Calculus CP
Investigating Limits as
Name: ____________________________
Date: _____________________________
x 
Directions: For each function given, answer the questions and fill in the chart. Then answer the follow-up
questions. You will need a calculator and your notes on horizontal and slant asymptotes.
Investigation #1:
Function #1:
1)
2)
3)
4)
𝑓(𝑥) =
27
𝑥+1
What is the degree of the top? _____________
What is the degree of the bottom? _____________
What is the horizontal asymptote of this function? __________________
Complete the table below by filling in each x-value into the function and evaluating. You may use
your calculator. It will be easiest to have the value of the function in decimals. Round to three
decimal places.
x-value
f(x)
1
10
100
200
300
5) What whole number does the function approach as the x-values increase towards infinity?
Function #2: f (x ) 
1)
2)
3)
4)
2x
x2  1
What is the degree of the top? _____________
What is the degree of the bottom? _____________
What is the horizontal asymptotes of this function? __________________
Complete the table below by filling in each x-value into the function and evaluating. You may use
your calculator. It will be easiest to have the value of the function in decimals. Round to three
decimal places.
x-value
f(x)
1
10
100
200
300
5) What whole number does the function approach as the x-value increases towards infinity?
Conclusion:
1) When the function is bottom heavy, what can you conclude about the horizontal asymptote and
the value the functions are approaching as x approaches infinity?
Investigation #2:
Function #1:
1)
2)
3)
4)
f ( x) 
4 x  3
2 x  15
What is the degree of the top? _____________
What is the degree of the bottom? _____________
What is the horizontal asymptote of this function? Make sure to reduce it. _______________
Complete the table below by filling in each x-value into the function and evaluating. You may use
your calculator. It will be easiest to have the value of the function in decimals. Round to three
decimal places.
x-value
f(x)
1
10
100
200
300
5) What whole number does the function approach as the x-values increase towards infinity?
Function #2:
1)
2)
3)
4)
f ( x) 
6 x2  5
2x2  7 x
What is the degree of the top? _____________
What is the degree of the bottom? _____________
What is the horizontal asymptote of this function? Make sure to reduce it. _______________
Complete the table below by filling in each x-value into the function and evaluating. You may use
your calculator. It will be easiest to have the value of the function in decimals. Round to three
decimal places.
x-value
f(x)
1
10
100
200
300
5) What whole number does the function approach as the x-values increase towards infinity?
Conclusion:
1) When the degree is the same, what can you conclude about the horizontal asymptote and the
value the functions are approaching as x approaches infinity?
Investigation #3:
Function #1:
1)
2)
3)
4)
f ( x) 
x2 1
x3
What is the degree of the top? _____________
What is the degree of the bottom? _____________
What is the horizontal asymptote of this function? _______________
Complete the table below by filling in each x-value into the function and evaluating. You may use
your calculator. It will be easiest to have the value of the function in decimals. Round to three
decimal places.
x-value
f(x)
10
20
30
40
50
5) What happens to the value of the function as the x-value increases towards infinity?
Function #2:
1)
2)
3)
4)
𝑓(𝑥) =
−4𝑥 4
𝑥 2 +3
What is the degree of the top? _____________
What is the degree of the bottom? _____________
What is the horizontal asymptote of this function? _______________
Complete the table below by filling in each x-value into the function and evaluating. You may use
your calculator. It will be easiest to have the value of the function in decimals. Round to three
decimal places.
x-value
f(x)
10
20
30
40
50
5) What happens to the value of the function as the x-value increases towards infinity?
Conclusion:
1) When the function is top heavy, what can you conclude about the horizontal asymptote and the
value the functions are approaching as x approaches infinity?
Summary:
See if you can summarize the rules for finding limits as x approaches infinity:
1) If the function is bottom heavy, the limit will be _________________________________.
2) If the degrees are the same, the limit will be ___________________________________.
3) If the function is top heavy, the limit will be ____________________________________.
*Make sure you are looking at the HIGHEST EXPONENT in the numerator and denominator to
determine the degree and which case you have.
See if you can evaluate these limits using the rules from above!
2𝑥 2 −1
𝑥→∞ 3𝑥 4 +2
3𝑥
𝑥→∞ 7𝑥−1
=
2) lim
𝑥 2 +8
𝑥→∞ 𝑥
=
5) lim
1) lim
4) lim
=
5𝑥 2 −𝑥+2
𝑥→∞ 𝑥+2𝑥 2
=
−5𝑥 3 −4𝑥 2 +8
𝑥→∞ 6𝑥 2 +3𝑥+2
3) lim
=
𝑥+3
𝑥
𝑥→∞ 3 −1
6) lim
=