Question 1

Find
$$\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$$
for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 5 x ^ { 2 } }$.

Accepted Answer

A)  $\frac { \sqrt { 11 } } { 45 } - 81$ 
B)  $\frac { \sqrt { 3 } } { 5 } - 81$ 
C)  $\frac { 9 \sqrt { 11 } } { 5 }$ 
D)  $\frac { \sqrt { 3 } } { 5 } + 81$ 
E)  limit does not exist 
A)  $\frac { \sqrt { 11 } } { 45 } - 81$ 
B)  $\frac { \sqrt { 3 } } { 5 } - 81$ 
C)  $\frac { 9 \sqrt { 11 } } { 5 }$ 
D)  $\frac { \sqrt { 3 } } { 5 } + 81$ 
E)  limit does not exist

Question 2

Find 
$\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 4 - 6 x$.

Accepted Answer

A)  6 
B)  10 
C)  $- 6$ 
D)  4 
E)  limit does not exist 
A)  6 
B)  10 
C)  $- 6$ 
D)  4 
E)  limit does not exist

Question 3

Use the graph to find
$\lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right)$

Accepted Answer

A)  $\frac { 3 \sqrt { 3 } } { 2 }$ 
B)  $\infty$ 
C)  0 
D)  3 
E)  limit does not exist 
A)  $\frac { 3 \sqrt { 3 } } { 2 }$ 
B)  $\infty$ 
C)  0 
D)  3 
E)  limit does not exist

Question 4

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 5 n ^ { 3 } } { n ^ { 3 } + 3 }$ (assume $n$ begins with 1).

Accepted Answer

A)  4 
B)  5 
C)  $\frac { 23 } { 6 }$ 
D)  $\frac { 32 } { 5 }$ 
E)  the sequence diverges 
A)  4 
B)  5 
C)  $\frac { 23 } { 6 }$ 
D)  $\frac { 32 } { 5 }$ 
E)  the sequence diverges

Question 5

Complete the table and use the result to estimate
$$\lim _ { x \rightarrow - 3 } \frac { x + 3 } { x ^ { 2 } - x - 12 }$$
numerically.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & -3.1 & -3.01 & -3.001 & -3 & -2.999 & -2.99 & -2.9 \
\hline f(x) & & & & ? & & & \
\hline
\end{array}$

Accepted Answer

A)  $- \frac { 1 } { 7 }$ 
B)  7 
C)  $\infty$ 
D)  $- 7$ 
E)  limit does not exist 
A)  $- \frac { 1 } { 7 }$ 
B)  7 
C)  $\infty$ 
D)  $- 7$ 
E)  limit does not exist

Question 6

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.
$$f ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } , \quad f ^ { \prime } ( x ) = 12 x ^ { 3 } - 12 x$$

Accepted Answer

A)  $( - 1 , - 3 ) , ( 0,0 ) , ( 1 , - 3 )$ 
B)  $( - 1,3 ) , ( 1,12 )$ 
C)  $( - 1 , - 3 ) , ( 1 , - 3 )$ 
D)  $( 0,0 ) , ( 1,0 )$ 
E)  $( 0,0 ) , ( 1 , - 3 )$ 
A)  $( - 1 , - 3 ) , ( 0,0 ) , ( 1 , - 3 )$ 
B)  $( - 1,3 ) , ( 1,12 )$ 
C)  $( - 1 , - 3 ) , ( 1 , - 3 )$ 
D)  $( 0,0 ) , ( 1,0 )$ 
E)  $( 0,0 ) , ( 1 , - 3 )$

Question 7

Determine any point(s) of intersection of the following equations algebraically.
$$\begin{array} { l } 
28 = 19 x - y \
- x ^ { 2 } + y = 8 x
\end{array}$$

Accepted Answer

A)  $( 5,65 )$,(8, 128) 
B)  $( 1 , - 9 )$, $( 4,48 )$ 
C)  $( 3,29 )$ $( 6,86 )$ 
D)  $( 2,10 )$,(5, 67) 
E)  $( 4,48 )$$( 7,105 )$ 
A)  $( 5,65 )$,(8, 128) 
B)  $( 1 , - 9 )$, $( 4,48 )$ 
C)  $( 3,29 )$ $( 6,86 )$ 
D)  $( 2,10 )$,(5, 67) 
E)  $( 4,48 )$$( 7,105 )$

Question 8

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 8 - x$.

