Question 1

Find $\lim _ { x \rightarrow 3 } \frac { x - 3 } { x ^ { 2 } - 9 }$ .

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 2 } { 3 }$ 
E) limit does not exist 
A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 2 } { 3 }$ 
E) limit does not exist

Question 2

Find the first five terms of the sequence.   $\alpha _ { n } = \frac { n ^ { 2 } } { 2 n + 1 }$

Accepted Answer

A)  $\alpha _ { 1 } = \frac { 1 } { 3 } , \alpha _ { 2 } = \frac { 4 } { 5 } , \alpha _ { 3 } = \frac { 9 } { 7 } , \alpha _ { 4 } = \frac { 16 } { 9 } , \alpha _ { 5 } = \frac { 11 } { 5 }$ 
B)  $\alpha _ { 1 } = 3 , \alpha _ { 2 } = \frac { 4 } { 5 } , \alpha _ { 3 } = \frac { 7 } { 9 } , \alpha _ { 4 } = \frac { 16 } { 9 } , \alpha _ { 5 } = \frac { 25 } { 11 }$ 
C)  $\alpha _ { 1 } = \frac { 1 } { 3 } , \alpha _ { 2 } = \frac { 4 } { 5 } , \alpha _ { 3 } = \frac { 9 } { 7 } , \alpha _ { 4 } = \frac { 16 } { 9 } , \alpha _ { 5 } = \frac { 25 } { 11 }$ 
D)  $\alpha _ { 1 } = 3 , \alpha _ { 2 } = \frac { 4 } { 5 } , \alpha _ { 3 } = \frac { 7 } { 3 } , \alpha _ { 4 } = \frac { 16 } { 9 } , \alpha _ { 5 } = \frac { 25 } { 11 }$ 
E)  $\alpha _ { 1 } = 3 , \alpha _ { 2 } = \frac { 4 } { 5 } , \alpha _ { 3 } = \frac { 9 } { 7 } , \alpha _ { 4 } = \frac { 16 } { 9 } , \alpha _ { 5 } = \frac { 11 } { 5 }$ 
A)  $\alpha _ { 1 } = \frac { 1 } { 3 } , \alpha _ { 2 } = \frac { 4 } { 5 } , \alpha _ { 3 } = \frac { 9 } { 7 } , \alpha _ { 4 } = \frac { 16 } { 9 } , \alpha _ { 5 } = \frac { 11 } { 5 }$ 
B)  $\alpha _ { 1 } = 3 , \alpha _ { 2 } = \frac { 4 } { 5 } , \alpha _ { 3 } = \frac { 7 } { 9 } , \alpha _ { 4 } = \frac { 16 } { 9 } , \alpha _ { 5 } = \frac { 25 } { 11 }$ 
C)  $\alpha _ { 1 } = \frac { 1 } { 3 } , \alpha _ { 2 } = \frac { 4 } { 5 } , \alpha _ { 3 } = \frac { 9 } { 7 } , \alpha _ { 4 } = \frac { 16 } { 9 } , \alpha _ { 5 } = \frac { 25 } { 11 }$ 
D)  $\alpha _ { 1 } = 3 , \alpha _ { 2 } = \frac { 4 } { 5 } , \alpha _ { 3 } = \frac { 7 } { 3 } , \alpha _ { 4 } = \frac { 16 } { 9 } , \alpha _ { 5 } = \frac { 25 } { 11 }$ 
E)  $\alpha _ { 1 } = 3 , \alpha _ { 2 } = \frac { 4 } { 5 } , \alpha _ { 3 } = \frac { 9 } { 7 } , \alpha _ { 4 } = \frac { 16 } { 9 } , \alpha _ { 5 } = \frac { 11 } { 5 }$

Question 3

Find the limit (if it exists).  $\lim _ { x \rightarrow \infty } \left( \frac { 5 - 40 x } { x + 5 } \right)$

Accepted Answer

A) 40 
B) -40 
C) 5 
D) -8 
E) does not exist 
A) 40 
B) -40 
C) 5 
D) -8 
E) does not exist

Question 4

Use the limit process to find the slope of the graph of $\sqrt { x + 22 }$ at $( 3,5 )$ .

