Question 1

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

Accepted Answer

A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 1 } { 2 \sqrt { x - 7 } }$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { 2 \sqrt { x - 10 } }$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { 2 \sqrt { x - 8 } }$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 1 } { 2 \sqrt { x - 10 } }$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 1 } { 2 \sqrt { x - 8 } }$ 
A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 1 } { 2 \sqrt { x - 7 } }$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { 2 \sqrt { x - 10 } }$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { 2 \sqrt { x - 8 } }$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 1 } { 2 \sqrt { x - 10 } }$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 1 } { 2 \sqrt { x - 8 } }$

Question 2

Find the derivative of the function.   $f ( x ) = \frac { 1 } { \sqrt { x - 4 } }$

Accepted Answer

A)  $\frac { 1 } { 2 ( x - 4 ) ^ { \frac { 3 } { 2 } } }$ 
B)  $\frac { 1 } { 2 ( x + 4 ) ^ { \frac { 3 } { 2 } } }$ 
C)  $\frac { 1 } { ( x + 4 ) ^ { 2 } }$ 
D)  $\frac { 1 } { 2 \sqrt { x - 4 } }$ 
E)  $- \frac { 1 } { 2 ( x - 4 ) ^ { \frac { 3 } { 2 } } }$ 
A)  $\frac { 1 } { 2 ( x - 4 ) ^ { \frac { 3 } { 2 } } }$ 
B)  $\frac { 1 } { 2 ( x + 4 ) ^ { \frac { 3 } { 2 } } }$ 
C)  $\frac { 1 } { ( x + 4 ) ^ { 2 } }$ 
D)  $\frac { 1 } { 2 \sqrt { x - 4 } }$ 
E)  $- \frac { 1 } { 2 ( x - 4 ) ^ { \frac { 3 } { 2 } } }$

Question 3

Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 } \text { and } g ( x ) = \frac { \sqrt { x ^ { 2 } + 6 } } { 6 x ^ { 2 } }$ .

Accepted Answer

A)  $\frac { \sqrt { 31 } } { 150 } - 375$ 
B)  $\frac { \sqrt { 31 } } { 150 } + 375$ 
C)  $\frac { 1 } { 150 } - 375$ 
D)  $\frac { \sqrt { 31 } } { 375 } - 150$ 
E) limit does not exist 
A)  $\frac { \sqrt { 31 } } { 150 } - 375$ 
B)  $\frac { \sqrt { 31 } } { 150 } + 375$ 
C)  $\frac { 1 } { 150 } - 375$ 
D)  $\frac { \sqrt { 31 } } { 375 } - 150$ 
E) limit does not exist

Question 4

Use the graph to determine the limit visually (if it exists).Then identify another function h2(x)that agrees with the given function at all but one point.
  $$h ( x ) = \frac { x ^ { 2 } - 3 x } { x }$$       $$\lim _ { h } h ( x ) = \text { ? }$$

Accepted Answer

A)  $h _ { 2 } ( x ) = x + 3$ $\lim h ( x ) = 0$ 
B)  $h _ { 2 } ( x ) = 2 x - 3$ $\lim _ { x \rightarrow - 3 } h ( x ) = - 9$ 
C)  $h _ { 2 } ( x ) = 2 x + 3$ $\lim _ { x \rightarrow - 3 } h ( x ) = - 3$ 
D)  $h _ { 2 } ( x ) = x - 3$ $\lim _ { x \rightarrow - 3 } h ( x ) = - 6$ 
E)  $h _ { 2 } ( x ) = x - 3$ $\lim _ { x \rightarrow - 3 } h ( x ) = - 9$ 
A)  $h _ { 2 } ( x ) = x + 3$ $\lim h ( x ) = 0$ 
B)  $h _ { 2 } ( x ) = 2 x - 3$ $\lim _ { x \rightarrow - 3 } h ( x ) = - 9$ 
C)  $h _ { 2 } ( x ) = 2 x + 3$ $\lim _ { x \rightarrow - 3 } h ( x ) = - 3$ 
D)  $h _ { 2 } ( x ) = x - 3$ $\lim _ { x \rightarrow - 3 } h ( x ) = - 6$ 
E)  $h _ { 2 } ( x ) = x - 3$ $\lim _ { x \rightarrow - 3 } h ( x ) = - 9$

Question 5

Find a formula for the slope of the graph of $f ( x ) = 9 x ^ { 6 }$ .

