Question 1

Find $\lim _ { t \rightarrow 6 } \frac { t ^ { 3 } - 216 } { t - 6 }$ .

Accepted Answer

A) 18 
B) 72 
C) 108 
D) 6 
E) limit does not exist 
A) 18 
B) 72 
C) 108 
D) 6 
E) limit does not exist

Question 2

Use the limit process to find the slope of the graph of $6 x - 8 x ^ { 2 }$ at $( 7 , - 350 )$ .

Accepted Answer

A)  $- 10$ 
B)  $118$ 
C)  $62$ 
D)  $112$ 
E)  $- 106$ 
A)  $- 10$ 
B)  $118$ 
C)  $62$ 
D)  $112$ 
E)  $- 106$

Question 3

Find the derivative of $f ( x ) = \frac { 6 } { x ^ { 5 } }$ .

Accepted Answer

A)  $\frac { 6 } { x ^ { 4 } }$ 
B)  $- \frac { 30 } { x ^ { 6 } }$ 
C)  $\frac { 6 } { 5 x ^ { 4 } }$ 
D)  $\frac { 30 } { x ^ { 4 } }$ 
E)  $- \frac { 30 } { x ^ { 4 } }$ 
A)  $\frac { 6 } { x ^ { 4 } }$ 
B)  $- \frac { 30 } { x ^ { 6 } }$ 
C)  $\frac { 6 } { 5 x ^ { 4 } }$ 
D)  $\frac { 30 } { x ^ { 4 } }$ 
E)  $- \frac { 30 } { x ^ { 4 } }$

Question 4

Find the limit (if it exists).   $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 } - 4 } { x }$

Accepted Answer

A)  $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 } - 4 } { x } = - \frac { 1 } { 8 }$ 
B)  $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 - 4 } } { x } = \frac { 1 } { 4 }$ 
C)  $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 - 4 } } { x } = \frac { 1 } { 8 }$ 
D)  $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 } - 4 } { x } = \frac { 1 } { 20 }$ 
E)  $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 } - 4 } { x } = - \frac { 1 } { 20 }$ 
A)  $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 } - 4 } { x } = - \frac { 1 } { 8 }$ 
B)  $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 - 4 } } { x } = \frac { 1 } { 4 }$ 
C)  $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 - 4 } } { x } = \frac { 1 } { 8 }$ 
D)  $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 } - 4 } { x } = \frac { 1 } { 20 }$ 
E)  $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 } - 4 } { x } = - \frac { 1 } { 20 }$

Question 5

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

Accepted Answer

A)  $- \frac { 4 } { 3 }$ 
B) 0 
C)   -$\frac { 3 } { 4 }$ 
D) 3 
E) $\infty$ 
A)  $- \frac { 4 } { 3 }$ 
B) 0 
C)   -$\frac { 3 } { 4 }$ 
D) 3 
E) $\infty$

Question 6

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 4 i + 2 } { n ^ { 2 } }$ as a rational function S(n)and find $\lim S ( n )$ .

Accepted Answer

A)  $S ( n ) = \frac { 2 ( n + 1 ) + 2 } { n } , \lim _ { n \rightarrow \infty } S ( n ) = 2$ 
B)  $S ( n ) = \frac { 2 ( n + 1 ) + 2 } { n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
C)  $S ( n ) = \frac { 2 ( n + 1 ) + 2 } { n } , \lim _ { n \rightarrow \infty } S ( n ) = 4$ 
D)  $S ( n ) = \frac { 2 ( n + 1 ) 2 } { n } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
E)  $S ( n ) = \frac { 2 n ( n + 1 ) + 2 } { n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = 2$ 
A)  $S ( n ) = \frac { 2 ( n + 1 ) + 2 } { n } , \lim _ { n \rightarrow \infty } S ( n ) = 2$ 
B)  $S ( n ) = \frac { 2 ( n + 1 ) + 2 } { n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
C)  $S ( n ) = \frac { 2 ( n + 1 ) + 2 } { n } , \lim _ { n \rightarrow \infty } S ( n ) = 4$ 
D)  $S ( n ) = \frac { 2 ( n + 1 ) 2 } { n } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
E)  $S ( n ) = \frac { 2 n ( n + 1 ) + 2 } { n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = 2$

Question 7

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 9 x } { 1 + 5 x }$ (if it exists).

