomes
β as π₯ β β.
π(π₯)
) = 0, what do you notice about the growth rate of π(π₯)
π₯ββ 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₯ββ 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₯ββ
π(π₯)
)
π(π₯)
using lβHôpitalβs 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₯) = ππ₯ 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 ππ₯ becomes
β as π₯ β β.
B
The growth rate of ln π₯ is greater than the growth rate of ππ₯ .
C
The growth rate of ππ₯ is greater than the growth rate of ln π₯.
D
The growth rate of ππ₯ becomes 0, while the growth rate of ln π₯ becomes
β as π₯ β β.
E
The growth rate of ππ₯ 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₯ββ
A
The limit does not exist.
B
1
C
1
2
D
0
E
β
Evaluate lim (
π₯ββ
6
π(π₯)
) using lβHôpitalβs rule.
π(π₯)
π(π₯)
) using lβHôpitalβs 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.
```