Question 1

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 5 i ^ { 3 } } { n ^ { 5 } }$ as a rational function S(n)and find $\lim_ { n \rightarrow \infty }  S ( n )$ .

Accepted Answer

A)  $S ( n ) = \frac { 5 ( n + 1 ) ^ { 2 } } { 4 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 5 } { 4 }$ 
B)  $S ( n ) = \frac { 5 n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } , \lim _ { n \rightarrow \infty } S ( n ) = 5$ 
C)  $S ( n ) = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 20 } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
D)  $S ( n ) = \frac { 5 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
E)  $S ( n ) = \frac { 5 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } }$ the limit does not exist 
A)  $S ( n ) = \frac { 5 ( n + 1 ) ^ { 2 } } { 4 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 5 } { 4 }$ 
B)  $S ( n ) = \frac { 5 n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } , \lim _ { n \rightarrow \infty } S ( n ) = 5$ 
C)  $S ( n ) = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 20 } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
D)  $S ( n ) = \frac { 5 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
E)  $S ( n ) = \frac { 5 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } }$ the limit does not exist

Question 2

Select correct graph of the following function.  $y = \frac { 4 x + 5 } { x ^ { 2 } - 5 }$

Accepted Answer

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

Question 3

Find the limit (if it exists).Use a graphing utility to verify your result graphically.   $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 }$

Accepted Answer

A)  $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 } = \frac { 1 } { 16 }$   
B)  $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 } = \frac { 1 } { 8 }$   
C)  $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 } = \frac { 1 } { 64 }$   
D)  $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 } = \frac { 1 } { 64 }$   
E)  $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 } = \frac { 1 } { 16 }$    
A)  $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 } = \frac { 1 } { 16 }$   
B)  $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 } = \frac { 1 } { 8 }$   
C)  $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 } = \frac { 1 } { 64 }$   
D)  $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 } = \frac { 1 } { 64 }$   
E)  $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 } = \frac { 1 } { 16 }$

Question 4

Find the limit (if it exists).Use a graphing utility to verify your result graphically.   $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x } = \frac { 1 } { 6 }$     
B)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x } = - \frac { 1 } { 36 }$     
C)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x } = \frac { 1 } { 36 }$     
D)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x } = - \frac { 1 } { 42 }$     
E)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x } = \frac { 1 } { 42 }$     
A)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x } = \frac { 1 } { 6 }$     
B)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x } = - \frac { 1 } { 36 }$     
C)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x } = \frac { 1 } { 36 }$     
D)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x } = - \frac { 1 } { 42 }$     
E)  $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x } = \frac { 1 } { 42 }$

Question 5

Use the limit process to find the area of the region between f(x)= 10x + 6 and the x-axis on the interval [0,7].

Accepted Answer

A)  $287$ 
B)  $112$ 
C)  $245$ 
D)  $532$ 
E)  $251$ 
A)  $287$ 
B)  $112$ 
C)  $245$ 
D)  $532$ 
E)  $251$

Question 6

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

Accepted Answer

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

Question 7

Find the limit   $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x } = 2$ 
B)  $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x } = - 4$ 
C)  $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x } = 4$ 
D)  $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x } = - 6$ 
E)  $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x } = 6$ 
A)  $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x } = 2$ 
B)  $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x } = - 4$ 
C)  $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x } = 4$ 
D)  $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x } = - 6$ 
E)  $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x } = 6$

Question 8

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \ln \left( x ^ { 2 } \right) } { x - 4 }$

Accepted Answer

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

Question 9

Rewrite $\sum _ { i = 1 } ^ { n } \left[ \frac { 3 } { n } + \left( \frac { 4 i } { n ^ { 2 } } \right) \right] \left( \frac { 4 i } { n } \right)$ as a rational function S(n)and find $\lim _ { n \rightarrow \infty } S ( n )$ .

Accepted Answer

A)  $S ( n ) = \frac { 2 [ 9 n ( n + 1 ) + 4 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } \text {, limit dos not exist }$ 
B)  $S ( n ) = \frac { 2 [ 9 + 4 n ] } { 3 n } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 34 } { 3 }$ 
C)  $S ( n ) = \frac { 2 [ 9 n + 4 ( n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 34 } { 3 }$ 
D)  $S ( n ) = \frac { 2 [ 9 n ( n + 1 ) + 4 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 5 } { 6 }$ 
E)  $S ( n ) = \frac { 2 [ 9 n ( n + 1 ) + 4 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 34 } { 3 }$ 
A)  $S ( n ) = \frac { 2 [ 9 n ( n + 1 ) + 4 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } \text {, limit dos not exist }$ 
B)  $S ( n ) = \frac { 2 [ 9 + 4 n ] } { 3 n } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 34 } { 3 }$ 
C)  $S ( n ) = \frac { 2 [ 9 n + 4 ( n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 34 } { 3 }$ 
D)  $S ( n ) = \frac { 2 [ 9 n ( n + 1 ) + 4 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 5 } { 6 }$ 
E)  $S ( n ) = \frac { 2 [ 9 n ( n + 1 ) + 4 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 34 } { 3 }$

