Question 1

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 2

Find the slope of the graph of the following function at the point (1, -3) . $$- x ^ { 2 } - 2$$

Accepted Answer

A)  4 
B)  -2 
C)  6 
D)  -3 
E)  -4 
A)  4 
B)  -2 
C)  6 
D)  -3 
E)  -4

Question 3

Find the derivative of $f ( x ) = 4 x ^ { 2 } + 2 x + 3$.

Accepted Answer

A)  $8 x + 2$ 
B)  $4 x + 2$ 
C)  $8 x + 5$ 
D)  $4 x + 5$ 
E)  $8 x$ 
A)  $8 x + 2$ 
B)  $4 x + 2$ 
C)  $8 x + 5$ 
D)  $4 x + 5$ 
E)  $8 x$

Question 4

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 7 \right]$ (if it exists).

Accepted Answer

A)  $\frac { 8 } { 25 }$ 
B)  $- 7$ 
C)  7 
D)  $\infty$ 
E)  limit does not exist 
A)  $\frac { 8 } { 25 }$ 
B)  $- 7$ 
C)  7 
D)  $\infty$ 
E)  limit does not exist

Question 5

Estimate the following limit numerically, if it exists.
$$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }$$

Accepted Answer

A)  $\frac { 1 } { 7 }$ 
B)  0 
C)  $- \frac { 1 } { 5 }$ 
D)  $\frac { 1 } { 5 }$ 
E)  The limit does not exist. 
A)  $\frac { 1 } { 7 }$ 
B)  0 
C)  $- \frac { 1 } { 5 }$ 
D)  $\frac { 1 } { 5 }$ 
E)  The limit does not exist.

Question 6

Find the following limit, if it exists.
$$\lim _ { x \rightarrow 3 } x ^ { 7 }$$

Accepted Answer

A)  3 
B)  21 
C)  $- 3$ 
D)  2187 
E)  The limit does not exist. 
A)  3 
B)  21 
C)  $- 3$ 
D)  2187 
E)  The limit does not exist.

Question 7

$$\text { Consider the graph of the function and approximate } \lim _ { x \rightarrow 0 } \frac { \sin ( - 8 x ) } { x } \text {, if it exists. }$$

Accepted Answer

A)  -1 
B)  -4 
C)  -8 
D)  1 
E)  The limit does not exist. 
A)  -1 
B)  -4 
C)  -8 
D)  1 
E)  The limit does not exist.

Question 8

Complete the table and numerically estimate the limit as $x$ approaches infinity for $f ( x ) = x - \sqrt { x ^ { 2 } + 4 }$.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \
\hline f(x) & & & & & & & \
\hline
\end{array}$

Accepted Answer

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

Question 9

Find the following limit. Round your answer to two decimals.
$$\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }$$

Accepted Answer

A)  $0.22$ 
B)  $1.32$ 
C)  $0.44$ 
D)  $0.18$ 
E)  $1.41$ 
A)  $0.22$ 
B)  $1.32$ 
C)  $0.44$ 
D)  $0.18$ 
E)  $1.41$

Question 10

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x }$ (if it exists).

Accepted Answer

A)  0 
B)  1 
C)  $- 1$ 
D)  2 
E)  limit does not exist 
A)  0 
B)  1 
C)  $- 1$ 
D)  2 
E)  limit does not exist

Question 11

Find the following limit.
$$\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1$$

Accepted Answer

A)  52 
B)  36 
C)  37 
D)  - 23 
E)  - 35 
A)  52 
B)  36 
C)  37 
D)  - 23 
E)  - 35

Question 12

Find
$$\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }$$
by direct substitution.

Accepted Answer

A)  $\frac { 8 } { \sqrt { 10 } }$ 
B)  $10 \sqrt { 2 }$ 
C)  $40 \sqrt { 10 }$ 
D)  1 
E)  $\frac { 40 } { \sqrt { 10 } }$ 
A)  $\frac { 8 } { \sqrt { 10 } }$ 
B)  $10 \sqrt { 2 }$ 
C)  $40 \sqrt { 10 }$ 
D)  1 
E)  $\frac { 40 } { \sqrt { 10 } }$

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

A)  -3 
B)  9 
C)  7 
D)  -11 
E)  The limit does not exist. 
A)  -3 
B)  9 
C)  7 
D)  -11 
E)  The limit does not exist.

