Limits at Infinity Practice Set
Instructions
Determine the value of each limit. If the limit does not exist, indicate whether it
approaches infinity, negative infinity, or simply DNE (Does Not Exist).
Part I: Rational Functions and Polynomials (Q1 - Q10)
Limit as x approaches infinity of (4x + 7) / (2x - 3)
Limit as x approaches infinity of (x squared - 6x + 2) / (5x cubed + x)
Limit as x approaches negative infinity of (3x squared + 1) / (x squared - 4)
Limit as x approaches negative infinity of (x cubed + 8) / (x squared + x)
Limit as x approaches infinity of (1 - 3t) / (t squared + 2t - 5)
Limit as x approaches negative infinity of (2x to the power of 5 - x) / (x to the power
of 5 + 3x squared)
Limit as x approaches infinity of (x to the power of 4 + 2x) / (2x to the power of 3 -
1)
Limit as t approaches negative infinity of (t cubed + 5t) / (t to the power of 4 + 6)
Limit as x approaches infinity of (pi * x squared + 3) / (e * x squared - 7)
Limit as x approaches infinity of (x + 1) to the power of 3 / (x cubed + 1)
Part II: Functions with Radicals and Absolute Values (Q11 - Q20)
Limit as x approaches infinity of Square root(x squared + 9) / (2x)
Limit as x approaches negative infinity of Square root(4x squared + x) / (x + 1)
Limit as x approaches infinity of (3x) / Square root(x squared - 2x + 1)
Limit as x approaches negative infinity of (5x - 2) / Square root(x squared + 4)
Limit as x approaches infinity of (Square root(x squared + x) - x)
Limit as x approaches negative infinity of (Square root(x squared + 2x) + x)
Limit as x approaches infinity of (Absolute value of x) / (x + 5)
Limit as x approaches negative infinity of (x) / (Absolute value of x - 4)
Limit as x approaches infinity of (Square root(9x squared + 1)) / (3x - 5)
Limit as x approaches negative infinity of (Square root(16x squared + 2x)) / (2x + 3)
Part III: Exponential, Combinatorial, and Challenging Forms (Q21 -
Q30)
Limit as x approaches infinity of (e to the power of x) / (e to the power of x + 1)
Limit as x approaches negative infinity of (e to the power of 2x) / (3e to the power
of x + 5)
Limit as x approaches infinity of (ln(x)) / (x)
Limit as x approaches infinity of (x squared) / (e to the power of x)
Limit as x approaches negative infinity of (x cubed - x squared) / (e to the power of
x)
Limit as x approaches infinity of (cos(x)) / (x)
Limit as x approaches negative infinity of (e to the power of x) / (x squared)
Limit as x approaches infinity of (x to the power of 4) / (x cubed - 2x to the power of
4)
Limit as x approaches negative infinity of (x cubed - 2x squared + 5)
Limit as x approaches infinity of (Square root(x to the power of 4 + 1)) / (x squared
+ x)
Solutions and Detailed Explanations
Part I: Rational Functions and Polynomials
Limit as x approaches infinity of (4x + 7) / (2x - 3)
Degree: Numerator Degree (1) = Denominator Degree (1).
Result: Ratio of leading coefficients: 4 / 2 = 2.
Limit as x approaches infinity of (x squared - 6x + 2) / (5x cubed + x)
Degree: Numerator Degree (2) < Denominator Degree (3).
Result: 0.
Limit as x approaches negative infinity of (3x squared + 1) / (x squared - 4)
Degree: Numerator Degree (2) = Denominator Degree (2).
Result: Ratio of leading coefficients: 3 / 1 = 3.
Limit as x approaches negative infinity of (x cubed + 8) / (x squared + x)
Degree: Numerator Degree (3) > Denominator Degree (2).
Result: The limit is based on the ratio of leading terms: x cubed / x squared = x. As x
-> negative infinity, the result is negative infinity.
Limit as t approaches infinity of (1 - 3t) / (t squared + 2t - 5)
Degree: Numerator Degree (1) < Denominator Degree (2).
Result: 0.
Limit as x approaches negative infinity of (2x to the power of 5 - x) / (x to the power
of 5 + 3x squared)
Degree: Numerator Degree (5) = Denominator Degree (5).
Result: Ratio of leading coefficients: 2 / 1 = 2.
Limit as x approaches infinity of (x to the power of 4 + 2x) / (2x to the power of 3 -
1)
Degree: Numerator Degree (4) > Denominator Degree (3).
Result: The limit is based on the ratio of leading terms: x to the power of 4 / (2x
cubed) = x/2. As x -> infinity, the result is infinity.
Limit as t approaches negative infinity of (t cubed + 5t) / (t to the power of 4 + 6)
Degree: Numerator Degree (3) < Denominator Degree (4).
Result: 0.
Limit as x approaches infinity of (pi * x squared + 3) / (e * x squared - 7)
Degree: Numerator Degree (2) = Denominator Degree (2).
Result: Ratio of leading coefficients: pi / e.
Limit as x approaches infinity of (x + 1) to the power of 3 / (x cubed + 1)
Analysis: Expand the numerator: (x + 1) cubed = x cubed + 3x squared + 3x + 1. Both
degrees are 3.
Result: Ratio of leading coefficients: 1 / 1 = 1.
Part II: Functions with Radicals and Absolute Values
Limit as x approaches infinity of Square root(x squared + 9) / (2x)
Analysis: Divide numerator and denominator by
x
(since
x → ∞
,
x=Square root (x2)
).
Simplification: Square root(1 + 9/x squared) / 2.
Result: Square root(1 + 0) / 2 = 1/2.
