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Math Problems
Algebra 2
Power rule
Simplify. Assume all variables are positive.
\newline
r
3
4
r
11
4
\frac{r^{\frac{3}{4}}}{r^{\frac{11}{4}}}
r
4
11
r
4
3
\newline
Write your answer in the form
A
A
A
or
A
B
\frac{A}{B}
B
A
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
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Simplify. Assume all variables are positive.
\newline
r
3
4
r
5
4
\frac{r^{\frac{3}{4}}}{r^{\frac{5}{4}}}
r
4
5
r
4
3
\newline
Write your answer in the form
A
A
A
or
A
B
\frac{A}{B}
B
A
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
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Simplify.
\newline
(
2
m
2
3
m
−
1
)
2
\left(\frac{2 m^{2}}{3 m^{-1}}\right)^{2}
(
3
m
−
1
2
m
2
)
2
\newline
Write your answer using only positive exponents.
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Which of the following statements about the graph of
y
=
12
(
0.75
)
x
y=12(0.75)^{x}
y
=
12
(
0.75
)
x
is true?
\newline
Choose
1
1
1
answer:
\newline
(A) As
x
x
x
increases,
y
y
y
increases at an increasing rate.
\newline
(B) As
x
x
x
increases,
y
y
y
increases at a decreasing rate.
\newline
(C) As
x
x
x
increases,
y
y
y
decreases at an increasing rate.
\newline
(D) As
x
x
x
increases,
y
y
y
decreases at a decreasing rate.
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6
5
p
+
k
q
=
4
5
q
=
3
5
p
−
2
5
\frac{6}{5}p + kq = \frac{4}{5} \\ q = \frac{3}{5}p - \frac{2}{5}
5
6
p
+
k
q
=
5
4
q
=
5
3
p
−
5
2
\newline
Consider the system of equations, where
k
k
k
is a constant. For which value of
k
k
k
is there no
(
p
,
q
)
(p,q)
(
p
,
q
)
solutions?
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Prove the identity.
\newline
cos
x
1
+
sin
x
=
sec
x
−
tan
x
\frac{\cos x}{1+\sin x}=\sec x-\tan x
1
+
sin
x
cos
x
=
sec
x
−
tan
x
\newline
Note that each Statement must be based on a Rule chosen from the Rule menu. To see a detail the right of the Rule.
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Which of the following relationships proves why
△
A
B
D
\triangle \mathrm{ABD}
△
ABD
and
△
C
B
D
\triangle \mathrm{CBD}
△
CBD
are congruent?
\newline
ASA
\newline
H
L
\mathrm{HL}
HL
\newline
SAS
\newline
Overlapping Triangles
\newline
Segments
A
B
A B
A
B
and
C
B
C B
CB
are congruent.
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f
(
x
)
=
2
x
5
+
x
4
−
18
x
3
−
17
x
2
+
20
x
+
12
f(x)=2 x^{5}+x^{4}-18 x^{3}-17 x^{2}+20 x+12
f
(
x
)
=
2
x
5
+
x
4
−
18
x
3
−
17
x
2
+
20
x
+
12
\newline
The function
f
f
f
is shown. If
x
−
3
x-3
x
−
3
is a factor of
f
f
f
, what is the value of
f
(
3
)
f(3)
f
(
3
)
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Calculus
1
1
1
: Exam
2
2
2
\newline
Name:
\newline
15
15
15
. (
8
8
8
pts) Use the first derivative test to find and classify all local extrema in the interval
(
−
4
,
5
)
(-4,5)
(
−
4
,
5
)
for the function
\newline
f
(
x
)
=
(
5
x
−
4
)
e
4
x
f(x)=(5 x-4) e^{4 x}
f
(
x
)
=
(
5
x
−
4
)
e
4
x
\newline
If there is more than one local maximum or minimum, list them all. If a local maximum or a local minimum does not occur on the interval, note that in your work. You may assume that
e
4
x
e^{4 x}
e
4
x
is a positive number for all values of
x
x
x
.
