Question 1

Find $ \lim _{x \rightarrow \infty} \frac{1-7 x}{1+9 x} $ (if it exists).

Accepted Answer

A)  0 
B)  $ -\frac{7}{9} $ 
C)  $ \frac{7}{9} $ 
D)  $ -\infty $ 
E)  limit does not exist 
A)  0 
B)  $ -\frac{7}{9} $ 
C)  $ \frac{7}{9} $ 
D)  $ -\infty $ 
E)  limit does not exist

Question 2

Find
$$\lim _ { x \rightarrow 7 } \frac { x - 8 } { x ^ { 2 } + 12 x + 27 }$$
by direct substitution.

Accepted Answer

A)  $\frac { 1 } { 160 }$ 
B)  $- \frac { 1 } { 160 }$ 
C)  0 
D)  $\infty$ 
E)  limit does not exist 
A)  $\frac { 1 } { 160 }$ 
B)  $- \frac { 1 } { 160 }$ 
C)  0 
D)  $\infty$ 
E)  limit does not exist

Question 3

Find
$$\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$$
for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 6 x ^ { 2 } }$.

Accepted Answer

A)  $\frac { \sqrt { 11 } } { 54 } - 81$ 
B)  $\frac { \sqrt { 3 } } { 6 } - 81$ 
C)  $\frac { 3 \sqrt { 11 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 6 } + 81$ 
E)  limit does not exist 
A)  $\frac { \sqrt { 11 } } { 54 } - 81$ 
B)  $\frac { \sqrt { 3 } } { 6 } - 81$ 
C)  $\frac { 3 \sqrt { 11 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 6 } + 81$ 
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

Find $ \lim _{t \rightarrow 3} \frac{t^{3}-27}{t-3} $.

Accepted Answer

A)  9 
B)  18 
C)  27 
D)  3 
E)  limit does not exist 
A)  9 
B)  18 
C)  27 
D)  3 
E)  limit does not exist

Question 6

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$

Accepted Answer

A)  14 
B)  $\frac { 7 } { 2 }$ 
C)  $\frac { \sqrt { 7 } } { 2 }$ 
D)  $2 \sqrt { 7 }$ 
E)  limit does not exist 
A)  14 
B)  $\frac { 7 } { 2 }$ 
C)  $\frac { \sqrt { 7 } } { 2 }$ 
D)  $2 \sqrt { 7 }$ 
E)  limit does not exist

Question 7

Approximate the area of the indicated region under the given curve using five rectangles.
$$f ( x ) = 5 - x ^ { 2 }$$

Accepted Answer

A)  $9.28$ 
B)  $4.64$ 
C)  $5.48$ 
D)  $4.83$ 
E)  $9.25$ 
A)  $9.28$ 
B)  $4.64$ 
C)  $5.48$ 
D)  $4.83$ 
E)  $9.25$

Question 8

Find $\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }$.

Accepted Answer

A)  $\frac { 1 } { 18 }$ 
B)  $\frac { 1 } { 9 }$ 
C)  $\frac { 9 } { 2 }$ 
D)  $\frac { 2 } { 9 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 18 }$ 
B)  $\frac { 1 } { 9 }$ 
C)  $\frac { 9 } { 2 }$ 
D)  $\frac { 2 } { 9 }$ 
E)  limit does not exist

Question 9

If $ f(x)=-2 x^{2}+x $, find the following limit, if it exists.
$\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

Accepted Answer

A)  $ -4 x+1 $ 
B)  $ -2 x+1 $ 
C)  $ 2 x+1 $ 
D)  $ -4 x $ 
E)  The limit does not exist. 
A)  $ -4 x+1 $ 
B)  $ -2 x+1 $ 
C)  $ 2 x+1 $ 
D)  $ -4 x $ 
E)  The limit does not exist.

Question 10

$\text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. }$

Accepted Answer

A)  3 
B)  $ \frac{1}{2} $ 
C)   -3 
D)  0 
E)  The slope is undefined. 
A)  3 
B)  $ \frac{1}{2} $ 
C)   -3 
D)  0 
E)  The slope is undefined.

