Question 1

Find the first five terms of the sequence. 
 $\alpha _ { n } = \frac { ( - 2 ) ^ { n } } { n }$

Accepted Answer

A)  $\alpha _ { 1 } = - 2$ , $a _ { 2 } =$ 2, $a _ { 3 } = -$ $\frac { 8 } { 3 }$ , $a _ { 4 } =$ 4, $a _ { 5 } = -$ $\frac { 32 } { 5 }$ 
B)  $a _ { 1 } = -$ $\frac { 1 } { 2 }$ , $a _ { 2 } =$ $\frac { 1 } { 2 }$ , $a _ { 3 } = -$ $\frac { 8 } { 3 }$ , $a _ { 4 } =$ 4, $a _ { 5 } = -$ $\frac { 32 } { 5 }$ 
C)  $a _ { 1 } = -$ $\frac { 1 } { 2 }$ , $a _ { 2 } =$ $\frac { 1 } { 2 }$ , $a _ { 3 } = -$ 3
8 , $a _ { 4 } =$ 4, $a _ { 5 } = -$ $\frac { 5 } { 32 }$ 
D)  $\alpha _ { 1 } = 2$ , $a _ { 2 } =$ 2, $a _ { 3 } = -$ $\frac { 8 } { 3 }$ , $a _ { 4 } =$ 4, $a _ { 5 } = -$ $\frac { 32 } { 5 }$ 
E)  $\alpha _ { 1 } = - 2$ , $a _ { 2 } =$ 2, $a _ { 3 } = -$ $\frac { 3 } {8}$  , $a _ { 4 } =$ 4, $a _ { 5 } = -$ $\frac { 5 } { 32 }$ 
A)  $\alpha _ { 1 } = - 2$ , $a _ { 2 } =$ 2, $a _ { 3 } = -$ $\frac { 8 } { 3 }$ , $a _ { 4 } =$ 4, $a _ { 5 } = -$ $\frac { 32 } { 5 }$ 
B)  $a _ { 1 } = -$ $\frac { 1 } { 2 }$ , $a _ { 2 } =$ $\frac { 1 } { 2 }$ , $a _ { 3 } = -$ $\frac { 8 } { 3 }$ , $a _ { 4 } =$ 4, $a _ { 5 } = -$ $\frac { 32 } { 5 }$ 
C)  $a _ { 1 } = -$ $\frac { 1 } { 2 }$ , $a _ { 2 } =$ $\frac { 1 } { 2 }$ , $a _ { 3 } = -$ 3
8 , $a _ { 4 } =$ 4, $a _ { 5 } = -$ $\frac { 5 } { 32 }$ 
D)  $\alpha _ { 1 } = 2$ , $a _ { 2 } =$ 2, $a _ { 3 } = -$ $\frac { 8 } { 3 }$ , $a _ { 4 } =$ 4, $a _ { 5 } = -$ $\frac { 32 } { 5 }$ 
E)  $\alpha _ { 1 } = - 2$ , $a _ { 2 } =$ 2, $a _ { 3 } = -$ $\frac { 3 } {8}$  , $a _ { 4 } =$ 4, $a _ { 5 } = -$ $\frac { 5 } { 32 }$

Question 2

Find the first five terms of the sequence.  $a _ { n } = \frac { n + 5 } { n ^ { 2 } + 5 }$

