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Math Problems
Algebra 2
Power rule
The graph of
y
=
40
(
1.15
)
x
y=40(1.15)^{x}
y
=
40
(
1.15
)
x
is shown. 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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Let
f
(
x
)
=
x
+
1
x
⋅
e
x
+
e
x
f(x) = \frac{x + 1}{x \cdot e^x + e^x}
f
(
x
)
=
x
⋅
e
x
+
e
x
x
+
1
when
x
≠
−
1
x \neq -1
x
=
−
1
.
Get tutor help
\newline
Find the vertical and horizontal asymptotes of the function:
\newline
f
(
x
)
=
(
2
x
−
1
)
(
3
x
+
1
)
(
x
−
2
)
(
x
+
4
)
f(x)=\frac{(2 x-1)(3 x+1)}{(x-2)(x+4)}
f
(
x
)
=
(
x
−
2
)
(
x
+
4
)
(
2
x
−
1
)
(
3
x
+
1
)
\newline
The fields below accept a list of numbers or formulas separated by semicolons (e.g.
2
;
4
;
6
2 ; 4 ; 6
2
;
4
;
6
or
x
+
1
;
x
−
1
)
x+1 ; x-1)
x
+
1
;
x
−
1
)
. The order of the list does not matter.
\newline
Vertical asymptotes:
\newline
x
=
x=
x
=
\newline
Horizontal asymptotes:
\newline
y
=
y=
y
=
\newline
Get tutor help
4
4
4
y
−
3
-3
−
3
x=
40
40
40
\newline
4
4
4
y=
3
3
3
x
−
30
-30
−
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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Simplify. Assume all variables are positive.
\newline
w
3
2
w
5
2
\frac{w^{\frac{3}{2}}}{w^{\frac{5}{2}}}
w
2
5
w
2
3
\newline
\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
______
\newline
Get tutor help
Rewrite the expression without using a negative exponent.
\newline
1
−
2
y
−
2
\frac{1}{-2y^{-2}}
−
2
y
−
2
1
\newline
Simplify your answer as much as possible.
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Which of the following equations represent functions? Assume
x
x
x
is the input and
y
y
y
is the output.
\newline
Multi-select Choices:
\newline
(A)
x
=
−
1
x = -1
x
=
−
1
\newline
(B)
y
=
x
y = x
y
=
x
\newline
(C)
y
=
1
y = 1
y
=
1
\newline
(D)
x
+
y
=
1
x + y = 1
x
+
y
=
1
\newline
(E)
x
=
0
x = 0
x
=
0
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Which of the equations are true identities?
\newline
A.
(
a
+
b
)
(
2
a
+
1
)
=
a
(
2
a
+
2
b
+
1
)
(a+b)(2 a+1)=a(2 a+2 b+1)
(
a
+
b
)
(
2
a
+
1
)
=
a
(
2
a
+
2
b
+
1
)
\newline
B.
(
n
+
2
)
2
−
n
2
=
4
(
n
+
1
)
(n+2)^{2}-n^{2}=4(n+1)
(
n
+
2
)
2
−
n
2
=
4
(
n
+
1
)
\newline
Choose
1
1
1
answer:
\newline
(A) Only A
\newline
(B) Only B
\newline
(C) Both A and B
\newline
(D) Neither A nor B
Get tutor help
Which of the equations are true identities?
\newline
A.
(
x
2
+
2
y
)
(
x
2
−
3
y
)
=
x
4
−
6
y
2
\left(x^{2}+2 y\right)\left(x^{2}-3 y\right)=x^{4}-6 y^{2}
(
x
2
+
2
y
)
(
x
2
−
3
y
)
=
x
4
−
6
y
2
\newline
B.
k
3
−
r
3
=
(
k
2
+
r
)
(
k
−
r
2
)
k^{3}-r^{3}=\left(k^{2}+r\right)\left(k-r^{2}\right)
k
3
−
r
3
=
(
k
2
+
r
)
(
k
−
r
2
)
\newline
Choose
1
1
1
answer:
\newline
(A) Only A
\newline
(B) Only B
\newline
(C) Both A and B
\newline
(D) Neither A nor B
Get tutor help
Which of the equations are true identities?
\newline
A.
n
(
n
−
2
)
(
n
+
2
)
=
n
3
−
4
n
n(n-2)(n+2)=n^{3}-4 n
n
(
n
−
2
)
(
n
+
2
)
=
n
3
−
4
n
\newline
B.
