Question 1

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 3 } - \sqrt { 3 } }$

Accepted Answer

A)  6 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $2 \sqrt { 3 }$ 
E)  limit does not exist 
A)  6 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $2 \sqrt { 3 }$ 
E)  limit does not exist

Question 2

Use the limit process to find the slope of the graph of the following function at the point $( 3 , - 19 )$.
$$g ( x ) = - 10 - 3 x$$

Accepted Answer

A)  $- 13$ 
B)  $- 19$ 
C)  $- 7$ 
D)  $- 3$ 
E)  3 
A)  $- 13$ 
B)  $- 19$ 
C)  $- 7$ 
D)  $- 3$ 
E)  3

Question 3

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 4

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 5

If 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 6

Use the limit process to find the slope of the graph of $4 x - 8 x ^ { 2 }$ at $( 3 , - 60 )$.

Accepted Answer

A)  $- 12$ 
B)  52 
C)  28 
D)  48 
E)  $- 44$ 
A)  $- 12$ 
B)  52 
C)  28 
D)  48 
E)  $- 44$

Question 7

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

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 8

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 9

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 10

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 11

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 12

Graph $f ( x ) = \left\{ \begin{array} { l c } 3 x + 2 & x < - 3 \ x - 4 & x \geq - 3 \end{array} \right.$ and find the limit of $f ( x )$ as $x$ approaches $- 3$

Accepted Answer

A)  $- 35$ 
B)  $- 7$ 
C)  $- 28$ 
D)  7 
E)  limit does not exist 
A)  $- 35$ 
B)  $- 7$ 
C)  $- 28$ 
D)  7 
E)  limit does not exist

Question 13

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 14

Consider the graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }$, 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 15

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

Accepted Answer

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

Question 16

Estimate the following limit numerically, if it exists.
$$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$$

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  0 
C)  $- \frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  The limit does not exist. 
A)  $\frac { 1 } { 6 }$ 
B)  0 
C)  $- \frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  The limit does not exist.

Question 17

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 18

Find the following limit, if it exists.
$$\lim _ { x \rightarrow \infty } \frac { 6 x ^ { 2 } - 4 } { - 3 - x ^ { 2 } }$$

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  $- 6$ 
C)  6 
D)  $- 4$ 
E)  The limit does not exist. 
A)  $- \frac { 1 } { 2 }$ 
B)  $- 6$ 
C)  6 
D)  $- 4$ 
E)  The limit does not exist.

Question 19

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 20

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$.

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  3 
C)  $\infty$ 
D)  $\frac { 1 } { 6 }$ 
E)  the slope is undefined at this point 
A)  $\frac { 1 } { 3 }$ 
B)  3 
C)  $\infty$ 
D)  $\frac { 1 } { 6 }$ 
E)  the slope is undefined at this point

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 3 } - \sqrt { 3 } }$

Use the limit process to find the slope of the graph of the following function at the point $( 3 , - 19 )$ . $g ( x ) = - 10 - 3 x$

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) ?

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

Use the limit process to find the slope of the graph of $4 x - 8 x ^ { 2 }$ at $( 3 , - 60 )$ .

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

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

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).

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

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

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

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

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

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$

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 the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 6 x ^ { 2 } - 4 } { - 3 - x ^ { 2 } }$

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

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$ .

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: Limits and an Introduction to Calculus

Find lim⁡x→0−xx+3−3\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 3 } - \sqrt { 3 } }limx→0−​x+3​−3​x​

Use the limit process to find the slope of the graph of the following function at the point (3,−19)( 3 , - 19 )(3,−19) . g(x)=−10−3xg ( x ) = - 10 - 3 xg(x)=−10−3x

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) ?

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 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)​

Use the limit process to find the slope of the graph of 4x−8x24 x - 8 x ^ { 2 }4x−8x2 at (3,−60)( 3 , - 60 )(3,−60) .

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 } , f ^ { \prime } ( x ) = - 12 x ^ { 3 } + 12 xf(x)=−3x4+6x2,f′(x)=−12x3+12x

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

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).

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

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

Graph f(x)={3x+2x<−3x−4x≥−3f ( x ) = \left\{ \begin{array} { l c } 3 x + 2 & x < - 3 \\ x - 4 & x \geq - 3 \end{array} \right.f(x)={3x+2x−4​x<−3x≥−3​ and find the limit of f(x)f ( x )f(x) as xxx approaches −3- 3−3

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).

Consider the graph of the function and approximate lim⁡x→0sin⁡(8x)x\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }limx→0​xsin(8x)​ , if it exists.

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

Estimate the following limit numerically, if it exists. lim⁡x→1x−1x2+4x−5\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }limx→1​x2+4x−5x−1​

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 the following limit, if it exists. lim⁡x→∞6x2−4−3−x2\lim _ { x \rightarrow \infty } \frac { 6 x ^ { 2 } - 4 } { - 3 - x ^ { 2 } }limx→∞​−3−x26x2−4​

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

Use the limit process to find the slope of the graph of x+4\sqrt { x + 4 }x+4​ at (5,3)( 5,3 )(5,3) .

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 $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 3 } - \sqrt { 3 } }$

Use the limit process to find the slope of the graph of the following function at the point $( 3 , - 19 )$ . $g ( x ) = - 10 - 3 x$

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) ?

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

Use the limit process to find the slope of the graph of $4 x - 8 x ^ { 2 }$ at $( 3 , - 60 )$ .

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

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

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).

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

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

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

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

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

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$

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 the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 6 x ^ { 2 } - 4 } { - 3 - x ^ { 2 } }$

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

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$ .