Accepted Answer

A)  $\frac { 9 } { 5 }$ 
B)  $- \frac { 9 } { 5 }$ 
C) -$\infty$ 
D) 0 
E) does not exist 
A)  $\frac { 9 } { 5 }$ 
B)  $- \frac { 9 } { 5 }$ 
C) -$\infty$ 
D) 0 
E) does not exist

Question 8

Use the limit process to find the area of the region between f(x)= x2 + 7 and the x-axis on the interval [0,3].

Accepted Answer

A)  $16$ 
B)  $30$ 
C)  $9$ 
D)  $48$ 
E)  $34$ 
A)  $16$ 
B)  $30$ 
C)  $9$ 
D)  $48$ 
E)  $34$

Question 9

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ .  $f ( x ) = 2 x + 1$

Accepted Answer

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

Question 10

Find the limit (if it exists).Use a graphing utility to verify your result graphically.   $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 }$

Accepted Answer

A)  $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 } = - 16$   
B)  $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 } = - 63$   
C)  $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 } = 63$   
D)  $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 } = 16$    
E)  $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 } = 7$   
A)  $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 } = - 16$   
B)  $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 } = - 63$   
C)  $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 } = 63$   
D)  $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 } = 16$    
E)  $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 } = 7$

Question 11

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \frac { x } { x + 1 } - 1 } { x + 2 }$

Accepted Answer

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

Question 12

Find the limit (if it exists).   $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x }$

Accepted Answer

A)  $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x } = - 8$ 
B)  $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x } = 72$ 
C)  $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x } = 64$ 
D)  $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x } = 8$ 
E)  $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x } = - 64$ 
A)  $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x } = - 8$ 
B)  $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x } = 72$ 
C)  $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x } = 64$ 
D)  $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x } = 8$ 
E)  $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x } = - 64$

Question 13

Use the limit process to find the slope of the graph of the function at the specified point.Use a graphing utility to confirm your result.   $g ( x ) = \frac { 1 } { x - 3 } , \quad \left( 6 , \frac { 1 } { 3 } \right)$

Accepted Answer

A)  $m = 3$ 
B)  $m = - 3$ 
C)  $m = 9$ 
D)  $m = - \frac { 1 } { 9 }$ 
E)  $m = \frac { 1 } { 6 }$ 
A)  $m = 3$ 
B)  $m = - 3$ 
C)  $m = 9$ 
D)  $m = - \frac { 1 } { 9 }$ 
E)  $m = \frac { 1 } { 6 }$

Question 14

Use the limit process to find the slope of the graph of $\sqrt { x + 7 }$ at $( 2,3 )$ .

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $3$ 
C)  $\infty$ 
D)  $\frac { 1 } { 6 }$ 
E) the slope is undefined at this point 
A)  $\frac { 1 } { 3 }$ 
B)  $3$ 
C)  $\infty$ 
D)  $\frac { 1 } { 6 }$ 
E) the slope is undefined at this point

Question 15

Evaluate the sum using the summation formula and property.   $\sum _ { i = 1 } ^ { 45 } 4$

Accepted Answer

A) 160 
B) 200 
C) 190 
D) 180 
E) 170 
A) 160 
B) 200 
C) 190 
D) 180 
E) 170

Question 16

Find the limit (if it exists).   $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y }$

Accepted Answer

A)  $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y } = \frac { \sqrt { 7 } } { 7 }$ 
B)  $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y } = \frac { \sqrt { 7 } } { 14 }$ 
C)  $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y } = - \frac { \sqrt { 7 } } { 7 }$ 
D)  $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y } = \frac { 14 \sqrt { 7 } } { 7 }$ 
E)  $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y } = - \frac { \sqrt { 7 } } { 14 }$ 
A)  $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y } = \frac { \sqrt { 7 } } { 7 }$ 
B)  $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y } = \frac { \sqrt { 7 } } { 14 }$ 
C)  $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y } = - \frac { \sqrt { 7 } } { 7 }$ 
D)  $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y } = \frac { 14 \sqrt { 7 } } { 7 }$ 
E)  $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y } = - \frac { \sqrt { 7 } } { 14 }$