Question 11

$\text { Rewrite } \sum_{i=1}^{n}\left[\frac{12}{n}+\left(\frac{11 i}{n^{2}}\right)\right]\left(\frac{8 i}{n}\right) \text { as a rational function } S(n) \text { and find } \lim _{n \rightarrow \infty} S(n) \text {. }$

Accepted Answer

A) $S(n)=\frac{4[36 n+11(n+1)]}{3 n^{2}}, \lim _{n \rightarrow \infty} S(n)=\frac{232}{3}$ 
B) $S(n)=\frac{4[36 n(n+1)+11(n+1)(2 n+1)]}{3 n^{2}}, \lim _{n \rightarrow \infty} S(n)=\frac{232}{3}$ 
C) $S(n)=\frac{4[36 n(n+1)+11(n+1)(2 n+1)]}{3 n^{2}}, \lim _{n \rightarrow \infty} S(n)=\frac{5}{6}$ 
D) $S(n)=\frac{4[36+11 n]}{3 n}, \lim _{n \rightarrow \infty} S(n)=\frac{232}{3}$ 
E) $S(n)=\frac{4[36 n(n+1)+11(n+1)(2 n+1)]}{3 n^{2}}, \text { limit does not exist }$ 
A) $S(n)=\frac{4[36 n+11(n+1)]}{3 n^{2}}, \lim _{n \rightarrow \infty} S(n)=\frac{232}{3}$ 
B) $S(n)=\frac{4[36 n(n+1)+11(n+1)(2 n+1)]}{3 n^{2}}, \lim _{n \rightarrow \infty} S(n)=\frac{232}{3}$ 
C) $S(n)=\frac{4[36 n(n+1)+11(n+1)(2 n+1)]}{3 n^{2}}, \lim _{n \rightarrow \infty} S(n)=\frac{5}{6}$ 
D) $S(n)=\frac{4[36+11 n]}{3 n}, \lim _{n \rightarrow \infty} S(n)=\frac{232}{3}$ 
E) $S(n)=\frac{4[36 n(n+1)+11(n+1)(2 n+1)]}{3 n^{2}}, \text { limit does not exist }$

Question 12

Complete the table and use the result to estimate $\lim _ { x \rightarrow 5 } ( 2 x + 4 )$ numerically.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & 4.9 & 4.99 & 4.999 & 5 & 5.001 & 5.01 & 5.1 \
\hline f(x) & & & & ? & & & \
\hline
\end{array}$

Accepted Answer

A)  9 
B)  10 
C)  14 
D)  6 
E)  limit does not exist 
A)  9 
B)  10 
C)  14 
D)  6 
E)  limit does not exist

Question 13

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 6 x } { 1 + 8 x }$ (if it exists).

Accepted Answer

A)  0 
B)  $- \frac { 3 } { 4 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $- \infty$ 
E)  limit does not exist 
A)  0 
B)  $- \frac { 3 } { 4 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $- \infty$ 
E)  limit does not exist

Question 14

Determine any points on the graph of the following function at which the tangent line is horizontal. $f(x)=x^{2}+8 x-8$

Accepted Answer

A)  $ (4,40) $ 
B)  $ (4,0) $ 
C)  $ (-4,0) $ 
D)  $ (0,-8) $ 
E)  $ \quad(-4,-24) $ 
A)  $ (4,40) $ 
B)  $ (4,0) $ 
C)  $ (-4,0) $ 
D)  $ (0,-8) $ 
E)  $ \quad(-4,-24) $

Question 15

Use the limit process to find the slope of the graph of $\sqrt { x + 12 }$ at $( 4,4 )$.

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  4 
C)  $\infty$ 
D)  $\frac { 1 } { 8 }$ 
E)  the slope is undefined at this point 
A)  $\frac { 1 } { 4 }$ 
B)  4 
C)  $\infty$ 
D)  $\frac { 1 } { 8 }$ 
E)  the slope is undefined at this point

Question 16

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the $x$-axis on the interval $[ 1,8 ]$.

Accepted Answer

A)  $\frac { 1087 } { 2 }$ 
B)  $\frac { 2023 } { 24 }$ 
C)  $\frac { 191 } { 2 }$ 
D)  574 
E)  $\frac { 1267 } { 24 }$ 
A)  $\frac { 1087 } { 2 }$ 
B)  $\frac { 2023 } { 24 }$ 
C)  $\frac { 191 } { 2 }$ 
D)  574 
E)  $\frac { 1267 } { 24 }$

Question 17

The cost function for a certain model of a digital camera given by $C = 12.00 x + 48,450$, where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 100$. Round your answer to the nearest cent.

