Deck 13: A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Question 1

Use the graph shown to determine if the limit exists. If it does, find its value.

- $$\lim _ { x \rightarrow 1 } f ( x )$$

A) 1
B) 3
C) 0
D) does not exist

Answer 1

3

Question 2

Use the TABLE feature of a graphing utility to find the limit.

- $\lim _ { x \rightarrow \theta } \left( e ^ { x } - e ^ { - x } \right)$

A) 2
B) 0
C) 1
D) does not exist

Answer 2

0

Question 3

Use the grid to graph the function. Find the limit, if it exists

- $$\lim _ { x \rightarrow 8 } f ( x ) , \quad f ( x ) = 1 - x ^ { 2 }$$

A) 1
B) -2
C) -8
D) does not exist

Answer 3

-8

Question 4

Find the limit algebraically.

- $$\lim _ { x \rightarrow - 1 } - 6$$

A) -6
B) 1
C) 0
D) 6

Question 5

Use the graph shown to determine if the limit exists. If it does, find its value.

- $$\lim _ { x \rightarrow 8 } f ( x )$$

A) 4
B) 3
C) 5
D) does not exist

Question 6

Use the grid to graph the function. Find the limit, if it exists

- $$\lim _ { x \rightarrow 2 } f ( x ) , \quad f ( x ) = 4 x + 2$$

A) -2
B) -6
C) -8
D) does not exist

Question 7

Use the TABLE feature of a graphing utility to find the limit.

- $\lim _ { h \rightarrow 3 } \left( \frac { h ^ { 2 } - 9 } { h ^ { 2 } + 3 h } \right)$

A) 2
B) -2
C) 0
D) does not exist

Question 8

Use the TABLE feature of a graphing utility to find the limit.

- $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } } { \cos x }$

A) -1
B) 0
C) 1
D) does not exist

Question 9

Use the grid to graph the function. Find the limit, if it exists

- $$\lim _ { x \rightarrow 0 } f ( x ) , \quad f ( x ) = \left\{ \begin{array} { r l } x + 1 & x < 0 \\4 x + 1 & x \geq 0\end{array} \right.$$

A) 0
B) 1
C) 5
D) does not exist

Question 10

Use the grid to graph the function. Find the limit, if it exists

- $$\lim _ { x \rightarrowπ / 2 } f ( x ) , f ( x ) = \sin x - 2$$

A) -1
B) 3
C) -2
D) does not exist

Question 11

Use the grid to graph the function. Find the limit, if it exists

- $$\lim _ { x \rightarrow 8 } f ( x ) , f ( x ) = 4 x - 1$$

A) 12
B) 11
C) 2
D) does not exist

Question 12

Use the graph shown to determine if the limit exists. If it does, find its value.

- $$\lim _ { x \rightarrow 4 } f ( x )$$

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

Question 13

Find the limit algebraically.

- $$\lim _ { x \rightarrow 2 } ( 2 x + 7 )$$

A) -18
B) 9
C) 3
D) does not exist

Question 14

Use the TABLE feature of a graphing utility to find the limit.

- $$\lim _ { x \rightarrow 0 } \frac { x + 2 } { 5 x + 2 }$$

A) $\frac { 2 } { 5 }$

B) 0

C) 1

D) does not exist

Question 15

Use the TABLE feature of a graphing utility to find the limit.

- $\lim _ { d \rightarrow } \left( \frac { d ^ { 3 } - 27 } { d - 3 } \right)$

A) 27
B) 9
C) 0
D) does not exist

Question 16

Use a graphing utility to find the indicated limit rounded to two decimal places.

- $$\lim _ { x \rightarrow - 1 } \frac { x ^ { 3 } - x ^ { 2 } + 3 x - 3 } { x ^ { 4 } - x ^ { 3 } + x - 1 }$$

A) 2.04
B) 2.96
C) 3.00
D) 2.00

Question 17

Use the TABLE feature of a graphing utility to find the limit.

- $\lim _ { x \rightarrow 2 } \left( x ^ { 2 } + 8 x - 2 \right)$

A) 0
B) -18
C) 18
D) does not exist

Question 18

Use the grid to graph the function. Find the limit, if it exists

- $$\lim _ { x \rightarrow 1 } f ( x ) , f ( x ) = | 5 x |$$

A) -5
B) 1
C) 5
D) does not exist

Question 19

Use the graph shown to determine if the limit exists. If it does, find its value.

- $$\lim _ { x \rightarrow 2 } f ( x )$$

A) 0
B) 1
C) 2
D) does not exist

Question 20

Find the limit algebraically.

- $$\lim _ { x \rightarrow 6 } x$$

A) 0
B) 6
C) -6
D) 1

Question 21

Find the limit algebraically.

- $$\lim _ { x \rightarrow 1 } \left( 3 x ^ { 2 } - 19 \right) ^ { 2 }$$

A) 100
B) -352
C) 256
D) 144

Question 22

Find the limit algebraically.

- $$\lim _ { x \rightarrow - 4 } \left( 4 x ^ { 3 } - 2 x + 123 \right) ^ { 2 / 3 }$$

A) -125
B) -5
C) 125
D) 25

Question 23

Find the limit algebraically.

- $$\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)$$

A) 15
B) 0
C) 29
D) does not exist

Question 24

Find the limit algebraically.

- $$\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)$$

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

Question 25

Find the limit algebraically.

- $$\lim _ { x \rightarrow 1 } \frac { 3 x ^ { 2 } + 7 x - 2 } { 3 x ^ { 2 } - 4 x + 2 }$$

A) $$- \frac { 7 } { 4 }$$
B) 0
C) 8
D) does not exist

Question 26

Find the limit algebraically.