Accepted Answer

A)  $\alpha _ { 1 } = 1 , \alpha _ { 2 } = \frac { 7 } { 9 } , \alpha _ { 3 } = \frac { 4 } { 7 } , \alpha _ { 4 } = \frac { 3 } { 7 } , \alpha _ { 5 } = \frac { 1 } { 3 }$ 
B)  $\alpha _ { 1 } = 1 , \alpha _ { 2 } = \frac { 9 } { 7 } , \alpha _ { 3 } = \frac { 4 } { 7 } , \alpha _ { 4 } = \frac { 3 } { 7 } , \alpha _ { 5 } = 3$ 
C)  $\alpha _ { 1 } = \frac { 1 } { 2 } , \alpha _ { 2 } = \frac { 9 } { 7 } , \alpha _ { 3 } = \frac { 4 } { 7 } , \alpha _ { 4 } = \frac { 3 } { 7 } , \alpha _ { 5 } = \frac { 1 } { 3 }$ 
D)  $\alpha _ { 1 } = 1 , \alpha _ { 2 } = \frac { 9 } { 7 } , \alpha _ { 3 } = \frac { 4 } { 7 } , \alpha _ { 4 } = \frac { 3 } { 7 } , \alpha _ { 5 } = \frac { 1 } { 3 }$ 
E)  $\alpha _ { 1 } = \frac { 1 } { 2 } , \quad \alpha _ { 2 } = \frac { 7 } { 9 } , \alpha _ { 3 } = \frac { 4 } { 7 } , \alpha _ { 4 } = \frac { 3 } { 7 } , \alpha _ { 5 } = 3$ 
A)  $\alpha _ { 1 } = 1 , \alpha _ { 2 } = \frac { 7 } { 9 } , \alpha _ { 3 } = \frac { 4 } { 7 } , \alpha _ { 4 } = \frac { 3 } { 7 } , \alpha _ { 5 } = \frac { 1 } { 3 }$ 
B)  $\alpha _ { 1 } = 1 , \alpha _ { 2 } = \frac { 9 } { 7 } , \alpha _ { 3 } = \frac { 4 } { 7 } , \alpha _ { 4 } = \frac { 3 } { 7 } , \alpha _ { 5 } = 3$ 
C)  $\alpha _ { 1 } = \frac { 1 } { 2 } , \alpha _ { 2 } = \frac { 9 } { 7 } , \alpha _ { 3 } = \frac { 4 } { 7 } , \alpha _ { 4 } = \frac { 3 } { 7 } , \alpha _ { 5 } = \frac { 1 } { 3 }$ 
D)  $\alpha _ { 1 } = 1 , \alpha _ { 2 } = \frac { 9 } { 7 } , \alpha _ { 3 } = \frac { 4 } { 7 } , \alpha _ { 4 } = \frac { 3 } { 7 } , \alpha _ { 5 } = \frac { 1 } { 3 }$ 
E)  $\alpha _ { 1 } = \frac { 1 } { 2 } , \quad \alpha _ { 2 } = \frac { 7 } { 9 } , \alpha _ { 3 } = \frac { 4 } { 7 } , \alpha _ { 4 } = \frac { 3 } { 7 } , \alpha _ { 5 } = 3$

Question 3

Find the limit (if it exists). $\lim _ { t \rightarrow + \infty } \frac { 6 t ^ { 2 } - 3 t + 1 } { - 9 t ^ { 2 } + 3 t + 3 }$

Accepted Answer

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

Question 4

Use the function and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.Use a graphing utility to verify your results.   $f ( x ) = 5 x ^ { 6 } + 6 x ^ { 5 } , \quad f ^ { \prime } ( x ) = 30 x ^ { 5 } + 30 x ^ { 4 }$

Accepted Answer

A)  $f$ has vertical tangents at $( - 1 , - 1 )$ and $( 0,0 )$ . 
B)  $f$ has horizontal tangents at $( 1,1 )$ and $( 0,0 )$ . 
C)  $f$ has horizontal tangents at $( - 1 , - 1 )$ and $( 0,0 )$ . 
D)  $f$ has vertical tangents at $( 1,1 )$ and $( 0,0 )$ . 
E)  $f$ has vertical tangents at $( - 1 , - 1 )$ and $( 1,1 )$ . 
A)  $f$ has vertical tangents at $( - 1 , - 1 )$ and $( 0,0 )$ . 
B)  $f$ has horizontal tangents at $( 1,1 )$ and $( 0,0 )$ . 
C)  $f$ has horizontal tangents at $( - 1 , - 1 )$ and $( 0,0 )$ . 
D)  $f$ has vertical tangents at $( 1,1 )$ and $( 0,0 )$ . 
E)  $f$ has vertical tangents at $( - 1 , - 1 )$ and $( 1,1 )$ .

