Question 1

Find the following limit if it exists: $\lim \ln ( x - 3 )$. Use $\pm \infty$ when appropriate.
$$x \rightarrow 3 ^ { + }$$

Accepted Answer

A)  $\infty$ 
B)  3 
C)  1 
D)  $- \infty$ 
E)  does not exist 
A)  $\infty$ 
B)  3 
C)  1 
D)  $- \infty$ 
E)  does not exist D

Question 2

A sphere has a volume of 4.76 cubic inches. What is the radius of the sphere? Round your answer to four decimal places.

Accepted Answer

A)  $1.0435$ inches 
B)  $1.6565$ inches 
C)  $1.0660$ inches 
D)  $2.1320$ inches 
E)  $1.9335$ inches 
A)  $1.0435$ inches 
B)  $1.6565$ inches 
C)  $1.0660$ inches 
D)  $2.1320$ inches 
E)  $1.9335$ inches A

Question 3

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 0 } \frac { \sin ^ { 3 } x } { x ^ { 3 } }$$\begin{array}{|c|c|c|c|c|c|c|}
\hline x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \
\hline f(x) & & & & & & \
\hline
\end{array}$

Accepted Answer

A)  $- 0.5$ 
B)  0 
C)  1 
D)  $0.5$ 
E)  $- 1$ 
A)  $- 0.5$ 
B)  0 
C)  1 
D)  $0.5$ 
E)  $- 1$ C

Question 4

Determine the limit (if it exists). $$\lim _ { x \rightarrow 0 } \frac { 12 ( 1 - \cos x ) } { x ^ { 2 } }$$

Accepted Answer

A)  6 
B)  48 
C)  10 
D)  24 
E)  does not exist 
A)  6 
B)  48 
C)  10 
D)  24 
E)  does not exist

Question 5

$\text { Consider the function } f ( x ) = 6 x - x ^ { 2 } \text { and the point } P ( 2,8 ) \text { on the graph of } f \text {. }$ Graph $f$ and the secant line passing through $P ( 2,8 )$ and $Q ( x , f ( x ) )$ for $x = 3$.

Accepted Answer

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

Question 6

Find the limit. $\lim _ { x \rightarrow - 4 } 9 x ^ { 2 } + 36 x$

Accepted Answer

A)  108 
B)  $- 108$ 
C)  288 
D)  $- 288$ 
E)  0 
A)  108 
B)  $- 108$ 
C)  288 
D)  $- 288$ 
E)  0

Question 7

$\text { Suppose that } \lim _ { x \rightarrow c } f ( x ) = - 11 \text { and } \lim _ { x \rightarrow c } g ( x ) = - 3 \text {. Find the following limit. }$ $\lim _ { x \rightarrow c } [ f ( x ) - g ( x ) ]$

Accepted Answer

A)  $- 11$ 
B)  $- 8$ 
C)  33 
D)  $- 14$ 
E)  0 
A)  $- 11$ 
B)  $- 8$ 
C)  33 
D)  $- 14$ 
E)  0

Question 8

$\text { Let } f ( x ) = - x ^ { 2 } - 5 \text { and } g ( x ) = 2 x \text {. Find the limit }$ $\lim _ { x \rightarrow - 2 } g ( f ( x ) )$

Accepted Answer

A)  $- 18$ 
B)  25 
C)  21 
D)  8 
E)  9 
A)  $- 18$ 
B)  25 
C)  21 
D)  8 
E)  9

Question 9

$\text { Find the } x \text {-values (if any) at which the function } f ( x ) = \frac { x } { x ^ { 2 } - 100 } \text { is not continuous. }$ Which of the discontinuities are removable?

Accepted Answer

A)  10 and $- 10$, removable 
B)  discontinuous everywhere 
C)  continuous everywhere 
D)  10 and $- 10$, not removable 
E)  0 , removable 
A)  10 and $- 10$, removable 
B)  discontinuous everywhere 
C)  continuous everywhere 
D)  10 and $- 10$, not removable 
E)  0 , removable

Question 10

Find the limit (if it exists). $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$, where $f ( x ) = \left\{ \begin{array} { c c } x ^ { 3 } + 10 , & x < 1 \ x + 10 , & x \geq 1 \end{array} \right.$

Accepted Answer

A)  Limit does not exist. 
B)  0 
C)  10 
D)  11 
E)  30 
A)  Limit does not exist. 
B)  0 
C)  10 
D)  11 
E)  30

Question 11

Find the limit. $\lim _ { x \rightarrow 2 } \cos \frac { \pi x } { 3 }$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
E)  0 
A)  $\frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
E)  0

Question 12

Find the limit (if it exists). $$\lim _ { x \rightarrow 5 } \frac { \sqrt { x + 4 } - 3 } { x - 5 }$$

Accepted Answer

A)  6 
B)  1 
C)  0 
D)  $\frac { 1 } { 6 }$ 
E)  Limit does not exist. 
A)  6 
B)  1 
C)  0 
D)  $\frac { 1 } { 6 }$ 
E)  Limit does not exist.

