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

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

Exam 11: Limits and an Introduction to Calculus

Use the graph below to find $\lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 } , if it exists.$

(Multiple Choice)

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Question 21

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

(Multiple Choice)

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Question 22

Find $\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }$ (if it exists).

(Multiple Choice)

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Question 23

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

(Multiple Choice)

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Question 24

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

(Multiple Choice)

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Question 25

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

(Multiple Choice)

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Question 26

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

(Multiple Choice)

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Question 27

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

(Multiple Choice)

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Question 28

Determine any points on the graph of the following function at which the tangent line is horizontal. $f ( x ) = x ^ { 2 } + 6 x - 8$

(Multiple Choice)

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Question 29

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

(Multiple Choice)

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Question 30

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

(Multiple Choice)

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Question 31

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 ).

(Multiple Choice)

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Question 32

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x } , if it exists.$

(Multiple Choice)

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Question 33

The cost function for a certain model of a digital camera given by $C = 13.50 x + 48,950$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 200$ . Round your answer to the nearest cent.

(Multiple Choice)

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Question 34

. Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 15 + y } - \sqrt { 15 } } { y }$

(Multiple Choice)

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Question 35

Use the graph to find $\lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 }$ , if it exists. $Use the graph to find \lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 } , if it exists.$

(Multiple Choice)

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Question 36

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

(Multiple Choice)

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Question 37

Use asymptotes to match $f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 2 } + 1 }$ with its graph.

(Multiple Choice)

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Question 38

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)

(Multiple Choice)

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Question 39

Find $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }$ by direct substitution.

(Multiple Choice)

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Question 40

Use the graph below to find $\lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 } , if it exists.$

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

Find $\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }$ (if it exists).

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

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

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

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

Determine any points on the graph of the following function at which the tangent line is horizontal. $f ( x ) = x ^ { 2 } + 6 x - 8$

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

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

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 ).

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x } , if it exists.$

The cost function for a certain model of a digital camera given by $C = 13.50 x + 48,950$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 200$ . Round your answer to the nearest cent.

. Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 15 + y } - \sqrt { 15 } } { y }$

Use the graph to find $\lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 }$ , if it exists. $Use the graph to find \lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 } , if it exists.$

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

Use asymptotes to match $f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 2 } + 1 }$ with its graph.

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 $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }$ by direct substitution.

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

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Exam 11: Limits and an Introduction to Calculus

Use the graph below to find lim⁡x→−4∣x+4∣x+4\lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 }limx→−4​x+4∣x+4∣​ , if it exists.

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

Find lim⁡x→∞65x\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }limx→∞​5x6​ (if it exists).

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 the slope of the graph of the following function at the point (−1,1)( - 1,1 )(−1,1) . 3x2−23 x ^ { 2 } - 23x2−2

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.

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 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​

Determine any points on the graph of the following function at which the tangent line is horizontal. f(x)=x2+6x−8f ( x ) = x ^ { 2 } + 6 x - 8f(x)=x2+6x−8

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) .

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

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 ).

Consider the following graph of the function and approximate lim⁡x→0e−2x−1x\lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x }limx→0​xe−2x−1​ , if it exists.

The cost function for a certain model of a digital camera given by C=13.50x+48,950C = 13.50 x + 48,950C=13.50x+48,950 , where CCC is the cost (in dollars) and xxx is the number of cameras produced. Find the average cost per unit when x=200x = 200x=200 . Round your answer to the nearest cent.

. Find lim⁡y→015+y−15y\lim _ { y \rightarrow 0 } \frac { \sqrt { 15 + y } - \sqrt { 15 } } { y }limy→0​y15+y​−15​​

Use the graph to find lim⁡x→−11x+1\lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 }limx→−1​x+11​ , if it exists.

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

Use asymptotes to match f(x)=4x2x2+1f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 2 } + 1 }f(x)=x2+14x2​ with its graph.

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 lim⁡x→6x+7x−2\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }limx→6​x−2x+7​​ by direct substitution.

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

Use the graph below to find $\lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 } , if it exists.$

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

Find $\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }$ (if it exists).

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

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

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

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

Determine any points on the graph of the following function at which the tangent line is horizontal. $f ( x ) = x ^ { 2 } + 6 x - 8$

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

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

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 ).

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x } , if it exists.$

The cost function for a certain model of a digital camera given by $C = 13.50 x + 48,950$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 200$ . Round your answer to the nearest cent.

. Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 15 + y } - \sqrt { 15 } } { y }$

Use the graph to find $\lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 }$ , if it exists. $Use the graph to find \lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 } , if it exists.$

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

Use asymptotes to match $f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 2 } + 1 }$ with its graph.

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 $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }$ by direct substitution.