Accepted Answer

A)  $\frac { 1 } { 10 }$ 
B)  $\infty$ 
C) the slope is undefined at this point 
D)  $5$ 
E)  $\frac { 1 } { 5 }$ 
A)  $\frac { 1 } { 10 }$ 
B)  $\infty$ 
C) the slope is undefined at this point 
D)  $5$ 
E)  $\frac { 1 } { 5 }$

Question 5

Select the correct graph of the following function.  $f ( x ) = x - \sqrt { x ^ { 2 } + 7 }$

Accepted Answer

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

Question 6

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 7 i ^ { 3 } } { n ^ { 8 } }$ as a rational function S(n)and find $\lim _ { n \rightarrow \infty } S ( n )$ .

Accepted Answer

A)  $S ( n ) = \frac { 7 ( n + 1 ) ^ { 2 } } { 4 n ^ { 6 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
B)  $S ( n ) = \frac { 7 n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } , \lim _ { n \rightarrow \infty } S ( n ) = 7$ 
C)  $S ( n ) = \frac { 7 ( n + 1 ) ^ { 2 } } { 4 n ^ { 6 } } , \text { the limit does not exist }$ 
D)  $S ( n ) = \frac { 7 ( n + 1 ) ^ { 2 } } { 4 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 7 } { 4 }$ 
E) $S ( n ) = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 28 } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
A)  $S ( n ) = \frac { 7 ( n + 1 ) ^ { 2 } } { 4 n ^ { 6 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
B)  $S ( n ) = \frac { 7 n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } , \lim _ { n \rightarrow \infty } S ( n ) = 7$ 
C)  $S ( n ) = \frac { 7 ( n + 1 ) ^ { 2 } } { 4 n ^ { 6 } } , \text { the limit does not exist }$ 
D)  $S ( n ) = \frac { 7 ( n + 1 ) ^ { 2 } } { 4 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 7 } { 4 }$ 
E) $S ( n ) = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 28 } , \lim _ { n \rightarrow \infty } S ( n ) = 0$

Question 7

Use the first six terms to predict the limit of the sequence   $\alpha _ { n } = \frac { n + 2 } { n ^ { 3 } + 2 }$ (assume n begins with 1).

Accepted Answer

A) 0 
B) 2 
C) 1 
D)  $\frac { 1 } { 2 }$ 
E) the sequence diverges 
A) 0 
B) 2 
C) 1 
D)  $\frac { 1 } { 2 }$ 
E) the sequence diverges

Question 8

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not.  $f ( x ) = \frac { 4 } { x - 3 }$ , $\lim _ { x \rightarrow 3 } f ( x )$

Accepted Answer

A)    $\lim _ { x \rightarrow 3 } f ( x )$ =3 
B)     $\lim _ { x \rightarrow 3 } f ( x )$ =0 
C)    $\lim _ { x \rightarrow 3 } f ( x )$ =4 
D)    $\lim _ { x \rightarrow 3 } f ( x )$ does not exist 
E)     $\lim _ { x \rightarrow 3 } f ( x )$ does not exist 
A)    $\lim _ { x \rightarrow 3 } f ( x )$ =3 
B)     $\lim _ { x \rightarrow 3 } f ( x )$ =0 
C)    $\lim _ { x \rightarrow 3 } f ( x )$ =4 
D)    $\lim _ { x \rightarrow 3 } f ( x )$ does not exist 
E)     $\lim _ { x \rightarrow 3 } f ( x )$ does not exist

Question 9

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function.   $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 }$

Accepted Answer

A)  $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 } = - \frac { 1 } { 25 }$     
B)  $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 } = \frac { 1 } { 10 }$   
C)  $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 } = \frac { 1 } { 25 }$     
D)  $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 } = \frac { 1 } { 11 }$     
E)  $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 } = - \frac { 1 } { 10 }$    
A)  $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 } = - \frac { 1 } { 25 }$     
B)  $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 } = \frac { 1 } { 10 }$   
C)  $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 } = \frac { 1 } { 25 }$     
D)  $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 } = \frac { 1 } { 11 }$     
E)  $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 } = - \frac { 1 } { 10 }$

Question 10

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.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = 6 - x ^ { 2 }&[2,5]\
\end{array}$

Accepted Answer

A) A = 12.33 square units 
B) A = 10.33 square units 
C) A = 9.33 square units 
D) A = 13.33 square units 
E) A = 21 square units 
A) A = 12.33 square units 
B) A = 10.33 square units 
C) A = 9.33 square units 
D) A = 13.33 square units 
E) A = 21 square units