Question 10

Find the limit (if it exists).  $\lim _ { x \rightarrow \infty } \left( \frac { 3 } { x ^ { 2 } } + 4 \right)$

Accepted Answer

A) 4 
B) -4 
C) 3 
D) 7 
E) does not exist 
A) 4 
B) -4 
C) 3 
D) 7 
E) does not exist

Question 11

Determine what statement is true. ​
The exact area of a region is given by the limit of the sum of n rectangles as n approaches
​

Accepted Answer

A) 0 
B) ​infinity 
A) 0 
B) ​infinity

Question 12

Use the first six terms to predict the limit of the sequence $\alpha _ { n } = \frac { 7 n ^ { 3 } } { n ^ { 3 } + 4 }$ (assume n begins with 1).

Accepted Answer

A) 7 
B)  $\frac { 42 } { 5 }$ 
C)  $\frac { 33 } { 8 }$ 
D) 6 
E) the sequence diverges 
A) 7 
B)  $\frac { 42 } { 5 }$ 
C)  $\frac { 33 } { 8 }$ 
D) 6 
E) the sequence diverges

Question 13

Approximate the limit accurate to three decimal places.  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x } \approx - 54.598$ 
B)  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x } \approx 54.598$ 
C)  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x } \approx - 90.017$ 
D)  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x } \approx 90.017$ 
E)  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x } \approx 32.786$ 
A)  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x } \approx - 54.598$ 
B)  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x } \approx 54.598$ 
C)  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x } \approx - 90.017$ 
D)  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x } \approx 90.017$ 
E)  $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x } \approx 32.786$

Question 14

Find the limit. 
 $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x } = - 9$ 
B)  $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x } = 8$ 
C)  $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x } = 7$ 
D)  $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x } = - 7$ 
E)  $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x } = 9$ 
A)  $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x } = - 9$ 
B)  $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x } = 8$ 
C)  $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x } = 7$ 
D)  $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x } = - 7$ 
E)  $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x } = 9$

Question 15

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.  $f ( x ) = \sqrt { x - 3 } , \quad ( 52,7 )$

Accepted Answer

A)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 3 } } \
\text { At } ( 52,7 ) , m = - \frac { 1 } { 14 }
\end{array}$ 
B)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 3 } } \
\operatorname { At } ( 52,7 ) , m = \frac { 1 } { 14 }
\end{array}$ 
C)  $\begin{array} { l } 
m = \frac { 1 } { \sqrt { x - 3 } } \
\operatorname { At } ( 52,7 ) , m = \frac { 1 } { 14 }
\end{array}$ 
D)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 3 } } \
\operatorname { At } ( 52,7 ) , m = \frac { 1 } { 14 }
\end{array}$ 
E)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 3 } } \
\text { At } ( 52,7 ) , m = - \frac { 1 } { 14 }
\end{array}$ 
A)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 3 } } \
\text { At } ( 52,7 ) , m = - \frac { 1 } { 14 }
\end{array}$ 
B)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 3 } } \
\operatorname { At } ( 52,7 ) , m = \frac { 1 } { 14 }
\end{array}$ 
C)  $\begin{array} { l } 
m = \frac { 1 } { \sqrt { x - 3 } } \
\operatorname { At } ( 52,7 ) , m = \frac { 1 } { 14 }
\end{array}$ 
D)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 3 } } \
\operatorname { At } ( 52,7 ) , m = \frac { 1 } { 14 }
\end{array}$ 
E)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 3 } } \
\text { At } ( 52,7 ) , m = - \frac { 1 } { 14 }
\end{array}$

Question 16

Find the limit (if it exists).Use a graphing utility to verify your result graphically.   $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 }$

Accepted Answer

A)  $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = 4$     
B)  $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = - 2$     
C)  $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = 2$     
D)  $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = - 10$     
E)  $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = 10$     
A)  $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = 4$     
B)  $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = - 2$     
C)  $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = 2$     
D)  $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = - 10$     
E)  $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = 10$