Question 15

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 5 n ^ { 3 } + 8 } { n + 3 }$ (assume $n$ begins with 1).

Accepted Answer

A)  $\frac { 5 } { 3 }$ 
B)  111 
C)  79 
D)  0 
E)  the sequence diverges 
A)  $\frac { 5 } { 3 }$ 
B)  111 
C)  79 
D)  0 
E)  the sequence diverges

Question 16

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.
$$f ( x ) = - 2 x ^ { 4 } + 4 x ^ { 2 } , f ^ { \prime } ( x ) = - 8 x ^ { 3 } + 8 x$$

Accepted Answer

A)  $( - 1,2 ) , ( 0,0 ) , ( 1,2 )$ 
B)  $( - 1 , - 2 ) , ( 1 , - 8 )$ 
C)  $( - 1,2 ) , ( 1,2 )$ 
D)  $( 0,0 ) , ( 1,0 )$ 
E)  $( 0,0 ) , ( 1,2 )$ 
A)  $( - 1,2 ) , ( 0,0 ) , ( 1,2 )$ 
B)  $( - 1 , - 2 ) , ( 1 , - 8 )$ 
C)  $( - 1,2 ) , ( 1,2 )$ 
D)  $( 0,0 ) , ( 1,0 )$ 
E)  $( 0,0 ) , ( 1,2 )$

Question 17

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]$ (if it exists).

Accepted Answer

A)  $\frac { 9 } { 25 }$ 
B)  $- 8$ 
C)  8 
D)  $\infty$ 
E)  limit does not exist 
A)  $\frac { 9 } { 25 }$ 
B)  $- 8$ 
C)  8 
D)  $\infty$ 
E)  limit does not exist

Question 18

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$.

Accepted Answer

A)  24 
B)  $\infty$ 
C)  12 
D)  0 
E)  limit does not exist 
A)  24 
B)  $\infty$ 
C)  12 
D)  0 
E)  limit does not exist

Question 19

Find
$\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$
for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }$.

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 15 } - 108$ 
B)  $\frac { \sqrt { 10 } } { 5 } - 108$ 
C)  $\frac { 36 \sqrt { 2 } } { 5 }$ 
D)  $\frac { \sqrt { 10 } } { 5 } + 108$ 
E)  limit does not exist 
A)  $\frac { \sqrt { 2 } } { 15 } - 108$ 
B)  $\frac { \sqrt { 10 } } { 5 } - 108$ 
C)  $\frac { 36 \sqrt { 2 } } { 5 }$ 
D)  $\frac { \sqrt { 10 } } { 5 } + 108$ 
E)  limit does not exist

Question 20

Use the limit process to find the slope of the graph of the following function at the point $( 5 , - 27 )$.
$$g ( x ) = - 7 - 4 x$$

Accepted Answer

A)  $- 11$ 
B)  $- 27$ 
C)  $- 3$ 
D)  $- 4$ 
E)  5 
A)  $- 11$ 
B)  $- 27$ 
C)  $- 3$ 
D)  $- 4$ 
E)  5

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

Find the slope of the graph of the following function at the point (1, -3) . $- x ^ { 2 } - 2$

Find the derivative of $f ( x ) = 4 x ^ { 2 } + 2 x + 3$ .

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 7 \right]$ (if it exists).

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }$

Find the following limit, if it exists. $\lim _ { x \rightarrow 3 } x ^ { 7 }$

$\text { Consider the graph of the function and approximate } \lim _ { x \rightarrow 0 } \frac { \sin ( - 8 x ) } { x } \text {, if it exists. }$ $\text { Consider the graph of the function and approximate } \lim _ { x \rightarrow 0 } \frac { \sin ( - 8 x ) } { x } \text {, if it exists. }$

Complete the table and numerically estimate the limit as $x$ approaches infinity for $f ( x ) = x - \sqrt { x ^ { 2 } + 4 }$ . x 1 1 1 1 1 1 1 f(x)

Find the following limit. Round your answer to two decimals. $\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }$

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x }$ (if it exists). $Use the graph to determine \lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x } (if it exists).$

Find the following limit. $\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1$

Find $\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }$ by direct substitution.

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

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

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 5 n ^ { 3 } + 8 } { n + 3 }$ (assume $n$ begins with 1).

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f ( x ) = - 2 x ^ { 4 } + 4 x ^ { 2 } , f ^ { \prime } ( x ) = - 8 x ^ { 3 } + 8 x$

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]$ (if it exists).