Question 17

Rewrite the sum as a rational function S(n). $\sum _ { i = 1 } ^ { n } \frac { i } { n ^ { 4 } }$

Accepted Answer

A)   $S ( n ) = \frac { n + 1 } { 2 n ^ { 5 } }$ 
B)   $S ( n ) = \frac { n + 1 } { 2 n ^ { 6 } }$ 
C)  $S ( n ) = \frac { n + 1 } { 2 n ^ { 3 } }$ 
D)  $S ( n ) = \frac { n + 1 } { 2 n ^ { 7 } }$ 
E)  $S ( n ) = \frac { n + 1 } { 2 n ^ { 4 } }$ 
A)   $S ( n ) = \frac { n + 1 } { 2 n ^ { 5 } }$ 
B)   $S ( n ) = \frac { n + 1 } { 2 n ^ { 6 } }$ 
C)  $S ( n ) = \frac { n + 1 } { 2 n ^ { 3 } }$ 
D)  $S ( n ) = \frac { n + 1 } { 2 n ^ { 7 } }$ 
E)  $S ( n ) = \frac { n + 1 } { 2 n ^ { 4 } }$

Question 18

Evaluate the sum using the summation formula and property. 
 $\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)$

Accepted Answer

A) 89,150 
B) 88,950 
C) 89,250 
D) 88,850 
E) 89,050 
A) 89,150 
B) 88,950 
C) 89,250 
D) 88,850 
E) 89,050

Question 19

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

Accepted Answer

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

Question 20

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \frac { 1 } { x + 1 } - \frac { 1 } { 2 } } { x - 1 }$

Accepted Answer

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

Find $\lim _ { t \rightarrow 6 } \frac { t ^ { 3 } - 216 } { t - 6 }$ .

Use the limit process to find the slope of the graph of $6 x - 8 x ^ { 2 }$ at $( 7 , - 350 )$ .

Find the derivative of $f ( x ) = \frac { 6 } { x ^ { 5 } }$ .

Find the limit (if it exists). $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 } - 4 } { x }$

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

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 4 i + 2 } { n ^ { 2 } }$ as a rational function S(n)and find $\lim S ( n )$ .

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 9 x } { 1 + 5 x }$ (if it exists).

Use the limit process to find the area of the region between f(x)= x² + 7 and the x-axis on the interval [0,3].

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = 2 x + 1$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \frac { x } { x + 1 } - 1 } { x + 2 }$

Find the limit (if it exists). $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x }$

Use the limit process to find the slope of the graph of the function at the specified point.Use a graphing utility to confirm your result. $g ( x ) = \frac { 1 } { x - 3 } , \quad \left( 6 , \frac { 1 } { 3 } \right)$

Use the limit process to find the slope of the graph of $\sqrt { x + 7 }$ at $( 2,3 )$ .

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 45 } 4$

Find the limit (if it exists). $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y }$

Rewrite the sum as a rational function S(n). $\sum _ { i = 1 } ^ { n } \frac { i } { n ^ { 4 } }$

Evaluate the sum using the summation formula and property. $\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)$

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

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \frac { 1 } { x + 1 } - \frac { 1 } { 2 } } { x - 1 }$

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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Exam 12: Limits and An Introduction To Calculus

Find lim⁡t→6t3−216t−6\lim _ { t \rightarrow 6 } \frac { t ^ { 3 } - 216 } { t - 6 }limt→6​t−6t3−216​ .

Use the limit process to find the slope of the graph of 6x−8x26 x - 8 x ^ { 2 }6x−8x2 at (7,−350)( 7 , - 350 )(7,−350) .

Find the derivative of f(x)=6x5f ( x ) = \frac { 6 } { x ^ { 5 } }f(x)=x56​ .

Find the limit (if it exists). lim⁡x→0x+16−4x\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 } - 4 } { x }limx→0​xx+16​−4​

Find the limit (if it exists). lim⁡y→∞3y4y2+4\lim _ { y \rightarrow \infty } \frac { 3 y ^ { 4 } } { y ^ { 2 } + 4 }limy→∞​y2+43y4​

Rewrite ∑i=1n4i+2n2\sum _ { i = 1 } ^ { n } \frac { 4 i + 2 } { n ^ { 2 } }∑i=1n​n24i+2​ as a rational function S(n)and find lim⁡S(n)\lim S ( n )limS(n) .