Question 11

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ .  $f ( x ) = 6 - 7 x$

Accepted Answer

A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = 7$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = 6$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 8$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 7$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = 8$ 
A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = 7$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = 6$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 8$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - 7$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = 8$

Question 12

Find the limit (if it exists).  $\lim _ { x \rightarrow \infty } \left( \frac { 6 x ^ { 2 } - 7 } { 5 - x ^ { 2 } } \right)$

Accepted Answer

A) 7 
B) -6 
C) -7 
D) 6 
E) does not exist 
A) 7 
B) -6 
C) -7 
D) 6 
E) does not exist

Question 13

Find the limit by direct substitution.   $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$

Accepted Answer

A)  $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$ = $\infty$ 
B)  $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$ = 14 
C)  $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$ = 70 
D)  $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$ = 9 
E)  $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$ = -2 
A)  $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$ = $\infty$ 
B)  $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$ = 14 
C)  $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$ = 70 
D)  $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$ = 9 
E)  $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$ = -2

Question 14

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ .  $f ( x ) = \frac { 1 } { x - 3 }$

Accepted Answer

A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 4 } { ( x - 3 ) ^ { - 2 } }$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 6 } { ( x - 3 ) ^ { 2 } }$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { ( x - 3 ) ^ { - 2 } }$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 1 } { ( x - 3 ) ^ { 2 } }$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { ( x - 3 ) ^ { 2 } }$ 
A)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 4 } { ( x - 3 ) ^ { - 2 } }$ 
B)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 6 } { ( x - 3 ) ^ { 2 } }$ 
C)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { ( x - 3 ) ^ { - 2 } }$ 
D)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \frac { 1 } { ( x - 3 ) ^ { 2 } }$ 
E)  $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = - \frac { 1 } { ( x - 3 ) ^ { 2 } }$

Question 15

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not.  $f ( x ) = \frac { \sqrt { x + 7 } - 1 } { x - 4 } , \lim _ { x \rightarrow 4 } f ( x )$

Accepted Answer

A)     $\lim _ { x \rightarrow 4 } f ( x )$ =7 
B)    $\lim _ { x \rightarrow 4 } f ( x )$ =6 
C)    $\lim _ { x \rightarrow 4 } f ( x )$ =6 
D)     $\lim _ { x \rightarrow 4 } f ( x )$ does not exist. 
E)    $\lim _ { x \rightarrow 4 } f ( x )$ does not exist. 
A)     $\lim _ { x \rightarrow 4 } f ( x )$ =7 
B)    $\lim _ { x \rightarrow 4 } f ( x )$ =6 
C)    $\lim _ { x \rightarrow 4 } f ( x )$ =6 
D)     $\lim _ { x \rightarrow 4 } f ( x )$ does not exist. 
E)    $\lim _ { x \rightarrow 4 } f ( x )$ does not exist.

Question 16

Select correct graph of the following function.  $y = 3 + \frac { 2 } { x }$

Accepted Answer

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

Question 17

Find the derivative of $f ( x ) = 5 x ^ { 2 } - 9 x + 5$ .

Accepted Answer

A)  $10 x - 9$ 
B)  $5 x - 9$ 
C)  $10 x - 4$ 
D)  $5 x - 4$ 
E)  $10 x$ 
A)  $10 x - 9$ 
B)  $5 x - 9$ 
C)  $10 x - 4$ 
D)  $5 x - 4$ 
E)  $10 x$

Question 18

Find the limit (if it exists).  $\lim _ { x \rightarrow + \infty } \frac { 4 x ^ { 3 } + 6 } { 11 x ^ { 3 } - 7 x ^ { 2 } + 7 }$

Accepted Answer

A)  $- \frac { 4 } { 11 }$ 
B)  $\frac { 11 } { 4 }$ 
C)  $\frac { 4 } { 11 }$ 
D)  $- \frac { 11 } { 4 }$ 
E) does not exist 
A)  $- \frac { 4 } { 11 }$ 
B)  $\frac { 11 } { 4 }$ 
C)  $\frac { 4 } { 11 }$ 
D)  $- \frac { 11 } { 4 }$ 
E) does not exist

Question 19

Find the derivative of $f ( x ) = \frac { 3 } { x ^ { 4 } }$ .