Question 13

$\text { Let } f ( x ) = \left\{ \begin{array} { l l } 
4 - x , & x \neq 1 \
0 , & x = 1
\end{array} \right. \text {. }$ Determine the following limit. (Hint: Use the graph to calculate the limit.)
$\lim _ { x \rightarrow 1 } f ( x )$

Accepted Answer

A)  5 
B)  4 
C)  3 
D)  0 
E)  does not exist 
A)  5 
B)  4 
C)  3 
D)  0 
E)  does not exist

Question 14

Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the Problem seems to require calculus, use a graphical or numerical approach to estimate the Solution.mFind the distance traveled in 16 seconds by an object traveling at a constant velocity of 20 feet Per second.

Accepted Answer

A)  calculus, $320 \mathrm { ft }$ 
B)  calculus, $340 \mathrm { ft }$ 
C)  precalculus, $320 \mathrm { ft }$ 
D)  calculus, $640 \mathrm { ft }$ 
E)  precalculus, $640 \mathrm { ft }$ 
A)  calculus, $320 \mathrm { ft }$ 
B)  calculus, $340 \mathrm { ft }$ 
C)  precalculus, $320 \mathrm { ft }$ 
D)  calculus, $640 \mathrm { ft }$ 
E)  precalculus, $640 \mathrm { ft }$

Question 15

Find the limit. $$\lim _ { x \rightarrow 5 } \cos \left( \frac { \pi x } { 6 } \right)$$

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  0 
C)  $\frac { 1 } { 2 }$ 
D)  $- \frac { \sqrt { 3 } } { 2 }$ 
E)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $- \frac { 1 } { 2 }$ 
B)  0 
C)  $\frac { 1 } { 2 }$ 
D)  $- \frac { \sqrt { 3 } } { 2 }$ 
E)  $\frac { \sqrt { 3 } } { 2 }$

Question 16

$\text { Determine whether } f ( x ) = \frac { x ^ { 10 } } { x ^ { 2 } - 9 } \text { approaches } \infty \text { or } - \infty \text { as } x \text { approaches } - 3 \text { from }$ the left and from the right by completing the tables below.
$\begin{array}{|c|c|c|c|c|}
\hline x & -3.5 & -3.1 & -3.01 & -3.001 \
\hline f(x) & & & & \
\hline
\end{array}$

$\begin{array}{|c|c|c|c|c|}
\hline x & -2.999 & -2.99 & -2.9 & -2.5 \
\hline f(x) & & & & \
\hline
\end{array}$

Accepted Answer

A)  $\lim _ { x \rightarrow - 3 ^ { - } } f ( x ) = - \infty , \lim _ { x \rightarrow - 3 ^ { + } } f ( x ) = \infty$ 
B) $\lim _{x \rightarrow-3^{-}} f(x)=\infty, \lim _{x \rightarrow-3^{+}} f(x)=-\infty$ 
C)  $\lim _ { x \rightarrow - 3 ^ { - } } f ( x ) =  - \infty , \lim _ { x \rightarrow - 3 ^ { + } } f ( x ) = \infty$ 
D) $\lim _ { x \rightarrow - 3 ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow - 3 ^ { + } } f ( x ) = -\infty$ 
A)  $\lim _ { x \rightarrow - 3 ^ { - } } f ( x ) = - \infty , \lim _ { x \rightarrow - 3 ^ { + } } f ( x ) = \infty$ 
B) $\lim _{x \rightarrow-3^{-}} f(x)=\infty, \lim _{x \rightarrow-3^{+}} f(x)=-\infty$ 
C)  $\lim _ { x \rightarrow - 3 ^ { - } } f ( x ) =  - \infty , \lim _ { x \rightarrow - 3 ^ { + } } f ( x ) = \infty$ 
D) $\lim _ { x \rightarrow - 3 ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow - 3 ^ { + } } f ( x ) = -\infty$

Question 17

Find all the vertical asymptotes (if any) of the graph of the function $f ( x ) = \frac { 5 } { ( x - 3 ) ^ { 2 } }$

Accepted Answer

A)  $x = - 3$ 
B)  $x = 5$ 
C)  $x = 3 , - 3$ 
D)  $x = 3$ 
E)  no vertical asymptotes 
A)  $x = - 3$ 
B)  $x = 5$ 
C)  $x = 3 , - 3$ 
D)  $x = 3$ 
E)  no vertical asymptotes