Find lim⁡x→∞1−9x1+5x\lim _ { x \rightarrow \infty } \frac { 1 - 9 x } { 1 + 5 x }limx→∞​1+5x1−9x​ (if it exists).

Use the limit process to find the area of the region between f(x)= x2 + 7 and the x-axis on the interval [0,3].

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)=2x+1f ( x ) = 2 x + 1f(x)=2x+1

Find the limit (if it exists).Use a graphing utility to verify your result graphically. lim⁡x→7x2+2x−63x−7\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 }limx→7​x−7x2+2x−63​

Select the correct graph for the following function using a graphing utility. f(x)=xx+1−1x+2f ( x ) = \frac { \frac { x } { x + 1 } - 1 } { x + 2 }f(x)=x+2x+1x​−1​

Find the limit (if it exists). lim⁡x→π8csc⁡xcot⁡x\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x }limx→π​cotx8cscx​

Use the limit process to find the slope of the graph of the function at the specified point.Use a graphing utility to confirm your result. g(x)=1x−3,(6,13)g ( x ) = \frac { 1 } { x - 3 } , \quad \left( 6 , \frac { 1 } { 3 } \right)g(x)=x−31​,(6,31​)

Use the limit process to find the slope of the graph of x+7\sqrt { x + 7 }x+7​ at (2,3)( 2,3 )(2,3) .

Evaluate the sum using the summation formula and property. ∑i=1454\sum _ { i = 1 } ^ { 45 } 4∑i=145​4

Find the limit (if it exists). lim⁡y→0y+7−7y\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y }limy→0​yy+7​−7​​

Rewrite the sum as a rational function S(n). ∑i=1nin4\sum _ { i = 1 } ^ { n } \frac { i } { n ^ { 4 } }∑i=1n​n4i​

Evaluate the sum using the summation formula and property. ∑j=125(j3−3j2)\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)∑j=125​(j3−3j2)

Find the derivative of the function. f(x)=−3x+2f ( x ) = - 3 x + 2f(x)=−3x+2

Select the correct graph for the following function using a graphing utility. f(x)=1x+1−12x−1f ( x ) = \frac { \frac { 1 } { x + 1 } - \frac { 1 } { 2 } } { x - 1 }f(x)=x−1x+11​−21​​

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

Find $\lim _ { t \rightarrow 6 } \frac { t ^ { 3 } - 216 } { t - 6 }$ .

Use the limit process to find the slope of the graph of $6 x - 8 x ^ { 2 }$ at $( 7 , - 350 )$ .

Find the derivative of $f ( x ) = \frac { 6 } { x ^ { 5 } }$ .

Find the limit (if it exists). $\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 16 } - 4 } { x }$

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

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 4 i + 2 } { n ^ { 2 } }$ as a rational function S(n)and find $\lim S ( n )$ .

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 9 x } { 1 + 5 x }$ (if it exists).

Use the limit process to find the area of the region between f(x)= x² + 7 and the x-axis on the interval [0,3].

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = 2 x + 1$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } + 2 x - 63 } { x - 7 }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \frac { x } { x + 1 } - 1 } { x + 2 }$

Find the limit (if it exists). $\lim _ { x \rightarrow \pi } \frac { 8 \csc x } { \cot x }$

Use the limit process to find the slope of the graph of the function at the specified point.Use a graphing utility to confirm your result. $g ( x ) = \frac { 1 } { x - 3 } , \quad \left( 6 , \frac { 1 } { 3 } \right)$

Use the limit process to find the slope of the graph of $\sqrt { x + 7 }$ at $( 2,3 )$ .

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 45 } 4$

Find the limit (if it exists). $\lim _ { y \rightarrow 0 } \frac { \sqrt { y + 7 } - \sqrt { 7 } } { y }$

Rewrite the sum as a rational function S(n). $\sum _ { i = 1 } ^ { n } \frac { i } { n ^ { 4 } }$

Evaluate the sum using the summation formula and property. $\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)$

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

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \frac { 1 } { x + 1 } - \frac { 1 } { 2 } } { x - 1 }$