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Given:
P
(
x
)
=
x
4
+
x
3
−
11
х
2
−
9
x
+
18
P(x) = x^4 + x^3 - 11х^2 - 9x + 18
P
(
x
)
=
x
4
+
x
3
−
11
х
2
−
9
x
+
18
with factors
(
x
+
3
)
(x + 3)
(
x
+
3
)
and
(
x
−
1
)
(x - 1)
(
x
−
1
)
. A. Write
P
(
x
)
P(x)
P
(
x
)
in factored form. B. Find the zeros of the function.
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Use the reduction of order method to find the general solutio of the following equations. One solution of the homogeneous is shown alongside each equation.
\newline
13
13
13
.
2
x
2
y
′
′
+
3
x
y
′
−
y
=
1
x
,
y
1
=
x
1
/
2
2 x^{2} y^{\prime \prime}+3 x y^{\prime}-y=\frac{1}{x}, \quad y_{1}=x^{1 / 2}
2
x
2
y
′′
+
3
x
y
′
−
y
=
x
1
,
y
1
=
x
1/2
.
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The graph of
y
=
2
x
2
−
4
x
−
4
y=2x^2-4x-4
y
=
2
x
2
−
4
x
−
4
is shown in the
x
y
xy
x
y
plane. Which of the following characteristics of the graph is displayed as a constant or coefficient in the equation as written?
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Try Again
\newline
One or more of your answers are incorrect.
\newline
First, rewrite
\newline
4
5
\frac{4}{5}
5
4
and
\newline
9
11
\frac{9}{11}
11
9
so that they have a common denominator. Then, use
\newline
<
<
<
,
=
=
=
, or
\newline
>
>
>
to order
\newline
4
5
\frac{4}{5}
5
4
and
\newline
9
11
\frac{9}{11}
11
9
.
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Iestion
7
7
7
of
11
11
11
, Step
1
1
1
of
1
1
1
\newline
Correct
\newline
Find a formula for the inverse of the following function, if possible.
\newline
A
(
x
)
=
1
x
−
1
A(x)=\frac{1}{x-1}
A
(
x
)
=
x
−
1
1
\newline
Answer
\newline
How to enter your answer (opens in new window)
\newline
Selecting a radio button will replace the entered answer value(s) with the radio button value. If the radio button is not selected, the entered an
\newline
A
−
1
(
x
)
=
A^{-1}(x)=
A
−
1
(
x
)
=
\newline
does not have an inverse functi
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iestion
7
7
7
of
11
11
11
, Step
1
1
1
of
1
1
1
\newline
Correct
\newline
nd a formula for the inverse of the following function, if possible.
\newline
A
(
x
)
=
1
x
−
1
A(x)=\frac{1}{x-1}
A
(
x
)
=
x
−
1
1
\newline
nswer
\newline
How to enter your answer (opens in new window)
\newline
Selecting a radio button will replace the entered answer value(s) with the radio button value. If the radio button is not selected, the entered an
\newline
A
−
1
(
x
)
=
A^{-1}(x)=
A
−
1
(
x
)
=
\newline
does not have an inverse functi
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Express the given expression without logs, in simplest form. Assume all variables represent positive values.
\newline
(
e
−
2
ln
12
y
)
\left(e^{-2 \ln 12 \sqrt{y}}\right)
(
e
−
2
l
n
12
y
)
\newline
Answer:
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Express the given expression without logs, in simplest form. Assume all variables represent positive values.
\newline
(
7
log
7
(
2
w
2
)
)
\left(7^{\log _{7}\left(2 w^{2}\right)}\right)
(
7
l
o
g
7
(
2
w
2
)
)
\newline
Answer:
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Express the given expression without logs, in simplest form. Assume all variables represent positive values.
\newline
(
4
log
4
(
5
y
)
)
\left(4^{\log _{4}(5 \sqrt{y})}\right)
(
4
l
o
g
4
(
5
y
)
)
\newline
Answer:
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Express the given expression without logs, in simplest form. Assume all variables represent positive values.