Accepted Answer

A)  $54 x$ 
B)  $x ^ { 6 }$ 
C)  $54 x ^ { 5 }$ 
D)  $54 x ^ { 4 } ( x - 1 )$ 
E)  $\infty$ 
A)  $54 x$ 
B)  $x ^ { 6 }$ 
C)  $54 x ^ { 5 }$ 
D)  $54 x ^ { 4 } ( x - 1 )$ 
E)  $\infty$

Question 6

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

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B) 2 
C) -2 
D) $\infty$ 
E) does not exist 
A)  $- \frac { 1 } { 2 }$ 
B) 2 
C) -2 
D) $\infty$ 
E) does not exist

Question 7

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point.   $f ( x ) = \sqrt { x - 1 } , \quad ( 17,4 )$

Accepted Answer

A)  $\begin{array} { l } 
m = \frac { 1 } { \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
B)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
C)   $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = - \frac { 1 } { 8 }
\end{array}$ 
D)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
E)   $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = - \frac { 1 } { 8 }
\end{array}$ 
A)  $\begin{array} { l } 
m = \frac { 1 } { \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
B)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
C)   $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = - \frac { 1 } { 8 }
\end{array}$ 
D)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = \frac { 1 } { 8 }
\end{array}$ 
E)   $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 1 } } \
\text { At } ( 17,4 ) , m = - \frac { 1 } { 8 }
\end{array}$

Question 8

Find the limit (if it exists).Use a graphing utility to verify your result graphically.   $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x } = \frac { 1 } { 12 }$   
B)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x } = - \frac { 1 } { 12 }$   
C)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x } = \frac { 1 } { 3 }$     
D)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x } = - \frac { 1 } { 9 }$    
E)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x } = \frac { 1 } { 9 }$     
A)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x } = \frac { 1 } { 12 }$   
B)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x } = - \frac { 1 } { 12 }$   
C)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x } = \frac { 1 } { 3 }$     
D)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x } = - \frac { 1 } { 9 }$    
E)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x } = \frac { 1 } { 9 }$

Question 9

Find the limit (if it exists).Use a graphing utility to verify your result graphically.   $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 }$

Accepted Answer

A)  $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 } = - \frac { 1 } { 25 }$   
B)  $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 } = \frac { 1 } { 10 }$   
C)  $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 } = - \frac { 1 } { 10 }$     
D)  $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 } = \frac { 1 } { 25 }$   
E)  $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 } = - \frac { 1 } { 5 }$     
A)  $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 } = - \frac { 1 } { 25 }$   
B)  $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 } = \frac { 1 } { 10 }$   
C)  $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 } = - \frac { 1 } { 10 }$     
D)  $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 } = \frac { 1 } { 25 }$   
E)  $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 } = - \frac { 1 } { 5 }$

Question 10

Find $\lim _ { x \rightarrow 8 } \frac { 3 x } { \sqrt { x + 5 } }$ by direct substitution.

Accepted Answer

A)  $\frac { 1 } { 8 }$ 
B)  $0$ 
C)  $24 \sqrt { 13 }$ 
D) 1 
E)  $\frac { 24 } { \sqrt { 13 } }$ 
A)  $\frac { 1 } { 8 }$ 
B)  $0$ 
C)  $24 \sqrt { 13 }$ 
D) 1 
E)  $\frac { 24 } { \sqrt { 13 } }$

Question 11

Find the limit of the sequence (if it exists).   $\alpha _ { n } = \frac { 4 n ^ { 2 } + 1 } { 2 n }$