Question 5

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ .   $f ( x ) = 4 - 2 x - x ^ { 2 }$

Accepted Answer

A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 2 + 2 x$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 4 - 2 x$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = 2 + 3 x$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 2 - 2 x$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 2 x ^ { 2 } + 2 x$ 
A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 2 + 2 x$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 4 - 2 x$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = 2 + 3 x$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 2 - 2 x$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 2 x ^ { 2 } + 2 x$

Question 6

Use the given information to evaluate the limit.  $\lim _ { x \rightarrow c } f ( x ) = 2$ , $\lim _ { x \rightarrow c } g ( x ) = 10$ $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ]$

Accepted Answer

A)  $\lim _ { x \rightarrow c } [ f ( x ) + ( g ( x ) ] =$ 14 
B)  $\lim _ { x \rightarrow c } [ f ( x ) + ( g ( x ) ] =$ 2 
C)  $\lim _ { x \rightarrow c } [ f ( x ) + ( g ( x ) ] =$ 12 
D)  $\lim _ { x \rightarrow c } [ f ( x ) + ( g ( x ) ] =$ 13 
E)  $\lim _ { x \rightarrow c } [ f ( x ) + ( g ( x ) ] =$ 10 
A)  $\lim _ { x \rightarrow c } [ f ( x ) + ( g ( x ) ] =$ 14 
B)  $\lim _ { x \rightarrow c } [ f ( x ) + ( g ( x ) ] =$ 2 
C)  $\lim _ { x \rightarrow c } [ f ( x ) + ( g ( x ) ] =$ 12 
D)  $\lim _ { x \rightarrow c } [ f ( x ) + ( g ( x ) ] =$ 13 
E)  $\lim _ { x \rightarrow c } [ f ( x ) + ( g ( x ) ] =$ 10

Question 7

Use the graph to determine the limit visually (if it exists).Then identify another function g2(x)that agrees with the given function at all but one point.
  $g ( x ) = \frac { x ^ { 3 } - x } { x - 1 }$      $$\lim _ { x \rightarrow - 1 } g ( x ) = ?$$

Accepted Answer

A)  $g _ { 2 } ( x ) = x ( x + 1 )$ $\lim _ { x \rightarrow - 1 } g ( x ) = 0$ 
B)  $g _ { 2 } ( x ) = 3 x ^ { 2 } + 1$ $\lim _ { x \rightarrow - 1 } g ( x ) = 4$ 
C)  $g _ { 2 } ( x ) = 3 x ^ { 2 } - 1$ $\lim _ { x \rightarrow - 1 } g ( x ) = 2$ 
D)  $g _ { 2 } ( x ) = x ( x - 1 )$ $\lim _ { x \rightarrow - 1 } g ( x ) = 2$ 
E)  $g _ { 2 } ( x ) = x ( x + 1 )$ $\lim _ { x \rightarrow - 1 } g ( x ) = 4$ 
A)  $g _ { 2 } ( x ) = x ( x + 1 )$ $\lim _ { x \rightarrow - 1 } g ( x ) = 0$ 
B)  $g _ { 2 } ( x ) = 3 x ^ { 2 } + 1$ $\lim _ { x \rightarrow - 1 } g ( x ) = 4$ 
C)  $g _ { 2 } ( x ) = 3 x ^ { 2 } - 1$ $\lim _ { x \rightarrow - 1 } g ( x ) = 2$ 
D)  $g _ { 2 } ( x ) = x ( x - 1 )$ $\lim _ { x \rightarrow - 1 } g ( x ) = 2$ 
E)  $g _ { 2 } ( x ) = x ( x + 1 )$ $\lim _ { x \rightarrow - 1 } g ( x ) = 4$

Question 8

Select correct graph of the following function.  $y = \frac { 5 x } { 1 - x ^ { 2 } }$

Accepted Answer

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

Question 9

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 4 } } { x - 3 }$ by direct substitution.