\newline
(
1
2
log
12
(
8
w
)
)
\left(12^{\log _{12}(8 \sqrt{w})}\right)
(
1
2
l
o
g
12
(
8
w
)
)
\newline
Answer:
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Express the given expression without logs, in simplest form. Assume all variables represent positive values.
\newline
(
1
2
log
12
(
10
w
)
)
\left(12^{\log _{12}(10 \sqrt{w})}\right)
(
1
2
l
o
g
12
(
10
w
)
)
\newline
Answer:
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Rewrite the expression as a product of four linear factors:
\newline
(
8
x
2
+
9
x
)
2
−
13
(
8
x
2
+
9
x
)
−
14
\left(8 x^{2}+9 x\right)^{2}-13\left(8 x^{2}+9 x\right)-14
(
8
x
2
+
9
x
)
2
−
13
(
8
x
2
+
9
x
)
−
14
\newline
Answer:
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Rewrite the expression as a product of four linear factors:
\newline
(
5
x
2
−
12
x
)
2
−
2
(
5
x
2
−
12
x
)
−
63
\left(5 x^{2}-12 x\right)^{2}-2\left(5 x^{2}-12 x\right)-63
(
5
x
2
−
12
x
)
2
−
2
(
5
x
2
−
12
x
)
−
63
\newline
Answer:
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Rewrite the expression as a product of four linear factors:
\newline
(
12
x
2
+
5
x
)
2
−
9
(
12
x
2
+
5
x
)
+
14
\left(12 x^{2}+5 x\right)^{2}-9\left(12 x^{2}+5 x\right)+14
(
12
x
2
+
5
x
)
2
−
9
(
12
x
2
+
5
x
)
+
14
\newline
Answer:
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Rewrite the expression as a product of four linear factors:
\newline
(
12
x
2
+
13
x
)
2
+
4
(
12
x
2
+
13
x
)
+
3
\left(12 x^{2}+13 x\right)^{2}+4\left(12 x^{2}+13 x\right)+3
(
12
x
2
+
13
x
)
2
+
4
(
12
x
2
+
13
x
)
+
3
\newline
Answer:
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Prove the identity.
\newline
cot
2
x
csc
x
+
1
=
csc
x
−
1
\frac{\cot ^{2} x}{\csc x+1}=\csc x-1
csc
x
+
1
cot
2
x
=
csc
x
−
1
\newline
Note that each Statement must be based on a Rule chosen from the Rule menu. To se the right of the Rule.
\newline
Statement
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Find a formula for the inverse of the following function, if possible.
\newline
G
(
x
)
=
4
x
+
2
5
G(x)=\sqrt[5]{4 x+2}
G
(
x
)
=
5
4
x
+
2
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We want to factor the following expression:
\newline
(
x
+
1
)
2
−
4
y
2
(x+1)^{2}-4y^{2}
(
x
+
1
)
2
−
4
y
2
\newline
Which pattern can we use to factor the expression?
\newline
U
U
U
and
V
V
V
are either constant integers or single-variable expressions.
\newline
Choose
1
1
1
answer:
\newline
(A)
(
U
+
V
)
2
(U+V)^{2}
(
U
+
V
)
2
or
(
U
−
V
)
2
(U-V)^{2}
(
U
−
V
)
2
\newline
(B)
(
U
+
V
)
(
U
−
V
)
(U+V)(U-V)
(
U
+
V
)
(
U
−
V
)
\newline
(C) We can't use any of the patterns.
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4
x
+
a
(
3
x
+
4
)
(
x
+
1
)
−
5
6
x
+
8
=
b
c
(
x
+
1
)
\frac{4 x+a}{(3 x+4)(x+1)}-\frac{5}{6 x+8}=\frac{b}{c(x+1)}
(
3
x
+
4
)
(
x
+
1
)
4
x
+
a
−
6
x
+
8
5
=
c
(
x
+
1
)
b
\newline
The given equation is true for all
x
>
−
1
x>-1
x
>
−
1
, where
a
,
b
a, b
a
,
b
, and
c
c
c
are nonzero constants. Which of the following must be an integer?