Accepted Answer

A)  $\$ 49,650.00$ 
B)  $\$ 484.50$ 
C)  $\$ 12.00$ 
D)  $\$ 496.50$ 
E)  $\$ 484.62$ 
A)  $\$ 49,650.00$ 
B)  $\$ 484.50$ 
C)  $\$ 12.00$ 
D)  $\$ 496.50$ 
E)  $\$ 484.62$

Question 18

Use the derivative of $f ( x ) = 3 x ^ { 3 } + 9 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Accepted Answer

A)  $( 1,12 )$ 
B)  $( 1,12 )$ and $( - 1 , - 12 )$ 
C)  $( 3,108 )$ and $( - 3,108 )$ 
D)  $( 0,0 )$ 
E)  $f ( x )$ has no points with a horizontal tangent line. 
A)  $( 1,12 )$ 
B)  $( 1,12 )$ and $( - 1 , - 12 )$ 
C)  $( 3,108 )$ and $( - 3,108 )$ 
D)  $( 0,0 )$ 
E)  $f ( x )$ has no points with a horizontal tangent line.

Question 19

If $f ( x ) = - 3 x ^ { 2 } - 4 x$, find the following limit, if it exists.
$$\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$$

Accepted Answer

A)  $- 6 x - 4$ 
B)  $- 3 x - 4$ 
C)  $2 x - 4$ 
D)  $- 6 x$ 
E)  The limit does not exist. 
A)  $- 6 x - 4$ 
B)  $- 3 x - 4$ 
C)  $2 x - 4$ 
D)  $- 6 x$ 
E)  The limit does not exist.

Question 20

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

Find $\lim _{x \rightarrow \infty} \frac{1-7 x}{1+9 x}$ (if it exists).

Find $\lim _ { x \rightarrow 7 } \frac { x - 8 } { x ^ { 2 } + 12 x + 27 }$ by direct substitution.

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 6 x ^ { 2 } }$ .

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).

Find $\lim _{t \rightarrow 3} \frac{t^{3}-27}{t-3}$ .

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 5 - x ^ { 2 }$ $Approximate the area of the indicated region under the given curve using five rectangles. f ( x ) = 5 - x ^ { 2 }$

Find $\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }$ .

If $f(x)=-2 x^{2}+x$ , find the following limit, if it exists. $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

$\text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. }$ $\text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. }$

$\text { Rewrite } \sum_{i=1}^{n}\left[\frac{12}{n}+\left(\frac{11 i}{n^{2}}\right)\right]\left(\frac{8 i}{n}\right) \text { as a rational function } S(n) \text { and find } \lim _{n \rightarrow \infty} S(n) \text {. }$

Complete the table and use the result to estimate $\lim _ { x \rightarrow 5 } ( 2 x + 4 )$ numerically. x 4.9 4.99 4.999 5 5.001 5.01 5.1 f(x) ?

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 6 x } { 1 + 8 x }$ (if it exists).

Determine any points on the graph of the following function at which the tangent line is horizontal. $f(x)=x^{2}+8 x-8$

Use the limit process to find the slope of the graph of $\sqrt { x + 12 }$ at $( 4,4 )$ .

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the $x$ -axis on the interval $[ 1,8 ]$ .

The cost function for a certain model of a digital camera given by $C = 12.00 x + 48,450$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 100$ . Round your answer to the nearest cent.

Use the derivative of $f ( x ) = 3 x ^ { 3 } + 9 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

If $f ( x ) = - 3 x ^ { 2 } - 4 x$ , find the following limit, if it exists. $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$

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

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

Filters

Exam 11: Calculus Practice Problems

Find lim⁡x→∞1−7x1+9x \lim _{x \rightarrow \infty} \frac{1-7 x}{1+9 x} limx→∞​1+9x1−7x​ (if it exists).

Find lim⁡x→7x−8x2+12x+27\lim _ { x \rightarrow 7 } \frac { x - 8 } { x ^ { 2 } + 12 x + 27 }limx→7​x2+12x+27x−8​ by direct substitution.