Accepted Answer

A)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = - 4$ 
B)  $\lim _ { n \rightarrow \infty } \alpha _ { n } = \frac { 1 } { 2 }$ 
C)  $\lim _ { n \rightarrow \infty }\alpha _ { n } = 4$ 
D)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 0$ 
E)  $\lim_ { n \rightarrow \infty } a _ { n } = \infty$ 
A)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = - 4$ 
B)  $\lim _ { n \rightarrow \infty } \alpha _ { n } = \frac { 1 } { 2 }$ 
C)  $\lim _ { n \rightarrow \infty }\alpha _ { n } = 4$ 
D)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 0$ 
E)  $\lim_ { n \rightarrow \infty } a _ { n } = \infty$

Question 12

Use the derivative of $f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 }$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Accepted Answer

A)  $( 1,5 )$ and $( - 1,1 )$ 
B)  $( - 1 , - 8 )$ 
C)  $( 0,0 )$ and $( - 1,1 )$ 
D)  $( 2,0 )$ 
E)  $f ( x )$ has no points with a horizontal tangent line. 
A)  $( 1,5 )$ and $( - 1,1 )$ 
B)  $( - 1 , - 8 )$ 
C)  $( 0,0 )$ and $( - 1,1 )$ 
D)  $( 2,0 )$ 
E)  $f ( x )$ has no points with a horizontal tangent line.

Question 13

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 3 - 6 x$ .

Accepted Answer

A) -6. 
B) 3 
C) -3 
D) limit does not exist 
E) -6 
A) -6. 
B) 3 
C) -3 
D) limit does not exist 
E) -6

Question 14

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 5 }$ by direct substitution.

Accepted Answer

A)  $\frac { \sqrt { 23 } } { 2 }$ 
B)  $\frac { 1 } { 7 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\sqrt { 10 }$ 
E)  $\frac { \sqrt { 10 } } { 2 }$ 
A)  $\frac { \sqrt { 23 } } { 2 }$ 
B)  $\frac { 1 } { 7 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\sqrt { 10 }$ 
E)  $\frac { \sqrt { 10 } } { 2 }$

Question 15

Find the derivative of the function.  $g ( x ) = 9 - \frac { 1 } { 5 } x$

Accepted Answer

A)  $- \frac { 1 } { 5 }$ 
B)  $5$ 
C)  $5 x$ 
D)  $\frac { 1 } { 5 }$ 
E)  $9$ 
A)  $- \frac { 1 } { 5 }$ 
B)  $5$ 
C)  $5 x$ 
D)  $\frac { 1 } { 5 }$ 
E)  $9$

Question 16

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

Accepted Answer

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

Question 17

Use the given information to evaluate the limit.  $\lim _ { x \rightarrow c } f ( x ) = 11$ $\lim _ { x\rightarrow c } \sqrt { f ( x ) }$

Accepted Answer

A)  $\lim _ { x \rightarrow c } \sqrt { f ( x ) } = 11$ 
B)  $\lim _ { x \rightarrow c } \sqrt { f ( x ) } = 3$ 
C)  $\lim _ { x \rightarrow c } \sqrt { f ( x ) } = \sqrt { 11 }$ 
D)  $\lim _ { x \rightarrow c } \sqrt { f ( x ) } = 2$ 
E)  $\lim _ { x \rightarrow c } \sqrt { f ( x ) } = 16$ 
A)  $\lim _ { x \rightarrow c } \sqrt { f ( x ) } = 11$ 
B)  $\lim _ { x \rightarrow c } \sqrt { f ( x ) } = 3$ 
C)  $\lim _ { x \rightarrow c } \sqrt { f ( x ) } = \sqrt { 11 }$ 
D)  $\lim _ { x \rightarrow c } \sqrt { f ( x ) } = 2$ 
E)  $\lim _ { x \rightarrow c } \sqrt { f ( x ) } = 16$

Question 18

Find the limit by direct substitution.   $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$

Accepted Answer

A)  $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$ =  $\infty$ 
B)  $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$ = 15 
C)  $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$ = 7 
D)  $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$ = 27 
E)  $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$ = -5 
A)  $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$ =  $\infty$ 
B)  $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$ = 15 
C)  $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$ = 7 
D)  $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$ = 27 
E)  $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$ = -5

