Deck 12: Limits and An Introduction To Calculus

Question 1

Evaluate the sum using the summation formula and property.
$\sum _ { k = 1 } ^ { 10 } \left( k ^ { 3 } + 9 \right)$

A)3,015
B)3,315
C)3,215
D)2,915
E)3,115

Answer 1

3,115

Question 2

Evaluate the sum using the summation formula and property.
$\sum _ { j = 1 } ^ { 20 } \left( j ^ { 2 } + j \right)$

A)2,980
B)3,080
C)2,880
D)3,180
E)3,280

Answer 2

3,080

Question 3

Find $\lim _{n \rightarrow \infty} S(n) \sum_{i=1}^{n}\left[4-2\left(\frac{i}{n}\right)\right]\left(\frac{1}{n}\right)$

A) $\lim_{n\rightarrow \infty } S ( n ) = 3$
B) $\lim _{n\rightarrow \infty }S ( n ) = 7$
C) $\lim_{n\rightarrow \infty } S ( n ) = 4$
D) $\lim_{n\rightarrow\infty } S ( n ) = 5$
E) $\lim_{n\rightarrow \infty } S ( n ) = 6$

Answer 3

$\lim_{n\rightarrow \infty } S ( n ) = 3$

Question 4

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

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

Question 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.
$\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = 3 x - 4&[2,5]\\\end{array}$

A)A = 21.5 square units
B)A = 21 square units
C)A = 19.5 square units
D)A = 20 square units
E)A = 20.5 square units

Question 6

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.
$\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = - 2 x + 5&[0,2]\\\end{array}$

A)A = 7 square units
B)A = 8 square units
C)A = 5square units
D)A = 4 square units
E)A = 6 square units

Question 7

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.
$\begin{array} {l } \text {Function }& \text {Interval }\\f(x)=3 x+1&[0,2]\end{array}$

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

Question 8

Evaluate the sum using the summation formula and property.
$\sum _ { i = 1 } ^ { 10 } i ^ { 3 }$

A)3,125
B)2,825
C)3,225
D)2,925
E)3,025

Question 9

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.
$\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = 9 - x ^ { 3 }&[1,2]\\\end{array}$

A)A = 5.375 square units
B)A = 5.5 square units
C)A = 5.25 square units
D)A = 5.625 square units
E)A = 5.125 square units

Question 10

Evaluate the sum using the summation formula and property.
$\sum _ { i = 1 } ^ { 30 } i ^ { 2 }$

A)9,355
B)9,455
C)9,655
D)9,255
E)9,555

Question 11

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.
$\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = 66 - x ^ { 3 }&[1,4]\\\end{array}$

A)A = 85.5 square units
B)A = 36.75 square units
C)A = 231.75 square units
D)A = 183 square units
E)A = 134.25 square units

Question 12

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

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

Question 13

Rewrite the sum as a rational function S(n). $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 3 } }$

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

Question 14

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 60 } 7$

A)410
B)400
C)420
D)430
E)440

Question 15

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.
$\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = 8 x - x ^ { 3 } &[0,1]\\\end{array}$

A)A = 4.5 square units
B)A = 3 square units
C)A = 4 square units
D)A = 3.5 square units
E)A = 3.75 square units

Question 16

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

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 17

Find $\lim _ { n \rightarrow \infty }S ( n )$ .
$\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 4 } }$

A) $\lim _ { n \rightarrow \infty } S ( n ) = \frac { 1 } { 6 }$
B) $\lim _ { n \rightarrow \infty } S ( n ) = \frac { 1 } { 5 }$
C) $\lim _ { n \rightarrow \infty } S ( n ) = \frac { 1 } { 4 }$
D) $\lim _ { n \rightarrow \infty } S ( n ) = \frac { 1 } { 8 }$
E) $\lim _ { n \rightarrow \infty } S ( n ) = \frac { 1 } { 7 }$

Question 18

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

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 19

Evaluate the sum using the summation formula and property.
$\sum _ { k = 1 } ^ { 35 } ( 2 k + 3 )$

A)1,465
B)1,565
C)1,165
D)1,365
E)1,265

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.
$\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = x ^ { 2 } + 2&[0,1]\\\end{array}$

A)A = 2 square units
B)A = 3.33 square units
C)A = 3 square units
D)A = 2.33 square units
E)A = 2.67 square units

Question 21

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 4 - x ^ { 2 }$

A) $7.28$
B) $3.64$
C) $4.48$
D) $3.83$
E) $7.25$

Question 22

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

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 23

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

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 24

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 10 } \left( x ^ { 2 } + 10 x \right)$ and the x-axis on the interval $[ 1,10 ]$ .Round the answer upto 1 decimal places.