Question 17

Find the derivative of the function.   $f ( x ) = 4 - 7 x ^ { 2 }$

Accepted Answer

A)  $- \frac { 1 } { 7 }$ 
B)  $14 x$ 
C)  $\frac { 1 } { 7 }$ 
D)  $- 14 x$ 
E)  $7 x$ 
A)  $- \frac { 1 } { 7 }$ 
B)  $14 x$ 
C)  $\frac { 1 } { 7 }$ 
D)  $- 14 x$ 
E)  $7 x$

Question 18

Find the derivative of the function.   $f ( x ) = - 3$

Accepted Answer

A)  $- 3$ 
B)  $3$ 
C)  $- 3 x$ 
D)  $0$ 
E)  $3 x$ 
A)  $- 3$ 
B)  $3$ 
C)  $- 3 x$ 
D)  $0$ 
E)  $3 x$

Question 19

The cost function for a certain model of personal digital assistant (PDA)is given by $C = 13.54 x + 45,740$ ,where C is measured in dollars and x is the number of PDAs produced.Write a model for the average cost per unit produced and also find the average costs per unit when  x  = 180 and  x = 1030.(Round your answer to two decimal places. )

Accepted Answer

A)  $\bar { C } = \text { average cost } = 13.54 - \frac { 45,740 } { x }$ $C ( 180 ) = \$ 267.65$ $C ( 1030 ) = \$ 57.95$ 
B)  $\bar { C } = \text { average cost } = 13.54 + \frac { 45,740 } { x }$ $\bar { C } ( 180 ) = \$ 57.95$ $C ( 1030 ) = \$ 267.65$ 
C)  $\bar { C } = \text { average cost } = 45,740 - \frac { 13.54 } { x }$ $C ( 180 ) = \$ 267.65$ $C ( 1030 ) = \$ 267.65$ 
D)  $\bar { C } = \text { average cost } = 13.54 + \frac { 45,740 } { x }$ $C ( 180 ) = \$ 267.65$ $C ( 1030 ) = \$ 57.95$ 
E)  $\bar { C } = \text { average cost } = 45,740 + \frac { 13.54 } { x }$ $C ( 180 ) = \$ 267.65$ $C ( 1030 ) = \$ 267.65$ 
A)  $\bar { C } = \text { average cost } = 13.54 - \frac { 45,740 } { x }$ $C ( 180 ) = \$ 267.65$ $C ( 1030 ) = \$ 57.95$ 
B)  $\bar { C } = \text { average cost } = 13.54 + \frac { 45,740 } { x }$ $\bar { C } ( 180 ) = \$ 57.95$ $C ( 1030 ) = \$ 267.65$ 
C)  $\bar { C } = \text { average cost } = 45,740 - \frac { 13.54 } { x }$ $C ( 180 ) = \$ 267.65$ $C ( 1030 ) = \$ 267.65$ 
D)  $\bar { C } = \text { average cost } = 13.54 + \frac { 45,740 } { x }$ $C ( 180 ) = \$ 267.65$ $C ( 1030 ) = \$ 57.95$ 
E)  $\bar { C } = \text { average cost } = 45,740 + \frac { 13.54 } { x }$ $C ( 180 ) = \$ 267.65$ $C ( 1030 ) = \$ 267.65$

Question 20

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
  $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f(x)=3 x+1&[0,2]
\end{array}$

Accepted Answer

A) A = 7 square units 
B) A = 11 square units 
C) A = 10 square units 
D) A = 8 square units 
E) A = 9 square units 
A) A = 7 square units 
B) A = 11 square units 
C) A = 10 square units 
D) A = 8 square units 
E) A = 9 square units

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 5 i ^ { 3 } } { n ^ { 5 } }$ as a rational function S(n)and find $\lim_ { n \rightarrow \infty } S ( n )$ .

Select correct graph of the following function. $y = \frac { 4 x + 5 } { x ^ { 2 } - 5 }$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 }$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x }$

Use the limit process to find the area of the region between f(x)= 10x + 6 and the x-axis on the interval [0,7].

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

Find the limit $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \ln \left( x ^ { 2 } \right) } { x - 4 }$

Rewrite $\sum _ { i = 1 } ^ { n } \left[ \frac { 3 } { n } + \left( \frac { 4 i } { n ^ { 2 } } \right) \right] \left( \frac { 4 i } { n } \right)$ as a rational function S(n)and find $\lim _ { n \rightarrow \infty } S ( n )$ .

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 3 } { x ^ { 2 } } + 4 \right)$

Determine what statement is true. The exact area of a region is given by the limit of the sum of n rectangles as n approaches

Use the first six terms to predict the limit of the sequence $\alpha _ { n } = \frac { 7 n ^ { 3 } } { n ^ { 3 } + 4 }$ (assume n begins with 1).