Accepted Answer

A)  $\frac { 4 } { 4 }$ 
B)  $\frac { \sqrt { 23 } } { 4 }$ 
C)  $\frac { \sqrt { 11 } } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\sqrt { 4 }$ 
A)  $\frac { 4 } { 4 }$ 
B)  $\frac { \sqrt { 23 } } { 4 }$ 
C)  $\frac { \sqrt { 11 } } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\sqrt { 4 }$

Question 10

Select the correct graph of the following function.  $y = \frac { 5 x } { 2 - x }$

Accepted Answer

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

Question 11

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 15 }$ .

Accepted Answer

A)  $\sqrt { x - 15 }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 15 } }$ 
C)  $\frac { \sqrt { x - 15 } } { 2 }$ 
D) limit does not exist 
E)  $\frac { 1 } { 15 \sqrt { x - 15 } }$ 
A)  $\sqrt { x - 15 }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 15 } }$ 
C)  $\frac { \sqrt { x - 15 } } { 2 }$ 
D) limit does not exist 
E)  $\frac { 1 } { 15 \sqrt { x - 15 } }$

Question 12

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { x + 4 } { x ^ { 2 } + 7 x + 10 }$

Accepted Answer

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

Question 13

Algebraically evaluate the limit (if it exists)by the appropriate technique(s).  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 }$

Accepted Answer

A)  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 } = \frac { 1 } { 22 }$ 
B)  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 } = - \frac { 1 } { 21 }$ 
C)  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 } = \frac { 1 } { 21 }$ 
D)  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 } = \frac { 1 } { 20 }$ 
E)  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 } = - \frac { 1 } { 20 }$ 
A)  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 } = \frac { 1 } { 22 }$ 
B)  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 } = - \frac { 1 } { 21 }$ 
C)  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 } = \frac { 1 } { 21 }$ 
D)  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 } = \frac { 1 } { 20 }$ 
E)  $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 } = - \frac { 1 } { 20 }$

Question 14

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

Accepted Answer

A)  $\frac { 12 } { x ^ { 4 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x + 4 } }$ 
C)  $2 x + 4$ 
D)  $\frac { 1 } { 2 \sqrt { x - 4 } }$ 
E)  $- \frac { 1 } { 2 \sqrt { x + 4 } }$ 
A)  $\frac { 12 } { x ^ { 4 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x + 4 } }$ 
C)  $2 x + 4$ 
D)  $\frac { 1 } { 2 \sqrt { x - 4 } }$ 
E)  $- \frac { 1 } { 2 \sqrt { x + 4 } }$

Question 15

Find the limit (if it exists).  $\lim _ { x \rightarrow - \infty } \frac { 6 } { 7 } x - \frac { 9 } { x ^ { 2 } }$

Accepted Answer

A) -6 
B) -7 
C) 6 
D) 0 
E) -$\infty$ 
A) -6 
B) -7 
C) 6 
D) 0 
E) -$\infty$

Question 16

Find the limit of the sequence (if it exists). $\alpha _ { n } = \frac { 2 n } { n ^ { 2 } + 7 }$

Accepted Answer

A)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 0$ 
B)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 2$ 
C)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 7$ 
D)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = - 2$ 
E) does not exist 
A)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 0$ 
B)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 2$ 
C)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = 7$ 
D)  $\lim_ { n \rightarrow \infty } \alpha _ { n } = - 2$ 
E) does not exist

Question 17

Use the limit process to find the slope of the graph of $7 x - 6 x ^ { 2 }$ at $( 3 , - 33 )$ .

Accepted Answer

A)  $36$ 
B)  $- 29$ 
C)  $25$ 
D)  $43$ 
E)  $- 5$ 
A)  $36$ 
B)  $- 29$ 
C)  $25$ 
D)  $43$ 
E)  $- 5$

Question 18

Evaluate $\sum _ { k = 1 } ^ { 20 } \left( k ^ { 3 } + 6 \right)$ using the summation formulas and properties.