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If
y
=
(
x
−
1
)
(
x
+
5
)
y=(x-1)(x+5)
y
=
(
x
−
1
)
(
x
+
5
)
is graphed in the
x
y
xy
x
y
-plane, which of the following characteristics of the graph is displayed as a constant in the equation?
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(
3
y
−
2
)
(
y
+
a
)
=
3
y
2
+
b
(3y-2)(y+a)=3y^{2}+b
(
3
y
−
2
)
(
y
+
a
)
=
3
y
2
+
b
?
\newline
If the given equation is true for all values of
y
y
y
, where
a
a
a
and
b
b
b
are constants, which of the following is the value of
b
b
b
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
38
-38
−
38
\newline
(B)
12
12
12
\newline
(C)
34
34
34
\newline
(D)
36
36
36
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4
y
−
3
x
=
40
4y-3x=40
4
y
−
3
x
=
40
\newline
4
y
=
3
x
−
30
4y=3x-30
4
y
=
3
x
−
30
\newline
Which of the following accurately describes all solutions to the system of equations shown?
\newline
Choose
1
1
1
answer:
\newline
(A)
x
=
0
x=0
x
=
0
and
y
=
0
y=0
y
=
0
\newline
(B)
x
=
5
3
x=\frac{5}{3}
x
=
3
5
and
y
=
45
4
y=\frac{45}{4}
y
=
4
45
\newline
(C) There are infinite solutions to the system.
\newline
(D) There are no solutions to the system.
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The equation
y
=
30
(
1
2
)
x
y=30\left(\frac{1}{2}\right)^x
y
=
30
(
2
1
)
x
is graphed in the
x
y
xy
x
y
-plane. Which of the following statements about the graph is true?
\newline
Choose
1
1
1
answer:
\newline
(A) As
x
x
x
increases,
y
y
y
increases at an increasing rate.
\newline
(B) As
x
x
x
increases,
y
y
y
increases at a decreasing rate.
\newline
(C) As
x
x
x
increases,
y
y
y
decreases at an increasing rate.
\newline
(D) As
x
x
x
increases,
y
y
y
decreases at a decreasing rate.
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If
y
=
38
(
1.04
)
x
y=38(1.04)^x
y
=
38
(
1.04
)
x
is graphed in the
x
y
xy
x
y
-plane, which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?
\newline
Choose
1
1
1
answer:
\newline
(A)
x
x
x
-intercept
\newline
(B)
y
y
y
-intercept
\newline
(C) Slope
\newline
(D) The value
y
y
y
approaches as
x
x
x
becomes very large
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Which statement best describes the limit shown below?
\newline
lim
x
→
∞
−
7
ln
x
3
x
38
+
5
x
38
\lim _{x \rightarrow \infty} \frac{-7 \ln x}{3 x^{38}+5 x^{38}}
x
→
∞
lim
3
x
38
+
5
x
38
−
7
ln
x
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
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Which statement best describes the limit shown below?
\newline
lim
x
→
∞
−
6
log
5
x
+
x
89
x
18
+
5
x
\lim _{x \rightarrow \infty} \frac{-6 \log _{5} x+x^{89}}{x^{18}+5^{x}}
x
→
∞
lim
x
18
+
5
x
−
6
lo
g
5
x
+
x
89
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
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Which statement best describes the limit shown below?
\newline
lim
x
→
∞
4
e
x
−
x
34
\lim _{x \rightarrow \infty} \frac{4 e^{x}}{-x^{34}}
x
→
∞
lim
−
x
34
4
e
x
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
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Which statement best describes the limit shown below?
\newline
lim
x
→
∞
x
10
−
2
x
10
−
8
log
3
x
\lim _{x \rightarrow \infty} \frac{x^{10}}{-2 x^{10}-8 \log _{3} x}
x
→
∞
lim
−
2
x
10
−
8
lo
g
3
x
x
10
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
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Which statement best describes the limit shown below?
\newline
lim
x
→
∞
x
19
2
x
19
−
10
e
x
\lim _{x \rightarrow \infty} \frac{x^{19}}{2 x^{19}-10 e^{x}}
x
→
∞
lim
2
x
19
−
10
e
x
x
19
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
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Which statement best describes the limit shown below?