Question 19

Use the derivative of $f ( x ) = 5 x ^ { 3 } + 15 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Accepted Answer

A)  $( 1,20 )$ 
B)  $( 1,20 )$ and $( - 1 , - 20 )$ 
C)  $( 5,700 )$ and $( - 5,700 )$ 
D) (0,0) 
E) f(x)has no points with a horizontal tangent line. 
A)  $( 1,20 )$ 
B)  $( 1,20 )$ and $( - 1 , - 20 )$ 
C)  $( 5,700 )$ and $( - 5,700 )$ 
D) (0,0) 
E) f(x)has no points with a horizontal tangent line.

Question 20

Find the limit of the sequence (if it exists). 
$\alpha _ { n } = \frac { ( - 4 ) ^ { n + 1 } } { n ^ { 2 } }$

Accepted Answer

A)  $\lim _ { n \rightarrow \infty } \alpha _ { n } = - 4$ 
B)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 0$ 
C)  $\lim _ { n \rightarrow \infty }\alpha _ { n } = 4$ 
D)  $\lim _ { n \rightarrow \infty } \alpha _ { n } = \frac { 1 } { 4 }$ 
E) does not exist 
A)  $\lim _ { n \rightarrow \infty } \alpha _ { n } = - 4$ 
B)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 0$ 
C)  $\lim _ { n \rightarrow \infty }\alpha _ { n } = 4$ 
D)  $\lim _ { n \rightarrow \infty } \alpha _ { n } = \frac { 1 } { 4 }$ 
E) does not exist

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

Find the derivative of the function. $f ( x ) = \frac { 1 } { \sqrt { x - 4 } }$

Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 } \text { and } g ( x ) = \frac { \sqrt { x ^ { 2 } + 6 } } { 6 x ^ { 2 } }$ .

Find a formula for the slope of the graph of $f ( x ) = 9 x ^ { 6 }$ .

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

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = \sqrt { x - 1 } , \quad ( 17,4 )$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x }$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 }$

Find $\lim _ { x \rightarrow 8 } \frac { 3 x } { \sqrt { x + 5 } }$ by direct substitution.

Find the limit of the sequence (if it exists). $\alpha _ { n } = \frac { 4 n ^ { 2 } + 1 } { 2 n }$

Use the derivative of $f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 }$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 3 - 6 x$ .

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 5 }$ by direct substitution.

Find the derivative of the function. $g ( x ) = 9 - \frac { 1 } { 5 } x$

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

Use the given information to evaluate the limit. $\lim _ { x \rightarrow c } f ( x ) = 11$ $\lim _ { x\rightarrow c } \sqrt { f ( x ) }$

Find the limit by direct substitution. $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$

Use the derivative of $f ( x ) = 5 x ^ { 3 } + 15 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Find the limit of the sequence (if it exists). $\alpha _ { n } = \frac { ( - 4 ) ^ { n + 1 } } { n ^ { 2 } }$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Filters

Exam 12: Limits and An Introduction To Calculus

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)​ . f(x)=x−8f ( x ) = \sqrt { x - 8 }f(x)=x−8​

Find the derivative of the function. f(x)=1x−4f ( x ) = \frac { 1 } { \sqrt { x - 4 } }f(x)=x−4​1​

Find lim⁡x→5[g(x)−f(x)]\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]limx→5​[g(x)−f(x)] for f(x)=3x3 and g(x)=x2+66x2f ( x ) = 3 x ^ { 3 } \text { and } g ( x ) = \frac { \sqrt { x ^ { 2 } + 6 } } { 6 x ^ { 2 } }f(x)=3x3 and g(x)=6x2x2+6​​ .

Use the graph to determine the limit visually (if it exists).Then identify another function h2(x)that agrees with the given function at all but one point. h(x)=x2−3xxh ( x ) = \frac { x ^ { 2 } - 3 x } { x }h(x)=xx2−3x​ lim⁡hh(x)= ? \lim _ { h } h ( x ) = \text { ? }limh​h(x)= ?