Approximate the limit accurate to three decimal places. $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x }$

Find the limit. $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x }$

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. $f ( x ) = \sqrt { x - 3 } , \quad ( 52,7 )$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 }$

Find the derivative of the function. $f ( x ) = 4 - 7 x ^ { 2 }$

Find the derivative of the function. $f ( x ) = - 3$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=3x+1 [0,2]

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

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Exam 12: Limits and An Introduction To Calculus

Rewrite ∑i=1n5i3n5\sum _ { i = 1 } ^ { n } \frac { 5 i ^ { 3 } } { n ^ { 5 } }∑i=1n​n55i3​ as a rational function S(n)and find lim⁡n→∞S(n)\lim_ { n \rightarrow \infty } S ( n )limn→∞​S(n) .

Select correct graph of the following function. y=4x+5x2−5y = \frac { 4 x + 5 } { x ^ { 2 } - 5 }y=x2−54x+5​

Find the limit (if it exists).Use a graphing utility to verify your result graphically. lim⁡x→8x−8x2−64\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 }limx→8​x2−64x−8​

Find the limit (if it exists).Use a graphing utility to verify your result graphically. lim⁡x→01x−6+16x\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x }limx→0​xx−61​+61​​

Use the limit process to find the area of the region between f(x)= 10x + 6 and the x-axis on the interval [0,7].

Select the correct graph of the following function. y=x2x2+2y = \frac { x ^ { 2 } } { x ^ { 2 } + 2 }y=x2+2x2​

Find the limit lim⁡x→0e4x−1x\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x }limx→0​xe4x−1​

Select the correct graph for the following function using a graphing utility. f(x)=ln⁡(x2)x−4f ( x ) = \frac { \ln \left( x ^ { 2 } \right) } { x - 4 }f(x)=x−4ln(x2)​

Find the limit (if it exists). lim⁡x→∞(3x2+4)\lim _ { x \rightarrow \infty } \left( \frac { 3 } { x ^ { 2 } } + 4 \right)limx→∞​(x23​+4)

Determine what statement is true. ​ The exact area of a region is given by the limit of the sum of n rectangles as n approaches ​

Use the first six terms to predict the limit of the sequence αn=7n3n3+4\alpha _ { n } = \frac { 7 n ^ { 3 } } { n ^ { 3 } + 4 }αn​=n3+47n3​ (assume n begins with 1).

Approximate the limit accurate to three decimal places. lim⁡x→0(1+4x)1/x\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x }limx→0​(1+4x)1/x

Find the limit. lim⁡x→01−e−7xx\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x }limx→0​x1−e−7x​

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−3,(52,7)f ( x ) = \sqrt { x - 3 } , \quad ( 52,7 )f(x)=x−3​,(52,7)

Find the limit (if it exists).Use a graphing utility to verify your result graphically. lim⁡x→−4x2+10x+24x+4\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 }limx→−4​x+4x2+10x+24​

Find the derivative of the function. f(x)=4−7x2f ( x ) = 4 - 7 x ^ { 2 }f(x)=4−7x2

Find the derivative of the function. f(x)=−3f ( x ) = - 3f(x)=−3

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=3x+1 [0,2]

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

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 5 i ^ { 3 } } { n ^ { 5 } }$ as a rational function S(n)and find $\lim_ { n \rightarrow \infty } S ( n )$ .

Select correct graph of the following function. $y = \frac { 4 x + 5 } { x ^ { 2 } - 5 }$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 8 } \frac { x - 8 } { x ^ { 2 } - 64 }$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { x - 6 } + \frac { 1 } { 6 } } { x }$

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

Find the limit $\lim _ { x \rightarrow 0 } \frac { e ^ { 4 x } - 1 } { x }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \ln \left( x ^ { 2 } \right) } { x - 4 }$

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 3 } { x ^ { 2 } } + 4 \right)$

Determine what statement is true. The exact area of a region is given by the limit of the sum of n rectangles as n approaches

Use the first six terms to predict the limit of the sequence $\alpha _ { n } = \frac { 7 n ^ { 3 } } { n ^ { 3 } + 4 }$ (assume n begins with 1).

Approximate the limit accurate to three decimal places. $\lim _ { x \rightarrow 0 } ( 1 + 4 x ) ^ { 1 / x }$

Find the limit. $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x }$

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. $f ( x ) = \sqrt { x - 3 } , \quad ( 52,7 )$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 }$

Find the derivative of the function. $f ( x ) = 4 - 7 x ^ { 2 }$

Find the derivative of the function. $f ( x ) = - 3$