\newline
lim
x
→
∞
−
7
x
70
3
x
9
+
10
e
x
\lim _{x \rightarrow \infty} \frac{-7 x^{70}}{3 x^{9}+10 e^{x}}
x
→
∞
lim
3
x
9
+
10
e
x
−
7
x
70
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
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Which statement best describes the limit shown below?
\newline
lim
x
→
∞
x
2
+
x
68
x
68
\lim _{x \rightarrow \infty} \frac{x^{2}+x^{68}}{x^{68}}
x
→
∞
lim
x
68
x
2
+
x
68
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
Which statement best describes the limit shown below?
\newline
lim
x
→
∞
x
76
2
(
2
)
x
+
3
x
76
\lim _{x \rightarrow \infty} \frac{x^{76}}{2(2)^{x}+3 x^{76}}
x
→
∞
lim
2
(
2
)
x
+
3
x
76
x
76
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
Which statement best describes the limit shown below?
\newline
lim
x
→
∞
x
14
x
73
\lim _{x \rightarrow \infty} \frac{x^{14}}{x^{73}}
x
→
∞
lim
x
73
x
14
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
Which statement best describes the limit shown below?
\newline
lim
x
→
∞
9
x
56
−
10
e
x
−
10
x
97
\lim _{x \rightarrow \infty} \frac{9 x^{56}}{-10 e^{x}-10 x^{97}}
x
→
∞
lim
−
10
e
x
−
10
x
97
9
x
56
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
Which statement best describes the limit shown below?
\newline
lim
x
→
∞
−
10
x
13
−
2
x
2
\lim _{x \rightarrow \infty} \frac{-10 x^{13}}{-2 x^{2}}
x
→
∞
lim
−
2
x
2
−
10
x
13
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
Which statement best describes the limit shown below?
\newline
lim
x
→
∞
−
3
x
23
+
4
x
3
log
4
x
−
5
x
8
\lim _{x \rightarrow \infty} \frac{-3 x^{23}+4^{x}}{3 \log _{4} x-5 x^{8}}
x
→
∞
lim
3
lo
g
4
x
−
5
x
8
−
3
x
23
+
4
x
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
Which statement best describes the limit shown below?
\newline
lim
x
→
∞
2
x
16
−
3
x
24
−
2
x
24
+
log
5
x
\lim _{x \rightarrow \infty} \frac{2 x^{16}-3 x^{24}}{-2 x^{24}+\log _{5} x}
x
→
∞
lim
−
2
x
24
+
lo
g
5
x
2
x
16
−
3
x
24
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
Which statement best describes the limit shown below?
\newline
lim
x
→
∞
x
71
+
log
3
x
3
x
71
+
x
29
\lim _{x \rightarrow \infty} \frac{x^{71}+\log _{3} x}{3 x^{71}+x^{29}}
x
→
∞
lim
3
x
71
+
x
29
x
71
+
lo
g
3
x
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
Which statement best describes the limit shown below?
\newline
lim
x
→
∞
x
37
log
4
x
+
4
x
\lim _{x \rightarrow \infty} \frac{x^{37}}{\log _{4} x+4^{x}}
x
→
∞
lim
lo
g
4
x
+
4
x
x
37
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
Which statement best describes the limit shown below?
\newline
lim
x
→
∞
ln
x
+
10
x
20
x
88
+
e
x
\lim _{x \rightarrow \infty} \frac{\ln x+10 x^{20}}{x^{88}+e^{x}}
x
→
∞
lim
x
88
+
e
x
ln
x
+
10
x
20
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
Which statement best describes the limit shown below?
\newline
lim
x
→
∞
9
x
61
−
9
x
61
+
x
60
\lim _{x \rightarrow \infty} \frac{9 x^{61}}{-9 x^{61}+x^{60}}
x
→
∞
lim
−
9
x
61
+
x
60
9
x
61
\newline
The limit equals zero
\newline
The limit does not exist
\newline
The limit exists and does not equal zero
Get tutor help
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