Accepted Answer

A)  1 
B)  9 
C)  $- 1$ 
D)  8 
E)  limit does not exist 
A)  1 
B)  9 
C)  $- 1$ 
D)  8 
E)  limit does not exist

Question 9

Sketch the graph of the equation $2 x ^ { 3 } - 4$.

Accepted Answer

A) 
  
B) 
  
C) 
  
D)  
  
E) 
  
A) 
  
B) 
  
C) 
  
D)  
  
E)

Question 10

Solve the equation
$$\frac { x } { 9 } - \frac { x } { 27 } = 8$$

Accepted Answer

A)  $x = 216$ 
B)  $x = 36$ 
C)  $x = 14$ 
D)  $x = 12$ 
E)  $x = 108$ 
A)  $x = 216$ 
B)  $x = 36$ 
C)  $x = 14$ 
D)  $x = 12$ 
E)  $x = 108$

Question 11

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 7 }$.

Accepted Answer

A)  $\frac { 1 } { 7 \sqrt { x - 7 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 7 } }$ 
C)  $\frac { \sqrt { x - 7 } } { 2 }$ 
D)  $\sqrt { x - 7 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 7 \sqrt { x - 7 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 7 } }$ 
C)  $\frac { \sqrt { x - 7 } } { 2 }$ 
D)  $\sqrt { x - 7 }$ 
E)  limit does not exist

Question 12

The cost function for a certain graphing calculator is given by $C = 11.35 x + 53,000$ where $C$ is in dollars and $x$ is the number of calculators produced.
a. Write a model for the average cost per unit produced.
b. Find the average cost per unit when $x = 900$.
c. Determine the limit of the average cost function as $x$ approaches $\infty$.

Accepted Answer

The answer of The cost function for a certain graphing...

Question 13

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 6 }$.

Accepted Answer

A)  $\frac { 1 } { 6 \sqrt { x - 6 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 6 } }$ 
C)  $\frac { \sqrt { x - 6 } } { 2 }$ 
D)  $\sqrt { x - 6 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 6 \sqrt { x - 6 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 6 } }$ 
C)  $\frac { \sqrt { x - 6 } } { 2 }$ 
D)  $\sqrt { x - 6 }$ 
E)  limit does not exist

Question 14

Find the following limit of the sequence as $n$ approaches infinity, if it exists.
$$a _ { n } = \frac { 1 } { n } \left( n + \frac { 3 } { n } \left[ \frac { n ( n + 1 ) } { 5 } \right] \right)$$

Accepted Answer

A)  $\frac { 3 } { 5 }$ 
B)  $\frac { 13 } { 5 }$ 
C)  $\frac { 8 } { 5 }$ 
D)  $\frac { 9 } { 5 }$ 
E)  The limit does not exist. 
A)  $\frac { 3 } { 5 }$ 
B)  $\frac { 13 } { 5 }$ 
C)  $\frac { 8 } { 5 }$ 
D)  $\frac { 9 } { 5 }$ 
E)  The limit does not exist.

Question 15

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 15 } - \sqrt { 15 } }$

Accepted Answer

A)  30 
B)  $\frac { 15 } { 2 }$ 
C)  $\frac { \sqrt { 15 } } { 2 }$ 
D)  $2 \sqrt { 15 }$ 
E)  limit does not exist 
A)  30 
B)  $\frac { 15 } { 2 }$ 
C)  $\frac { \sqrt { 15 } } { 2 }$ 
D)  $2 \sqrt { 15 }$ 
E)  limit does not exist

Question 16

Find the following limit, if it exists.
$$\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }$$

Accepted Answer

A)  $- 6$ 
B)  12 
C)  6 
D)  $- 12$ 
E)  The limit does not exist. 
A)  $- 6$ 
B)  12 
C)  6 
D)  $- 12$ 
E)  The limit does not exist.

