# 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 and , evaluate using lβHΓ΄pitalβs rule.

Q2:

Evaluate using lβHΓ΄pitalβs rule.

From the answer to the limit, what can you say about the growth rate of compared to as ?

• AThe growth rate of is smaller than that of as .
• BThe growth rate of becomes , while the growth rate of becomes 0 as .
• CThe growth rate of is equal to that of as .
• DThe growth rate of is greater than that of as .
• EThe growth rate of becomes 0, while the growth rate of becomes as .

Q3:

If , what do you notice about the growth rate of compared to as ?

• AThe growth rate of is greater than that of as .
• BThe growth rate of is smaller than that of as .
• CThe growth rate of becomes 0, while the growth rate of becomes as .
• DThe growth rate of becomes , while the growth rate of becomes 0 as .
• EThe growth rate of is equal to that of as .

Q4:

If , what do you notice about the growth rate of compared to as ?

• AThe growth rate of becomes 0, while the growth rate of becomes as .
• BThe growth rate of is greater than that of as .
• CThe growth rate of is smaller than that of as .
• DThe growth rate of is equal to that of as .
• EThe growth rate of becomes , while the growth rate of becomes 0 as .

Q5:

Compare the growth rate of the two functions and using limits as .

• AThe growth rate of is equal to the growth rate of .
• BThe growth rate of is greater than the growth rate of .
• CThe growth rate of becomes , while the growth rate of becomes 0 as .
• DThe growth rate of is smaller than the growth rate of .
• EThe growth rate of becomes 0, while the growth rate of becomes as .

Q6:

For the functions and , evaluate using lβHΓ΄pitalβs rule.

Q7:

Compare the growth rate of the two functions and using limits as .

• AThe growth rate of is greater than the growth rate of .
• BThe growth rate of is greater than the growth rate of .
• CThe growth rate of is equal to the growth rate of .
• DThe growth rate of becomes 0, while the growth rate of becomes as .
• EThe growth rate of becomes 0, while the growth rate of becomes as .

Q8:

Compare the growth rate of the two functions and using limits as .

• AThe growth rate of becomes 0, while the growth rate of becomes as .
• BThe growth rate of is greater than the growth rate of .
• CThe growth rate of becomes 0, while the growth rate of becomes as .
• DThe growth rate of is equal to the growth rate of .
• EThe growth rate of is greater than the growth rate of .

Q9:

Compare the growth rate of the two functions and using limits as .

• AThe growth rate of becomes , while the growth rate of becomes 0 as .
• BThe growth rate of is greater than the growth rate of .
• CThe growth rate of is smaller than the growth rate of .
• DThe growth rate of is equal to the growth rate of .
• EThe growth rate of becomes 0, while the growth rate of becomes as .

Q10:

Consider the functions and .

Evaluate using lβHΓ΄pitalβs rule.

Evaluate using lβHΓ΄pitalβs rule.

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

What do you get from the two results about the growth rates of and as ?

• AThe growth rate of becomes 0, while the growth rate of becomes as .
• BThe growth rate of is greater than that of as .
• CThe growth rate is the same for both as .
• DThe growth rate of becomes 0, while the growth rate of becomes as .
• EThe growth rate of is greater than that of as .

Q11:

Given that and , use to determine whether or is dominant.

• A is the dominant function.
• BNeither nor is the dominant function.
• C is the dominant function.

Q12:

Given that and , find whether or is the dominant function.

• A is the dominant function.
• B is the dominant function.
• CNeither nor is the dominant function.

Q13:

Compare the growth rates of the functions and taking the limits as .

• AThe growth rate of is greater than the growth rate of .
• BThe growth rate of is greater than the growth rate of .
• CThe growth rate of is equal to the growth rate of .

Q14:

Given that and , use to determine whether or is dominant.

• A
• B
• C

Q15:

Given that and , use to determine whether or is dominant.

• ANeither nor is the dominant function.
• B is the dominant function.
• C is the dominant function.