(
x
+
1
)
2
−
2
x
+
y
2
=
x
2
+
y
2
+
1
(x+1)^{2}-2 x+y^{2}=x^{2}+y^{2}+1
(
x
+
1
)
2
−
2
x
+
y
2
=
x
2
+
y
2
+
1
\newline
Choose
1
1
1
answer:
\newline
(A) Only A
\newline
(B) Only B
\newline
(C) Both A and B
\newline
(D) Neither A nor B
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How does
h
(
x
)
=
1
0
x
h(x) = 10^x
h
(
x
)
=
1
0
x
change over the interval from
x
=
5
x = 5
x
=
5
to
x
=
6
x = 6
x
=
6
?
\newline
Choices:
\newline
(A)
h
(
x
)
h(x)
h
(
x
)
increases by
10
%
10\%
10%
\newline
(B)
h
(
x
)
h(x)
h
(
x
)
increases by
900
%
900\%
900%
\newline
(C)
h
(
x
)
h(x)
h
(
x
)
increases by
10
10
10
\newline
(D)
h
(
x
)
h(x)
h
(
x
)
decreases by
10
10
10
Get tutor help
- Let
g
g
g
be a function such that
g
(
5
)
=
7
g(5)=7
g
(
5
)
=
7
and
g
′
(
5
)
=
−
2
g^{\prime}(5)=-2
g
′
(
5
)
=
−
2
.
\newline
- Let
h
h
h
be the function
h
(
x
)
=
x
h(x)=x
h
(
x
)
=
x
.
\newline
Evaluate
d
d
x
[
g
(
x
)
⋅
h
(
x
)
]
\frac{d}{d x}[g(x) \cdot h(x)]
d
x
d
[
g
(
x
)
⋅
h
(
x
)]
at
x
=
5
x=5
x
=
5
.
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Let
y
=
x
cos
(
x
)
y=\sqrt{x} \cos (x)
y
=
x
cos
(
x
)
.
\newline
Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
x
+
sin
(
x
)
-\frac{1}{\sqrt{x}}+\sin (x)
−
x
1
+
sin
(
x
)
\newline
(B)
cos
(
x
)
2
x
−
x
sin
(
x
)
\frac{\cos (x)}{2 \sqrt{x}}-\sqrt{x} \sin (x)
2
x
c
o
s
(
x
)
−
x
sin
(
x
)
\newline
(C)
−
2
x
cos
(
x
)
−
x
sin
(
x
)
-2 \sqrt{x} \cos (x)-\sqrt{x} \sin (x)
−
2
x
cos
(
x
)
−
x
sin
(
x
)
\newline
(D)
−
sin
(
x
)
x
-\frac{\sin (x)}{\sqrt{x}}
−
x
s
i
n
(
x
)
Get tutor help
Which statement is true about the value of the expression below?
\newline
(
−
2
3
)
−
2
(-2^{3})^{-2}
(
−
2
3
)
−
2
\newline
(A) It is between
0
0
0
and
1
1
1
.
\newline
(B) It is less than
−
1
-1
−
1
.
\newline
(C) It is between
−
1
-1
−
1
and
0
0
0
.
\newline
(D) It is greater than
1
1
1
.
Get tutor help
(
2
x
+
5
)
(
−
m
x
+
9
)
=
0
(2 x+5)(-m x+9)=0
(
2
x
+
5
)
(
−
m
x
+
9
)
=
0
\newline
In the given equation,
m
m
m
is a constant. If the equation has the solutions
x
=
−
5
2
x=-\frac{5}{2}
x
=
−
2
5
and
x
=
3
2
x=\frac{3}{2}
x
=
2
3
, what is the value of
m
m
m
?
\newline
□
\square
□
Get tutor help
Simplify. Assume all variables are positive.
\newline
b
7
4
b
11
4
⋅
b
7
4
\frac{b^{\frac{7}{4}}}{b^{\frac{11}{4}} \cdot b^{\frac{7}{4}}}
b
4
11
⋅
b
4
7
b
4
7
\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
______
Get tutor help
Simplify. Assume all variables are positive.
\newline
b
8
3
b
4
3
⋅
b
2
3
\frac{b^{\frac{8}{3}}}{b^{\frac{4}{3}} \cdot b^{\frac{2}{3}}}
b
3
4
⋅
b
3
2
b
3
8
\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
______
Get tutor help
Simplify. Assume all variables are positive.
\newline
u
11
4
u
7
4
⋅
u
5
4
\frac{u^{\frac{11}{4}}}{u^{\frac{7}{4}} \cdot u^{\frac{5}{4}}}
u
4
7
⋅
u
4
5
u
4
11
\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
______
Get tutor help
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
______
Get tutor help
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
______
Get tutor help
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.
Get tutor help
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.
Get tutor help
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?
Get tutor help
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:
Get tutor help
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:
Get tutor help
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:
Get tutor help
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:
Get tutor help
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:
Get tutor help
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:
Get tutor help
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:
Get tutor help
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:
Get tutor help
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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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
Get tutor help
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
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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
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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
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2
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