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+26x2g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 6 x ^ { 2 } }g(x)=6x2x2+2​​ .

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

Find lim⁡t→3t3−27t−3 \lim _{t \rightarrow 3} \frac{t^{3}-27}{t-3} limt→3​t−3t3−27​ .

Find lim⁡x→0−xx+7−7\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }limx→0−​x+7​−7​x​

Approximate the area of the indicated region under the given curve using five rectangles. f(x)=5−x2f ( x ) = 5 - x ^ { 2 }f(x)=5−x2

Find lim⁡x→9−x−9x2−81\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }limx→9−​x2−81x−9​ .

If f(x)=−2x2+x f(x)=-2 x^{2}+x f(x)=−2x2+x , find the following limit, if it exists. 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)​

Use the figure below to approximate the slope of the curve at the point (x,y). \text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. } Use the figure below to approximate the slope of the curve at the point (x,y).

Complete the table and use the result to estimate lim⁡x→5(2x+4)\lim _ { x \rightarrow 5 } ( 2 x + 4 )limx→5​(2x+4) numerically. x 4.9 4.99 4.999 5 5.001 5.01 5.1 f(x) ?

Find lim⁡x→∞1−6x1+8x\lim _ { x \rightarrow \infty } \frac { 1 - 6 x } { 1 + 8 x }limx→∞​1+8x1−6x​ (if it exists).

Determine any points on the graph of the following function at which the tangent line is horizontal. f(x)=x2+8x−8f(x)=x^{2}+8 x-8f(x)=x2+8x−8

Use the limit process to find the slope of the graph of x+12\sqrt { x + 12 }x+12​ at (4,4)( 4,4 )(4,4) .

Use the limit process to find the area of the region between f(x)=18(x2+8x)f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)f(x)=81​(x2+8x) and the xxx -axis on the interval [1,8][ 1,8 ][1,8] .

The cost function for a certain model of a digital camera given by C=12.00x+48,450C = 12.00 x + 48,450C=12.00x+48,450 , where CCC is the cost (in dollars) and xxx is the number of cameras produced. Find the average cost per unit when x=100x = 100x=100 . Round your answer to the nearest cent.

Use the derivative of f(x)=3x3+9xf ( x ) = 3 x ^ { 3 } + 9 xf(x)=3x3+9x to determine any points on the graph of f(x)f ( x )f(x) at which the tangent line is horizontal.

If f(x)=−3x2−4xf ( x ) = - 3 x ^ { 2 } - 4 xf(x)=−3x2−4x , find the following limit, if it exists. 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)​

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​ .

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

Filters

Find $\lim _{x \rightarrow \infty} \frac{1-7 x}{1+9 x}$ (if it exists).

Find $\lim _ { x \rightarrow 7 } \frac { x - 8 } { x ^ { 2 } + 12 x + 27 }$ by direct substitution.

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 6 x ^ { 2 } }$ .

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).

Find $\lim _{t \rightarrow 3} \frac{t^{3}-27}{t-3}$ .

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 5 - x ^ { 2 }$ $Approximate the area of the indicated region under the given curve using five rectangles. f ( x ) = 5 - x ^ { 2 }$

Find $\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }$ .

If $f(x)=-2 x^{2}+x$ , find the following limit, if it exists. $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

$\text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. }$ $\text { Use the figure below to approximate the slope of the curve at the point }(x, y) \text {. }$

Complete the table and use the result to estimate $\lim _ { x \rightarrow 5 } ( 2 x + 4 )$ numerically. x 4.9 4.99 4.999 5 5.001 5.01 5.1 f(x) ?

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 6 x } { 1 + 8 x }$ (if it exists).

Determine any points on the graph of the following function at which the tangent line is horizontal. $f(x)=x^{2}+8 x-8$

Use the limit process to find the slope of the graph of $\sqrt { x + 12 }$ at $( 4,4 )$ .

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the $x$ -axis on the interval $[ 1,8 ]$ .

The cost function for a certain model of a digital camera given by $C = 12.00 x + 48,450$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 100$ . Round your answer to the nearest cent.

Use the derivative of $f ( x ) = 3 x ^ { 3 } + 9 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

If $f ( x ) = - 3 x ^ { 2 } - 4 x$ , find the following limit, if it exists. $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$

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