Accepted Answer

A)  $\frac { 12 } { x ^ { 3 } }$ 
B)  $\frac { 3 } { 4 x ^ { 3 } }$ 
C)  $- \frac { 12 } { x ^ { 3 } }$ 
D)  $\frac { 3 } { x ^ { 3 } }$ 
E)  $- \frac { 12 } { x ^ { 5 } }$ 
A)  $\frac { 12 } { x ^ { 3 } }$ 
B)  $\frac { 3 } { 4 x ^ { 3 } }$ 
C)  $- \frac { 12 } { x ^ { 3 } }$ 
D)  $\frac { 3 } { x ^ { 3 } }$ 
E)  $- \frac { 12 } { x ^ { 5 } }$

Question 20

Use the limit process to find the area of the region between $f ( x ) = 6 x + 2$ and the x-axis on the interval [0,7].

Accepted Answer

A) 308 
B) 161 
C) 56 
D) 147 
E) 149 
A) 308 
B) 161 
C) 56 
D) 147 
E) 149

Find $\lim _ { x \rightarrow 3 } \frac { x - 3 } { x ^ { 2 } - 9 }$ .

Find the first five terms of the sequence. $\alpha _ { n } = \frac { n ^ { 2 } } { 2 n + 1 }$

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 5 - 40 x } { x + 5 } \right)$

Use the limit process to find the slope of the graph of $\sqrt { x + 22 }$ at $( 3,5 )$ .

Select the correct graph of the following function. $f ( x ) = x - \sqrt { x ^ { 2 } + 7 }$

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 7 i ^ { 3 } } { n ^ { 8 } }$ as a rational function S(n)and find $\lim _ { n \rightarrow \infty } S ( n )$ .

Use the first six terms to predict the limit of the sequence $\alpha _ { n } = \frac { n + 2 } { n ^ { 3 } + 2 }$ (assume n begins with 1).

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \frac { 4 } { x - 3 }$ , $\lim _ { x \rightarrow 3 } f ( x )$

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function. $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 }$

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)=6- [2,5]

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = 6 - 7 x$

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 6 x ^ { 2 } - 7 } { 5 - x ^ { 2 } } \right)$

Find the limit by direct substitution. $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = \frac { 1 } { x - 3 }$

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \frac { \sqrt { x + 7 } - 1 } { x - 4 } , \lim _ { x \rightarrow 4 } f ( x )$

Select correct graph of the following function. $y = 3 + \frac { 2 } { x }$

Find the derivative of $f ( x ) = 5 x ^ { 2 } - 9 x + 5$ .

Find the limit (if it exists). $\lim _ { x \rightarrow + \infty } \frac { 4 x ^ { 3 } + 6 } { 11 x ^ { 3 } - 7 x ^ { 2 } + 7 }$

Find the derivative of $f ( x ) = \frac { 3 } { x ^ { 4 } }$ .

Use the limit process to find the area of the region between $f ( x ) = 6 x + 2$ and the x-axis on the interval [0,7].

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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Exam 12: Limits and An Introduction To Calculus

Find lim⁡x→3x−3x2−9\lim _ { x \rightarrow 3 } \frac { x - 3 } { x ^ { 2 } - 9 }limx→3​x2−9x−3​ .

Find the first five terms of the sequence. αn=n22n+1\alpha _ { n } = \frac { n ^ { 2 } } { 2 n + 1 }αn​=2n+1n2​

Find the limit (if it exists). lim⁡x→∞(5−40xx+5)\lim _ { x \rightarrow \infty } \left( \frac { 5 - 40 x } { x + 5 } \right)limx→∞​(x+55−40x​)

Use the limit process to find the slope of the graph of x+22\sqrt { x + 22 }x+22​ at (3,5)( 3,5 )(3,5) .

Select the correct graph of the following function. f(x)=x−x2+7f ( x ) = x - \sqrt { x ^ { 2 } + 7 }f(x)=x−x2+7​

Rewrite ∑i=1n7i3n8\sum _ { i = 1 } ^ { n } \frac { 7 i ^ { 3 } } { n ^ { 8 } }∑i=1n​n87i3​ as a rational function S(n)and find lim⁡n→∞S(n)\lim _ { n \rightarrow \infty } S ( n )limn→∞​S(n) .

Use the first six terms to predict the limit of the sequence αn=n+2n3+2\alpha _ { n } = \frac { n + 2 } { n ^ { 3 } + 2 }αn​=n3+2n+2​ (assume n begins with 1).