Find a formula for the slope of the graph of f(x)=9x6f ( x ) = 9 x ^ { 6 }f(x)=9x6 .

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

Find a formula for the slope of the graph of fff at the point (x,f(x))( x , f ( x ) )(x,f(x)) .Then use it to find the slope at the given point. f(x)=x−1,(17,4)f ( x ) = \sqrt { x - 1 } , \quad ( 17,4 )f(x)=x−1​,(17,4)

Find the limit (if it exists).Use a graphing utility to verify your result graphically. lim⁡x→01x+3−13x\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x }limx→0​xx+31​−31​​

Find the limit (if it exists).Use a graphing utility to verify your result graphically. lim⁡x→55−xx2−25\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 }limx→5​x2−255−x​

Find lim⁡x→83xx+5\lim _ { x \rightarrow 8 } \frac { 3 x } { \sqrt { x + 5 } }limx→8​x+5​3x​ by direct substitution.

Find the limit of the sequence (if it exists). αn=4n2+12n\alpha _ { n } = \frac { 4 n ^ { 2 } + 1 } { 2 n }αn​=2n4n2+1​

Use the derivative of f(x)=2x3+3x2f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 }f(x)=2x3+3x2 to determine any points on the graph of f(x)f ( x )f(x) at which the tangent line is horizontal.

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)=3−6xf ( x ) = 3 - 6 xf(x)=3−6x .

Find lim⁡x→7x+3x−5\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 5 }limx→7​x−5x+3​​ by direct substitution.

Find the derivative of the function. g(x)=9−15xg ( x ) = 9 - \frac { 1 } { 5 } xg(x)=9−51​x

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

Use the given information to evaluate the limit. lim⁡x→cf(x)=11\lim _ { x \rightarrow c } f ( x ) = 11limx→c​f(x)=11 lim⁡x→cf(x)\lim _ { x\rightarrow c } \sqrt { f ( x ) }limx→c​f(x)​

Find the limit by direct substitution. lim⁡x→4(11−x2)\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)limx→4​(11−x2)

Use the derivative of f(x)=5x3+15xf ( x ) = 5 x ^ { 3 } + 15 xf(x)=5x3+15x to determine any points on the graph of f(x)f ( x )f(x) at which the tangent line is horizontal.

Find the limit of the sequence (if it exists). αn=(−4)n+1n2\alpha _ { n } = \frac { ( - 4 ) ^ { n + 1 } } { n ^ { 2 } }αn​=n2(−4)n+1​

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

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

Find the derivative of the function. $f ( x ) = \frac { 1 } { \sqrt { x - 4 } }$

Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 } \text { and } g ( x ) = \frac { \sqrt { x ^ { 2 } + 6 } } { 6 x ^ { 2 } }$ .

Find a formula for the slope of the graph of $f ( x ) = 9 x ^ { 6 }$ .

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

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = \sqrt { x - 1 } , \quad ( 17,4 )$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x + 3 } - \frac { 1 } { 3 } } { x }$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 5 } \frac { 5 - x } { x ^ { 2 } - 25 }$

Find $\lim _ { x \rightarrow 8 } \frac { 3 x } { \sqrt { x + 5 } }$ by direct substitution.

Find the limit of the sequence (if it exists). $\alpha _ { n } = \frac { 4 n ^ { 2 } + 1 } { 2 n }$

Use the derivative of $f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 }$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 3 - 6 x$ .

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 5 }$ by direct substitution.

Find the derivative of the function. $g ( x ) = 9 - \frac { 1 } { 5 } x$

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

Use the given information to evaluate the limit. $\lim _ { x \rightarrow c } f ( x ) = 11$ $\lim _ { x\rightarrow c } \sqrt { f ( x ) }$

Find the limit by direct substitution. $\lim _ { x \rightarrow 4 } \left( 11 - x ^ { 2 } \right)$

Use the derivative of $f ( x ) = 5 x ^ { 3 } + 15 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Find the limit of the sequence (if it exists). $\alpha _ { n } = \frac { ( - 4 ) ^ { n + 1 } } { n ^ { 2 } }$