A) $1049.5$
B) $132.3$
C) $149.5$
D) $1098$
E) $82.8$

Question 25

Approximate the area of the indicated region under the given curve using five rectangles. $$f ( x ) = 3 - x ^ { 2 }$$

A)7.28
B)3.48
C)2.83
D)5.28
E)5.25

Question 26

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

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

Question 27

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

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 28

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 $\lim _ { n \rightarrow \infty } S ( n )$ .

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 29

Evaluate $\sum _ { k = 1 } ^ { 20 } \left( k ^ { 3 } + 4 \right)$ using the summation formulas and properties.

A)8,004
B)176,400
C)44,180
D)176,480
E)44,100

Question 30

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the x-axis on the interval [1,8].Round your answer upto 1 decimal places.

A) $52.8$
B) $84.3$
C) $543.5$
D) $574$
E) $95.5$

Question 31

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].

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

Question 32

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.
$\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = 8 x - x ^ { 3 } &[0,2]\\\end{array}$

A)A = 10 square units
B)A = 8 square units
C)A = 6 square units
D)A = 12 square units
E)A = 14 square units

Question 33

Evaluate $\sum _ { j = 1 } ^ { 20 } \left( j ^ { 3 } - 4 j ^ { 2 } \right)$ using the summation formulas and properties.

A)44,100
B)42,500
C)44,096
D)44,020
E)32,620

Question 34

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
​

A)0
B)​infinity

Question 35

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 10 i + 10 } { n ^ { 2 } }$ as a rational function S(n)and find $\lim_ { n \rightarrow \infty } S ( n )$ .

A) $S ( n ) = \frac { 5 ( n + 1 ) + 10 } { n } , \lim _ { n \rightarrow \infty } S ( n ) = 0$
B) $S ( n ) = \frac { 5 ( n + 1 ) + 10 } { n } , \lim _ { n \rightarrow \infty } S ( n ) = 5$
C) $S ( n ) = \frac { 5 ( n + 1 ) + 10 } { n } , \lim _ { n \rightarrow \infty } S ( n ) = 10$
D) $S ( n ) = \frac { 5 ( n + 1 ) + 10 } { n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$
E) $S ( n ) = \frac { 5 n ( n + 1 ) + 10 } { n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = 5$

Question 36

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].

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

Question 37

Use the limit process to find the area of the region between $f ( x ) = x ^ { 2 } + 2$ and the x-axis on the interval [0,9].

A)747
B)243
C)731
D)261
E)245

Question 38

Determine what statement is true.
The sum of the first n positive integers is

A) $\frac { n ( n + 1 ) } { 2 }$
B) $\frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 }$

Question 39

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].

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

Question 40

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

A) $105,622$
B) $105,550$
C) $103,750$
D) $105,625$
E) $89,050$

Question 41

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 32 x - 1 } { 4 x + 4 } \right)$

A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 32 }$
C)8
D)-8
E)does not exist

Question 42

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 4 } { x } - 6 \right)$

A)10
B)6
C)-6
D)4
E)does not exist

Question 43

Select the correct function for the graph using horizontal asymptotes as aids.

A) $6 + \frac { 4 } { x ^ { 2 } }$
B) $\frac { 6 } { x ^ { 2 } }$
C) $6 - \frac { 4 } { x ^ { 2 } }$
D) $6 - \frac { 4 } { x }$
E) $6 + \frac { 4 } { x }$

Question 44

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

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

Question 45

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 3 + 5 x } { 5 + 5 x } \right)$

A) $\frac { 3 } { 5 }$
B)-1
C) $- \frac { 3 } { 5 }$
D)1
E)does not exist

Question 46

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

A) $\frac { 3 } { 7 }$
B) $- \frac { 7 } { 3 }$
C)- $\frac { 7 } { 3 }$
D) $- \frac { 3 } { 7 }$
E)does not exist

Question 47

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

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

Question 48

Select the correct function for the graph using horizontal asymptotes as aids.

A) $\frac { 6 x ^ { 2 } } { x ^ { 2 } + 5 }$
B) $\frac { 5 x ^ { 2 } } { x ^ { 2 } - 6 }$
C) $\frac { x ^ { 2 } } { x ^ { 2 } - 6 }$
D) $\frac { x ^ { 2 } } { x ^ { 2 } + 6 }$
E) $\frac { 5 x ^ { 2 } } { x ^ { 2 } + 6 }$

Question 49

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

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

Question 50

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

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

Question 51

Select the correct function for the graph using horizontal asymptotes as aids.