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 } .$

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }$ .

Use the limit process to find the slope of the graph of the following function at the point $( 5 , - 27 )$ . $g ( x ) = - 7 - 4 x$

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

Review of Graphs, Equations, and Inequalities

Filters

Exam 11: Limits and an Introduction to Calculus

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

Find the slope of the graph of the following function at the point (1, -3) . −x2−2- x ^ { 2 } - 2−x2−2

Find the derivative of f(x)=4x2+2x+3f ( x ) = 4 x ^ { 2 } + 2 x + 3f(x)=4x2+2x+3 .

Find lim⁡x→−∞[x(5+x)2+7]\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 7 \right]limx→−∞​[(5+x)2x​+7] (if it exists).

Estimate the following limit numerically, if it exists. lim⁡x→1x−1x2+5x−6\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }limx→1​x2+5x−6x−1​

Find the following limit, if it exists. lim⁡x→3x7\lim _ { x \rightarrow 3 } x ^ { 7 }limx→3​x7

Complete the table and numerically estimate the limit as xxx approaches infinity for f(x)=x−x2+4f ( x ) = x - \sqrt { x ^ { 2 } + 4 }f(x)=x−x2+4​ . x 1 1 1 1 1 1 1 f(x)

Find the following limit. Round your answer to two decimals. lim⁡x→3x2+72x2\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }limx→3​2x2x2+7​​

Use the graph to determine lim⁡x→0x2−xx\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x }limx→0​xx2−x​ (if it exists).

Find the following limit. lim⁡x→−43x2+3x+1\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1limx→−4​3x2+3x+1

Find lim⁡x→85xx+2\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }limx→8​x+2​5x​ by direct substitution.

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

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

Use the first six terms to predict the limit of the sequence an=5n3+8n+3a _ { n } = \frac { 5 n ^ { 3 } + 8 } { n + 3 }an​=n+35n3+8​ (assume nnn begins with 1).

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)=−2x4+4x2,f′(x)=−8x3+8xf ( x ) = - 2 x ^ { 4 } + 4 x ^ { 2 } , f ^ { \prime } ( x ) = - 8 x ^ { 3 } + 8 xf(x)=−2x4+4x2,f′(x)=−8x3+8x

Find lim⁡x→−∞[x(5+x)2+8]\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]limx→−∞​[(5+x)2x​+8] (if it exists).

Use the graph to find lim⁡x→34x2−36x−3\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }limx→3​x−34x2−36​ .

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)=4x3f ( x ) = 4 x ^ { 3 }f(x)=4x3 and g(x)=x2+95x2g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }g(x)=5x2x2+9​​ .

Use the limit process to find the slope of the graph of the following function at the point (5,−27)( 5 , - 27 )(5,−27) . g(x)=−7−4xg ( x ) = - 7 - 4 xg(x)=−7−4x

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

Review of Graphs, Equations, and Inequalities

Filters

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

Find the slope of the graph of the following function at the point (1, -3) . $- x ^ { 2 } - 2$

Find the derivative of $f ( x ) = 4 x ^ { 2 } + 2 x + 3$ .

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 7 \right]$ (if it exists).

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }$

Find the following limit, if it exists. $\lim _ { x \rightarrow 3 } x ^ { 7 }$

Complete the table and numerically estimate the limit as $x$ approaches infinity for $f ( x ) = x - \sqrt { x ^ { 2 } + 4 }$ . x 1 1 1 1 1 1 1 f(x)

Find the following limit. Round your answer to two decimals. $\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }$

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x }$ (if it exists). $Use the graph to determine \lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - x } { x } (if it exists).$

Find the following limit. $\lim _ { x \rightarrow - 4 } 3 x ^ { 2 } + 3 x + 1$

Find $\lim _ { x \rightarrow 8 } \frac { 5 x } { \sqrt { x + 2 } }$ by direct substitution.

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

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

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 5 n ^ { 3 } + 8 } { n + 3 }$ (assume $n$ begins with 1).

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f ( x ) = - 2 x ^ { 4 } + 4 x ^ { 2 } , f ^ { \prime } ( x ) = - 8 x ^ { 3 } + 8 x$

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]$ (if it exists).

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 } .$

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }$ .

Use the limit process to find the slope of the graph of the following function at the point $( 5 , - 27 )$ . $g ( x ) = - 7 - 4 x$