Accepted Answer

A)  $44,100$ 
B)  $8,006$ 
C)  $176,520$ 
D)  $44,220$ 
E)  $176,400$ 
A)  $44,100$ 
B)  $8,006$ 
C)  $176,520$ 
D)  $44,220$ 
E)  $176,400$

Question 19

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function.   $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } }$

Accepted Answer

A)  $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } } = \frac { 1 } { 7 }$   
B)  $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } } = - \frac { 1 } { 7 }$   
C)  $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } } = - \frac { 1 } { 8 }$   
D)  $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } } = \frac { 1 } { 6 }$   
E)  $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } } = - \frac { 1 } { 6 }$   
A)  $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } } = \frac { 1 } { 7 }$   
B)  $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } } = - \frac { 1 } { 7 }$   
C)  $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } } = - \frac { 1 } { 8 }$   
D)  $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } } = \frac { 1 } { 6 }$   
E)  $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } } = - \frac { 1 } { 6 }$

Find the first five terms of the sequence. $\alpha _ { n } = \frac { ( - 2 ) ^ { n } } { n }$

Find the first five terms of the sequence. $a _ { n } = \frac { n + 5 } { n ^ { 2 } + 5 }$

Find the limit (if it exists). $\lim _ { t \rightarrow + \infty } \frac { 6 t ^ { 2 } - 3 t + 1 } { - 9 t ^ { 2 } + 3 t + 3 }$

Use the function and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.Use a graphing utility to verify your results. $f ( x ) = 5 x ^ { 6 } + 6 x ^ { 5 } , \quad f ^ { \prime } ( x ) = 30 x ^ { 5 } + 30 x ^ { 4 }$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = 4 - 2 x - x ^ { 2 }$

Use the given information to evaluate the limit. $\lim _ { x \rightarrow c } f ( x ) = 2$ , $\lim _ { x \rightarrow c } g ( x ) = 10$ $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ]$

Select correct graph of the following function. $y = \frac { 5 x } { 1 - x ^ { 2 } }$

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 4 } } { x - 3 }$ by direct substitution.

Select the correct graph of the following function. $y = \frac { 5 x } { 2 - x }$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 15 }$ .

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { x + 4 } { x ^ { 2 } + 7 x + 10 }$

Algebraically evaluate the limit (if it exists)by the appropriate technique(s). $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 }$

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

Find the limit (if it exists). $\lim _ { x \rightarrow - \infty } \frac { 6 } { 7 } x - \frac { 9 } { x ^ { 2 } }$

Find the limit of the sequence (if it exists). $\alpha _ { n } = \frac { 2 n } { n ^ { 2 } + 7 }$

Use the limit process to find the slope of the graph of $7 x - 6 x ^ { 2 }$ at $( 3 , - 33 )$ .

Evaluate $\sum _ { k = 1 } ^ { 20 } \left( k ^ { 3 } + 6 \right)$ using the summation formulas and properties.

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function. $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } }$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

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Exam 12: Limits and An Introduction To Calculus

Find the first five terms of the sequence. αn=(−2)nn\alpha _ { n } = \frac { ( - 2 ) ^ { n } } { n }αn​=n(−2)n​

Find the first five terms of the sequence. an=n+5n2+5a _ { n } = \frac { n + 5 } { n ^ { 2 } + 5 }an​=n2+5n+5​

Find the limit (if it exists). lim⁡t→+∞6t2−3t+1−9t2+3t+3\lim _ { t \rightarrow + \infty } \frac { 6 t ^ { 2 } - 3 t + 1 } { - 9 t ^ { 2 } + 3 t + 3 }limt→+∞​−9t2+3t+36t2−3t+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)​ . f(x)=4−2x−x2f ( x ) = 4 - 2 x - x ^ { 2 }f(x)=4−2x−x2

Use the given information to evaluate the limit. lim⁡x→cf(x)=2\lim _ { x \rightarrow c } f ( x ) = 2limx→c​f(x)=2 , lim⁡x→cg(x)=10\lim _ { x \rightarrow c } g ( x ) = 10limx→c​g(x)=10 lim⁡x→c[f(x)+g(x)]\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ]limx→c​[f(x)+g(x)]

Select correct graph of the following function. y=5x1−x2y = \frac { 5 x } { 1 - x ^ { 2 } }y=1−x25x​

Find lim⁡x→7x+4x−3\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 4 } } { x - 3 }limx→7​x−3x+4​​ by direct substitution.