A) $\frac { 6 x ^ { 2 } } { x ^ { 2 } + 8 }$
B) $\frac { x ^ { 2 } } { x ^ { 2 } - 6 }$
C) $\frac { x ^ { 2 } } { x ^ { 2 } + 6 }$
D) $\frac { 8 x ^ { 2 } } { x ^ { 2 } + 6 }$
E) $\frac { 8 x ^ { 2 } } { x ^ { 2 } - 6 }$

Question 52

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

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

Question 53

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 4 + x } { 7 + x } \right)$

A)4
B)1
C)-1
D)-4
E)does not exist

Question 54

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 9 x ^ { 2 } - 1 } { 5 x ^ { 2 } - 5 } \right)$

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

Question 55

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

A)15
B)-6
C)6
D)0
E)does not exist

Question 56

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

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

Question 57

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

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

Question 58

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

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

Question 59

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

A) $- \frac { 1 } { 2 }$
B)0
C)2
D) $\frac { 1 } { 2 }$
E) $\infty$

Question 60

Select the correct function for the graph using oblique asymptotes as aids.

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

Question 61

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

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

Question 62

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

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 63

The average typing speed S (in words per minute)for a student after t weeks of lessons is given by
$S = \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } }$
What is the limit of S as t approaches infinity?

A) $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = \frac { 1 } { 66 }$
B) $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = 66$
C) $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = 0$
D) $\lim _ { t \rightarrow \infty } \frac { 160 t ^ { 2 } } { 66 + t ^ { 2 } } = 160$
E)does not exist

Question 64

Select the correct graph of the following function. $f ( x ) = 8 \left( 5 x - \sqrt { 25 x ^ { 2 } + x } \right)$

A)
B)
C)
D)
E

Question 65

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

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 66

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

A)
B)
C)
D)
E)

Question 67

Find the limit of the sequence.
$\frac { 1 } { n } \left( n + \frac { 1 } { n } \left[ \frac { n ( n + 3 ) } { 4 } \right] \right)$

A) $\frac { 4 } { 5 }$
B) $\frac { 5 } { 4 }$
C)3
D)4
E)does not exist

Question 68

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

A)
B)
C)
D)
E)

Question 69

Select the correct graph of the following function. $f ( x ) = 5 x - \sqrt { 25 x ^ { 2 } + 3 }$

A)
B)
C)
D)
E)

Question 70

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

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 71

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

A)
B)
C)
D)
E)

Question 72

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

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 73

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

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 74

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

A)
B)
C)
D)
E)

Question 75

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

A) $\alpha _ { 1 } = 4 , \alpha _ { 2 } = \frac { 1 } { 3 } , \alpha _ { 3 } = \frac { 3 } { 8 } , \alpha _ { 4 } = \frac { 2 } { 5 } , \alpha _ { 5 } = \frac { 5 } { 12 }$
B) $\alpha _ { 1 } = 4 , \alpha _ { 2 } = \frac { 1 } { 3 } , \alpha _ { 3 } = \frac { 8 } { 3 } , \alpha _ { 4 } = \frac { 2 } { 5 } , \alpha _ { 5 } = \frac { 5 } { 12 }$
C) $\alpha _ { 1 } = 4 , \alpha _ { 2 } = \frac { 1 } { 3 } , \alpha _ { 3 } = \frac { 3 } { 8 } , \alpha _ { 4 } = \frac { 2 } { 5 } , \alpha _ { 5 } = \frac { 12 } { 5 }$
D) $\alpha _ { 1 } = \frac { 1 } { 4 } , \alpha _ { 2 } = \frac { 1 } { 3 } , \alpha _ { 3 } = \frac { 3 } { 8 } , \alpha _ { 4 } = \frac { 2 } { 5 } , \alpha _ { 5 } = \frac { 5 } { 12 }$
E) $\alpha _ { 1 } = \frac { 1 } { 4 } , \alpha _ { 2 } = \frac { 1 } { 3 } , \alpha _ { 3 } = \frac { 8 } { 3 } , \alpha _ { 4 } = \frac { 2 } { 5 } , \alpha _ { 5 } = \frac { 12 } { 5 }$

Question 76

Find the limit (if it exists). $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( x + 5 ) ^ { 2 } } - 8 \right]$

A)5
B)-8
C)-5
D)0
E)does not exist

Question 77

Find the limit of the sequence (if it exists). $\alpha _ { n } = \frac { ( n + 6 ) ! } { n ! }$

A) $\lim_ { n \rightarrow \infty }\alpha _ { n } = - 6$
B) $\lim_ { n \rightarrow \infty } \alpha _ { n } = 0$
C) $\lim_ { n \rightarrow \infty } \alpha _ { n } = 1$
D) $\lim_ { n \rightarrow \infty } \alpha _ { n } = 6$
E) $\lim _ { n \rightarrow \infty }a _ { n } = \infty$

Question 78

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

A)
B)
C)
D)
E)

Question 79

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

A)
B)
C)
D)
E)

Question 80

Select correct graph of the following function. $y = 2 - \frac { 8 } { x ^ { 2 } }$

A)
B)
C)
D)
E)