Question 17

Find the $x$-intercept of the graph of the equation $y = 2 \sqrt { x - 4 }$.

Accepted Answer

A)  $( 0 , - 4 )$ 
B)  $( 0,4 )$ 
C)  $( 4,0 )$ 
D)  $( - 4,0 )$ 
E)  $( 0 , - 8 )$ 
A)  $( 0 , - 4 )$ 
B)  $( 0,4 )$ 
C)  $( 4,0 )$ 
D)  $( - 4,0 )$ 
E)  $( 0 , - 8 )$

Question 18

Estimate the following limit numerically, if it exists.
$$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }$$

Accepted Answer

A)  $\frac { 1 } { 8 }$ 
B)  0 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$ 
E)  The limit does not exist. 
A)  $\frac { 1 } { 8 }$ 
B)  0 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$ 
E)  The limit does not exist.

Question 19

Use the limit process to find the slope of the graph of $6 x - 4 x ^ { 2 }$ at $( 7 , - 154 )$.

Accepted Answer

A)  $- 2$ 
B)  62 
C)  34 
D)  56 
E)  $- 50$ 
A)  $- 2$ 
B)  62 
C)  34 
D)  56 
E)  $- 50$

Question 20

Find the $y$-intercept of the graph of the equation $y = 3 x + 15$.

Accepted Answer

A)  $( 0,3 )$ 
B)  $( 15,0 )$ 
C)  $( - 5,0 )$ 
D)  $( 0,15 )$ 
E)  $( 0 , - 5 )$ 
A)  $( 0,3 )$ 
B)  $( 15,0 )$ 
C)  $( - 5,0 )$ 
D)  $( 0,15 )$ 
E)  $( 0 , - 5 )$

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 5 x ^ { 2 } }$ .

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 4 - 6 x$ .

Use the graph to find $\lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right)$ $Use the graph to find \lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right)$

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 5 n ^ { 3 } } { n ^ { 3 } + 3 }$ (assume $n$ begins with 1).

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 3 } \frac { x + 3 } { x ^ { 2 } - x - 12 }$ numerically. x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9 f(x) ?

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } , \quad f ^ { \prime } ( x ) = 12 x ^ { 3 } - 12 x$

Determine any point(s) of intersection of the following equations algebraically. 28=19x-y -+y=8x

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 8 - x$ .

Sketch the graph of the equation $2 x ^ { 3 } - 4$ .

Solve the equation $\frac { x } { 9 } - \frac { x } { 27 } = 8$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 7 }$ .

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 6 }$ .

Find the following limit of the sequence as $n$ approaches infinity, if it exists. $a _ { n } = \frac { 1 } { n } \left( n + \frac { 3 } { n } \left[ \frac { n ( n + 1 ) } { 5 } \right] \right)$

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 15 } - \sqrt { 15 } }$

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }$

Find the $x$ -intercept of the graph of the equation $y = 2 \sqrt { x - 4 }$ .

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }$

Use the limit process to find the slope of the graph of $6 x - 4 x ^ { 2 }$ at $( 7 , - 154 )$ .

Find the $y$ -intercept of the graph of the equation $y = 3 x + 15$ .

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

Exam 12: Review of Graphs, Equations, and Inequalities

Find lim⁡x→3[g(x)−f(x)]\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]limx→3​[g(x)−f(x)] for f(x)=3x3f ( x ) = 3 x ^ { 3 }f(x)=3x3 and g(x)=x2+25x2g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 5 x ^ { 2 } }g(x)=5x2x2+2​​ .

Find lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​ for f(x)=4−6xf ( x ) = 4 - 6 xf(x)=4−6x .