Question 18

$\text { Find the limit (if it exists). Note that } f ( x ) = ( | x | ) \text { represents the greatest integer }$ function. $\lim _ { x \rightarrow - 6 ^ { + } } ( - 3 [ | x | ] - 8 )$

Accepted Answer

A)  13 
B)  $- 10$ 
C)  10 
D)  $- 13$ 
E)  does not exist 
A)  13 
B)  $- 10$ 
C)  10 
D)  $- 13$ 
E)  does not exist

Question 19

Use the graph as shown to determine the following limits, and discuss the continuity of the function at $x = - 4$ 
(i) $\lim _ { x \rightarrow - 4 ^ { + } } f ( x )$
(ii) $\lim _ { x \rightarrow - 4 ^ { - } } f ( x )$
(iii) $\lim _ { x \rightarrow - 4 } f ( x )$

Accepted Answer

A)  $3,3,3$, continuous 
B)  $2,2,2$, not continuous 
C)  $3,3,3$, not continuous 
D)  $- 4 , - 4 , - 4$, continuous 
E)  $2,2,2$, continuous 
A)  $3,3,3$, continuous 
B)  $2,2,2$, not continuous 
C)  $3,3,3$, not continuous 
D)  $- 4 , - 4 , - 4$, continuous 
E)  $2,2,2$, continuous

Question 20

$\text { Consider the function } f ( x ) = \sqrt { x } \text { and the point } P ( 9,3 ) \text { on the graph of } f \text {. }$ Estimate the slope $m$ of the tangent line of $f$ at $P ( 9,3 )$. Round your answer to four decimal places.

Accepted Answer

A)  $\mathrm { m } = 0.1667$ 
B)  $\mathrm { m } = 0.0832$ 
C)  $\mathrm { m } = 0.3800$ 
D)  $\mathrm { m } = 0.0556$ 
E)  $\mathrm { m } = 0.0833$ 
A)  $\mathrm { m } = 0.1667$ 
B)  $\mathrm { m } = 0.0832$ 
C)  $\mathrm { m } = 0.3800$ 
D)  $\mathrm { m } = 0.0556$ 
E)  $\mathrm { m } = 0.0833$

Find the following limit if it exists: $\lim \ln ( x - 3 )$ . Use $\pm \infty$ when appropriate. $x \rightarrow 3 ^ { + }$

A sphere has a volume of 4.76 cubic inches. What is the radius of the sphere? Round your answer to four decimal places.

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 0 } \frac { \sin ^ { 3 } x } { x ^ { 3 } }$ x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x)

Determine the limit (if it exists). $\lim _ { x \rightarrow 0 } \frac { 12 ( 1 - \cos x ) } { x ^ { 2 } }$

$\text { Consider the function } f ( x ) = 6 x - x ^ { 2 } \text { and the point } P ( 2,8 ) \text { on the graph of } f \text {. }$ Graph $f$ and the secant line passing through $P ( 2,8 )$ and $Q ( x , f ( x ) )$ for $x = 3$ .

Find the limit. $\lim _ { x \rightarrow - 4 } 9 x ^ { 2 } + 36 x$

$\text { Suppose that } \lim _ { x \rightarrow c } f ( x ) = - 11 \text { and } \lim _ { x \rightarrow c } g ( x ) = - 3 \text {. Find the following limit. }$ $\lim _ { x \rightarrow c } [ f ( x ) - g ( x ) ]$

$\text { Let } f ( x ) = - x ^ { 2 } - 5 \text { and } g ( x ) = 2 x \text {. Find the limit }$ $\lim _ { x \rightarrow - 2 } g ( f ( x ) )$

$\text { Find the } x \text {-values (if any) at which the function } f ( x ) = \frac { x } { x ^ { 2 } - 100 } \text { is not continuous. }$ Which of the discontinuities are removable?

Find the limit (if it exists). $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ , where $f ( x ) = \left\{ \begin{array} { c c } x ^ { 3 } + 10 , & x < 1 \\ x + 10 , & x \geq 1 \end{array} \right.$

Find the limit. $\lim _ { x \rightarrow 2 } \cos \frac { \pi x } { 3 }$

Find the limit (if it exists). $\lim _ { x \rightarrow 5 } \frac { \sqrt { x + 4 } - 3 } { x - 5 }$

Find the limit. $\lim _ { x \rightarrow 5 } \cos \left( \frac { \pi x } { 6 } \right)$

Find all the vertical asymptotes (if any) of the graph of the function $f ( x ) = \frac { 5 } { ( x - 3 ) ^ { 2 } }$

$\text { Find the limit (if it exists). Note that } f ( x ) = ( | x | ) \text { represents the greatest integer }$ function. $\lim _ { x \rightarrow - 6 ^ { + } } ( - 3 [ | x | ] - 8 )$

$\text { Consider the function } f ( x ) = \sqrt { x } \text { and the point } P ( 9,3 ) \text { on the graph of } f \text {. }$ Estimate the slope $m$ of the tangent line of $f$ at $P ( 9,3 )$ . Round your answer to four decimal places.