Limit as x approaches negative infinity of Square root(4x squared + x) / (x + 1)
Analysis: Divide numerator and denominator by
x
. Since
x → − ∞
, use
x=−Square root(x2)
in the denominator simplification.
Simplification:
x
inside the radical becomes
x2
. Numerator
→Square root(4+1/x)
.
Denominator
→ x /x+1/x=1+1/x
. But since
x<0
, the ratio of leading terms is
−Square root (4)/1
.
Result:
−Square root (4)/1=−2
.
Limit as x approaches infinity of (3x) / Square root(x squared - 2x + 1)
Analysis: Divide by
x
. Since
x → ∞
,
x=Square root (x2)
.
Simplification: 3 / Square root(1 - 2/x + 1/x squared).
Result: 3 / Square root(1 - 0 + 0) = 3.
Limit as x approaches negative infinity of (5x - 2) / Square root(x squared + 4)
Analysis: Divide by
x
. Since
x → − ∞
, use
x=−Square root(x2)
.
Simplification: Numerator
→5−2/x
. Denominator
→ −Square root (1+4/x2)
.
Result:
5/(−Square root (1))=−5
.
Limit as x approaches infinity of (Square root(x squared + x) - x)
Analysis: Indeterminate form
∞ −∞
. Rationalize by multiplying by the conjugate:
(Square root(x squared + x) + x).
Simplification: Numerator
→(x2+x)− x2=x
. Denominator
→Square root(x2+x)+x
.
Divide by
x
.
Result:
1/(Square root (1+1/x)+1)=1/(1+1)=1/2
.
Limit as x approaches negative infinity of (Square root(x squared + 2x) + x)
Analysis: Indeterminate form
∞ −∞
. Rationalize.
Simplification: Numerator
→(x2+2x)− x2=2x
. Denominator
→Square root(x2+2x)− x
. Divide by
x
. Since
x → − ∞
,
Square root (x2)=− x
.
Result:
2/(−Square root (1)−1)=2/(−2)=−1
.
Limit as x approaches infinity of (Absolute value of x) / (x + 5)
Analysis: Since
x → ∞
, Absolute value of x =
x
.
Simplification:
x/(x+5)
. Degrees are equal.
Result:
1/1=1
.
Limit as x approaches negative infinity of (x) / (Absolute value of x - 4)
Analysis: Since
x → − ∞
, Absolute value of x =
− x
.
Simplification:
x/(− x − 4)
. Degrees are equal.
Result:
1/(−1)=−1
.
Limit as x approaches infinity of (Square root(9x squared + 1)) / (3x - 5)
Analysis: Divide by
x
.
x → ∞
.
Simplification: Square root(9 + 1/x squared) / (3 - 5/x).
Result: Square root(9) / 3 =
3/3=1
.
Limit as x approaches negative infinity of (Square root(16x squared + 2x)) / (2x + 3)
Analysis: Divide by
x
.
x → − ∞
. Use
x=−Square root(x2)
.
Simplification:
−Square root (16+2/x)/(2+3/x)
.
Result:
−Square root (16)/2=−4/2=−2
.
Part III: Exponential, Combinatorial, and Challenging Forms
Limit as x approaches infinity of (e to the power of x) / (e to the power of x + 1)
Analysis: Divide numerator and denominator by the dominant term, e to the power
of x.
Simplification:
1/(1+1/e to the power of x)
. As
x → ∞
,
1/e to the power of x →0
.
Result:
1/(1+0)=1
.
Limit as x approaches negative infinity of (e to the power of 2x) / (3e to the power
of x + 5)
Analysis: As
x → − ∞
, e to the power of
ax →0
for
a>0
.
Simplification: Numerator
→0
. Denominator
→3(0)+5=5
.
Result:
0/5=0
.
Limit as x approaches infinity of (ln(x)) / (x)
Analysis: LHopitals Rule (or knowledge of growth rates: Polynomial grows faster
than log). LHopitals is often introduced after limits, but the growth rate applies.
Result: 0.
Limit as x approaches infinity of (x squared) / (e to the power of x)
Analysis: Exponential growth dominates polynomial growth.
Result: 0.
Limit as x approaches negative infinity of (x cubed - x squared) / (e to the power of
x)
Analysis: As
x → − ∞
, e to the power of x
→0
(from the positive side). The numerator
x3− x2
is dominated by
x3
, which
→ −∞
.
Result:
− ∞ /0+¿=− ∞ ¿
.
Limit as x approaches infinity of (cos(x)) / (x)
Analysis: Use the Squeeze Theorem. Since
−1≤cos(x)≤1
, we have
−1/x ≤cos (x)/x ≤ 1/x
. Both bounding functions approach 0 as
x → ∞
.
Result: 0.
Limit as x approaches negative infinity of (e to the power of x) / (x squared)
Analysis: As
x → − ∞
, e to the power of x
→0
.
x2→ ∞
.
Result:
0/∞=0
.
Limit as x approaches infinity of (x to the power of 4) / (x cubed - 2x to the power of
4)
Degree: Numerator Degree (4) = Denominator Degree (4).
Result: Ratio of leading coefficients:
1/(−2)=−1/2
.
Limit as x approaches negative infinity of (x cubed - 2x squared + 5)
Analysis: Only consider the highest degree term:
x3
. As
x → − ∞
,
x3→ − ∞
.
Result: negative infinity.
Limit as x approaches infinity of (Square root(x to the power of 4 + 1)) / (x squared
+ x)
Analysis: Divide by
x2
(highest power in the denominator). In the numerator,
x2=Square root (x4)
.
Simplification: Square root(1 + 1/x to the power of 4) / (1 + 1/x).
Result: Square root(1) / 1 = 1.