Use the graph to find lim⁡x→03sin⁡(π3x)\lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right)limx→0​3sin(3xπ​)

Use the first six terms to predict the limit of the sequence an=5n3n3+3a _ { n } = \frac { 5 n ^ { 3 } } { n ^ { 3 } + 3 }an​=n3+35n3​ (assume nnn begins with 1).

Complete the table and use the result to estimate lim⁡x→−3x+3x2−x−12\lim _ { x \rightarrow - 3 } \frac { x + 3 } { x ^ { 2 } - x - 12 }limx→−3​x2−x−12x+3​ numerically. x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9 f(x) ?

Use the function below and its derivative to determine any points on the graph of fff at which the tangent line is horizontal. f(x)=3x4−6x2,f′(x)=12x3−12xf ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } , \quad f ^ { \prime } ( x ) = 12 x ^ { 3 } - 12 xf(x)=3x4−6x2,f′(x)=12x3−12x

Determine any point(s) of intersection of the following equations algebraically. 28=19x-y -+y=8x

Find lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​ for f(x)=8−xf ( x ) = 8 - xf(x)=8−x .

Sketch the graph of the equation 2x3−42 x ^ { 3 } - 42x3−4 .

Solve the equation x9−x27=8\frac { x } { 9 } - \frac { x } { 27 } = 89x​−27x​=8

Find lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​ for f(x)=x−7f ( x ) = \sqrt { x - 7 }f(x)=x−7​ .

Find lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​ for f(x)=x−6f ( x ) = \sqrt { x - 6 }f(x)=x−6​ .

Find the following limit of the sequence as nnn approaches infinity, if it exists. an=1n(n+3n[n(n+1)5])a _ { n } = \frac { 1 } { n } \left( n + \frac { 3 } { n } \left[ \frac { n ( n + 1 ) } { 5 } \right] \right)an​=n1​(n+n3​[5n(n+1)​])

Find lim⁡x→0−xx+15−15\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 15 } - \sqrt { 15 } }limx→0−​x+15​−15​x​

Find the following limit, if it exists. lim⁡x→−39−6x−3x23+x\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }limx→−3​3+x9−6x−3x2​

Find the xxx -intercept of the graph of the equation y=2x−4y = 2 \sqrt { x - 4 }y=2x−4​ .

Estimate the following limit numerically, if it exists. lim⁡x→1x−1x2+6x−7\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }limx→1​x2+6x−7x−1​

Use the limit process to find the slope of the graph of 6x−4x26 x - 4 x ^ { 2 }6x−4x2 at (7,−154)( 7 , - 154 )(7,−154) .

Find the yyy -intercept of the graph of the equation y=3x+15y = 3 x + 15y=3x+15 .

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 5 x ^ { 2 } }$ .

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 4 - 6 x$ .

Use the graph to find $\lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right)$ $Use the graph to find \lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right)$

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 5 n ^ { 3 } } { n ^ { 3 } + 3 }$ (assume $n$ begins with 1).

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 3 } \frac { x + 3 } { x ^ { 2 } - x - 12 }$ numerically. x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9 f(x) ?

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } , \quad f ^ { \prime } ( x ) = 12 x ^ { 3 } - 12 x$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 8 - x$ .

Sketch the graph of the equation $2 x ^ { 3 } - 4$ .

Solve the equation $\frac { x } { 9 } - \frac { x } { 27 } = 8$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 7 }$ .

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 6 }$ .

Find the following limit of the sequence as $n$ approaches infinity, if it exists. $a _ { n } = \frac { 1 } { n } \left( n + \frac { 3 } { n } \left[ \frac { n ( n + 1 ) } { 5 } \right] \right)$

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 15 } - \sqrt { 15 } }$

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }$

Find the $x$ -intercept of the graph of the equation $y = 2 \sqrt { x - 4 }$ .

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 6 x - 7 }$

Use the limit process to find the slope of the graph of $6 x - 4 x ^ { 2 }$ at $( 7 , - 154 )$ .

Find the $y$ -intercept of the graph of the equation $y = 3 x + 15$ .