Graphs and Models

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 2: A Preview of Calculus

Find the following limit if it exists: lim⁡ln⁡(x−3)\lim \ln ( x - 3 )limln(x−3) . Use ±∞\pm \infty±∞ when appropriate. x→3+x \rightarrow 3 ^ { + }x→3+

A sphere has a volume of 4.76 cubic inches. What is the radius of the sphere? Round your answer to four decimal places.

Complete the table and use the result to estimate the limit. lim⁡x→0sin⁡3xx3\lim _ { x \rightarrow 0 } \frac { \sin ^ { 3 } x } { x ^ { 3 } }limx→0​x3sin3x​ x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x)

Determine the limit (if it exists). lim⁡x→012(1−cos⁡x)x2\lim _ { x \rightarrow 0 } \frac { 12 ( 1 - \cos x ) } { x ^ { 2 } }limx→0​x212(1−cosx)​

Find the limit. lim⁡x→−49x2+36x\lim _ { x \rightarrow - 4 } 9 x ^ { 2 } + 36 xlimx→−4​9x2+36x

Let f(x)=−x2−5 and g(x)=2x. Find the limit \text { Let } f ( x ) = - x ^ { 2 } - 5 \text { and } g ( x ) = 2 x \text {. Find the limit } Let f(x)=−x2−5 and g(x)=2x. Find the limit lim⁡x→−2g(f(x))\lim _ { x \rightarrow - 2 } g ( f ( x ) )limx→−2​g(f(x))

Find the limit (if it exists). lim⁡x→1−f(x)\lim _ { x \rightarrow 1 ^ { - } } f ( x )limx→1−​f(x) , where f(x)={x3+10,x<1x+10,x≥1f ( x ) = \left\{ \begin{array} { c c } x ^ { 3 } + 10 , & x < 1 \\ x + 10 , & x \geq 1 \end{array} \right.f(x)={x3+10,x+10,​x<1x≥1​

Find the limit. lim⁡x→2cos⁡πx3\lim _ { x \rightarrow 2 } \cos \frac { \pi x } { 3 }limx→2​cos3πx​

Find the limit (if it exists). lim⁡x→5x+4−3x−5\lim _ { x \rightarrow 5 } \frac { \sqrt { x + 4 } - 3 } { x - 5 }limx→5​x−5x+4​−3​

Find the limit. lim⁡x→5cos⁡(πx6)\lim _ { x \rightarrow 5 } \cos \left( \frac { \pi x } { 6 } \right)limx→5​cos(6πx​)

Find all the vertical asymptotes (if any) of the graph of the function f(x)=5(x−3)2f ( x ) = \frac { 5 } { ( x - 3 ) ^ { 2 } }f(x)=(x−3)25​

Graphs and Models

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Find the following limit if it exists: $\lim \ln ( x - 3 )$ . Use $\pm \infty$ when appropriate. $x \rightarrow 3 ^ { + }$

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 0 } \frac { \sin ^ { 3 } x } { x ^ { 3 } }$ x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x)

Determine the limit (if it exists). $\lim _ { x \rightarrow 0 } \frac { 12 ( 1 - \cos x ) } { x ^ { 2 } }$

Find the limit. $\lim _ { x \rightarrow - 4 } 9 x ^ { 2 } + 36 x$

$\text { Let } f ( x ) = - x ^ { 2 } - 5 \text { and } g ( x ) = 2 x \text {. Find the limit }$ $\lim _ { x \rightarrow - 2 } g ( f ( x ) )$

Find the limit (if it exists). $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ , where $f ( x ) = \left\{ \begin{array} { c c } x ^ { 3 } + 10 , & x < 1 \\ x + 10 , & x \geq 1 \end{array} \right.$

Find the limit. $\lim _ { x \rightarrow 2 } \cos \frac { \pi x } { 3 }$

Find the limit (if it exists). $\lim _ { x \rightarrow 5 } \frac { \sqrt { x + 4 } - 3 } { x - 5 }$

Find the limit. $\lim _ { x \rightarrow 5 } \cos \left( \frac { \pi x } { 6 } \right)$

Find all the vertical asymptotes (if any) of the graph of the function $f ( x ) = \frac { 5 } { ( x - 3 ) ^ { 2 } }$