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. f(x)=4x−3f ( x ) = \frac { 4 } { x - 3 }f(x)=x−34​ , lim⁡x→3f(x)\lim _ { x \rightarrow 3 } f ( x )limx→3​f(x)

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function. lim⁡x→5−x−5x2−25\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 }limx→5−​x2−25x−5​

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)=6- [2,5]

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)=6−7xf ( x ) = 6 - 7 xf(x)=6−7x

Find the limit (if it exists). lim⁡x→∞(6x2−75−x2)\lim _ { x \rightarrow \infty } \left( \frac { 6 x ^ { 2 } - 7 } { 5 - x ^ { 2 } } \right)limx→∞​(5−x26x2−7​)

Find the limit by direct substitution. lim⁡x→−2(6x2+8x+1)\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)limx→−2​(6x2+8x+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)=1x−3f ( x ) = \frac { 1 } { x - 3 }f(x)=x−31​

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. f(x)=x+7−1x−4,lim⁡x→4f(x)f ( x ) = \frac { \sqrt { x + 7 } - 1 } { x - 4 } , \lim _ { x \rightarrow 4 } f ( x )f(x)=x−4x+7​−1​,limx→4​f(x)

Select correct graph of the following function. y=3+2xy = 3 + \frac { 2 } { x }y=3+x2​

Find the derivative of f(x)=5x2−9x+5f ( x ) = 5 x ^ { 2 } - 9 x + 5f(x)=5x2−9x+5 .

Find the limit (if it exists). lim⁡x→+∞4x3+611x3−7x2+7\lim _ { x \rightarrow + \infty } \frac { 4 x ^ { 3 } + 6 } { 11 x ^ { 3 } - 7 x ^ { 2 } + 7 }limx→+∞​11x3−7x2+74x3+6​

Find the derivative of f(x)=3x4f ( x ) = \frac { 3 } { x ^ { 4 } }f(x)=x43​ .

Use the limit process to find the area of the region between f(x)=6x+2f ( x ) = 6 x + 2f(x)=6x+2 and the x-axis on the interval [0,7].

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 $\lim _ { x \rightarrow 3 } \frac { x - 3 } { x ^ { 2 } - 9 }$ .

Find the first five terms of the sequence. $\alpha _ { n } = \frac { n ^ { 2 } } { 2 n + 1 }$

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 5 - 40 x } { x + 5 } \right)$

Use the limit process to find the slope of the graph of $\sqrt { x + 22 }$ at $( 3,5 )$ .

Select the correct graph of the following function. $f ( x ) = x - \sqrt { x ^ { 2 } + 7 }$

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 7 i ^ { 3 } } { n ^ { 8 } }$ as a rational function S(n)and find $\lim _ { n \rightarrow \infty } S ( n )$ .

Use the first six terms to predict the limit of the sequence $\alpha _ { n } = \frac { n + 2 } { n ^ { 3 } + 2 }$ (assume n begins with 1).

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \frac { 4 } { x - 3 }$ , $\lim _ { x \rightarrow 3 } f ( x )$

Graphically approximate the limit (if it exists)by using a graphing utility to graph the function. $\lim _ { x \rightarrow 5 ^ { - } } \frac { x - 5 } { x ^ { 2 } - 25 }$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = 6 - 7 x$

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 6 x ^ { 2 } - 7 } { 5 - x ^ { 2 } } \right)$

Find the limit by direct substitution. $\lim _ { x \rightarrow - 2 } \left( 6 x ^ { 2 } + 8 x + 1 \right)$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = \frac { 1 } { x - 3 }$

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \frac { \sqrt { x + 7 } - 1 } { x - 4 } , \lim _ { x \rightarrow 4 } f ( x )$

Select correct graph of the following function. $y = 3 + \frac { 2 } { x }$

Find the derivative of $f ( x ) = 5 x ^ { 2 } - 9 x + 5$ .

Find the limit (if it exists). $\lim _ { x \rightarrow + \infty } \frac { 4 x ^ { 3 } + 6 } { 11 x ^ { 3 } - 7 x ^ { 2 } + 7 }$

Find the derivative of $f ( x ) = \frac { 3 } { x ^ { 4 } }$ .

Use the limit process to find the area of the region between $f ( x ) = 6 x + 2$ and the x-axis on the interval [0,7].