Select the correct graph of the following function. y=5x2−xy = \frac { 5 x } { 2 - x }y=2−x5x​

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)=x−15f ( x ) = \sqrt { x - 15 }f(x)=x−15​ .

Select the correct graph for the following function using a graphing utility. f(x)=x+4x2+7x+10f ( x ) = \frac { x + 4 } { x ^ { 2 } + 7 x + 10 }f(x)=x2+7x+10x+4​

Algebraically evaluate the limit (if it exists)by the appropriate technique(s). lim⁡x→100+10−xx−100\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 }limx→100+​x−10010−x​​

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

Find the limit (if it exists). lim⁡x→−∞67x−9x2\lim _ { x \rightarrow - \infty } \frac { 6 } { 7 } x - \frac { 9 } { x ^ { 2 } }limx→−∞​76​x−x29​

Find the limit of the sequence (if it exists). αn=2nn2+7\alpha _ { n } = \frac { 2 n } { n ^ { 2 } + 7 }αn​=n2+72n​

Use the limit process to find the slope of the graph of 7x−6x27 x - 6 x ^ { 2 }7x−6x2 at (3,−33)( 3 , - 33 )(3,−33) .

Evaluate ∑k=120(k3+6)\sum _ { k = 1 } ^ { 20 } \left( k ^ { 3 } + 6 \right)∑k=120​(k3+6) using the summation formulas and properties.

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function. lim⁡x→3+3−x9−x2\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } }limx→3+​9−x23−x​

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

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Find the first five terms of the sequence. $\alpha _ { n } = \frac { ( - 2 ) ^ { n } } { n }$

Find the first five terms of the sequence. $a _ { n } = \frac { n + 5 } { n ^ { 2 } + 5 }$

Find the limit (if it exists). $\lim _ { t \rightarrow + \infty } \frac { 6 t ^ { 2 } - 3 t + 1 } { - 9 t ^ { 2 } + 3 t + 3 }$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = 4 - 2 x - x ^ { 2 }$

Use the given information to evaluate the limit. $\lim _ { x \rightarrow c } f ( x ) = 2$ , $\lim _ { x \rightarrow c } g ( x ) = 10$ $\lim _ { x \rightarrow c } [ f ( x ) + g ( x ) ]$

Select correct graph of the following function. $y = \frac { 5 x } { 1 - x ^ { 2 } }$

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 4 } } { x - 3 }$ by direct substitution.

Select the correct graph of the following function. $y = \frac { 5 x } { 2 - x }$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 15 }$ .

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { x + 4 } { x ^ { 2 } + 7 x + 10 }$

Algebraically evaluate the limit (if it exists)by the appropriate technique(s). $\lim _ { x \rightarrow 100 ^ { + } } \frac { 10 - \sqrt { x } } { x - 100 }$

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

Find the limit (if it exists). $\lim _ { x \rightarrow - \infty } \frac { 6 } { 7 } x - \frac { 9 } { x ^ { 2 } }$

Find the limit of the sequence (if it exists). $\alpha _ { n } = \frac { 2 n } { n ^ { 2 } + 7 }$

Use the limit process to find the slope of the graph of $7 x - 6 x ^ { 2 }$ at $( 3 , - 33 )$ .

Evaluate $\sum _ { k = 1 } ^ { 20 } \left( k ^ { 3 } + 6 \right)$ using the summation formulas and properties.

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function. $\lim _ { x \rightarrow 3 ^ { + } } \frac { 3 - x } { 9 - x ^ { 2 } }$