- $$\lim _ { x \rightarrow 1 } \frac { 2 x - 7 } { 4 x + 5 }$$

A) - 59
B) - 75
C) - 12
D) does not exist

Question 27

Find the limit algebraically.

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

A) Does not exist
B) 3
C) 1
D) 6

Question 28

Find the limit algebraically.

- $$\lim _ { x \rightarrow 1 } \left( x ^ { 2 } - 2 \right) ^ { 3 }$$

A) -3
B) -1
C) 3
D) 1

Question 29

Find the limit algebraically.

- $$\lim _ { x \rightarrow 1 } \frac { x ^ { 4 } - 1 } { x - 1 }$$

A) 2
B) 0
C) 4
D) Does not exist

Question 30

Find the limit algebraically.

- $$\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 2 x - 15 } { x + 3 }$$

A) Does not exist
B) 5
C) 0
D) -8

Question 31

Find the limit algebraically.

- $$\lim _ { x \rightarrow - 0 } ( \sqrt { x } - 2 )$$

A) -2
B) 2
C) 0
D) does not exist

Question 32

Find the limit algebraically.

- $$\lim _ { x \rightarrow 4 } \left( x ^ { 2 } + 4 x + 1 \right)$$

A) 31
B) -1
C) 1
D) 33

Question 33

Find the limit algebraically.

- $$\lim _ { x \rightarrow0 } ( x - \sqrt { 5 } ) ( x + \sqrt { 5 } )$$

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

Question 34

Find the limit algebraically.

- $$\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 5 x } { 5 x }$$

A) 0
B) -1
C) 5
D) Does not exist

Question 35

Find the limit algebraically.

- $$\lim _ { x \rightarrow 10 } - 8 x ^ { 3 }$$

A) -8
B) -1,000
C) -10
D) 8,000

Question 36

Find the limit algebraically.

- $$\lim _ { x \rightarrow 1 } \sqrt { 3 x - 2 }$$

A) -1
B) 1
C) 2
D) does not exist

Question 37

Find the limit algebraically.

- $$\lim _ { x \rightarrow 0 } \frac { 4 + x } { x ^ { 3 } }$$

A) -4
B) 4
C) Does not exist
D) 0

Question 38

Find the limit algebraically.

- $$\lim _ { x \rightarrow 6 } \frac { x + 6 } { ( x - 6 ) ^ { 2 } }$$

A) 0
B) 6
C) Does not exist
D) -6

Question 39

Find the limit algebraically.

- $$\lim _ { x \rightarrow9 } \left( 6 - 3 x ^ { 2 } \right)$$

A) -48
B) -237
C) 249
D) -21

Question 40

Find the limit algebraically.

- $$\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } - 4 } { x + 2 }$$

A) Does not exist
B) -2
C) 1
D) -4

Question 41

Find the limit algebraically.

- $$\lim _ { x \rightarrow 1 } \frac { x ^ { 3 } + 5 x ^ { 2 } + 3 x - 9 } { x - 1 }$$

A) -16
B) 0
C) 16
D) does not exist

Question 42

Use the graph of y = g(x) to answer the question.

-Does $\lim _ { x \rightarrow 4 } g ( x )$ xist? If it does, what is it?

A) yes; 0
B) yes; 3
C) yes; 4
D) does not exist

Question 43

Use the graph of y = g(x) to answer the question.
Find f(1).

A) 3
B) 4
C) 0
D) -1

Question 44

Find the limit as x approaches c of the average rate of change of the function from c to x.

-c = 3; $$f ( x ) = 3 x ^ { 2 } + 39 x$$

A) 108
B) 57
C) 30
D) 90

Question 45

Use the graph of y = g(x) to answer the question.

-What is the domain of g?

A) $\{ x \mid - 4 \leq x < 5 \}$
B) $\{ x \mid - 4 \leq x < 4$ or $4 < x \leq 5 \}$
C) $\{ x \mid - 4 < x < 5 \}$
D) $\{ x \mid - 4 < x < - 1$ or $- 1 < x < 2$ or $2 < x \leq 5 \}$

Question 46

Use the graph of y = g(x) to answer the question.

-Find the y-intercept(s), if any, of g.

A) 3
B) 0
C) -1
D) $$- \frac { 4 } { 3 }$$

Question 47

Use the graph of y = g(x) to answer the question.
Find f(-1).

A) 1
B) 0
C) 4
D) -1

Question 48

Use the graph of y = g(x) to answer the question.

-Find f(2)

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

Question 49

Find the limit as x approaches c of the average rate of change of the function from c to x.

-c = -3; $$f ( x ) = 2 x ^ { 2 } + 3$$

A) 18
B) -9
C) -12
D) Does not exist

Question 50

Find the limit algebraically.

- $$\lim _ { x \rightarrow 0 } \frac { 2 \sin x } { 4 x }$$

A) Does not exist
B) $\frac { 1 } { 2 }$
C) 1
D) 0

Question 51

Find the limit as x approaches c of the average rate of change of the function from c to x.
c = 3; f(x) = 5x + 3

A) -5
B) 3
C) 5
D) -3

Question 52

Find the limit as x approaches c of the average rate of change of the function from c to x.

-c = 4; $$f ( x ) = \frac { 3 } { x }$$

A) $- \frac { 3 } { 16 }$
B) $- 12$
C) $\frac { 3 } { 4 }$
D) Does not exist

Question 53

Use the graph of y = g(x) to answer the question.
Find f(-4).

A) -4
B) 0
C) 2
D) 1

Question 54

Use the graph of y = g(x) to answer the question.

- $$\text { Find } \lim _ { x \rightarrow 1 } g ( x ) \text {. }$$

A) 1
B) 0
C) 4
D) does not exist

Question 55

Find the limit as x approaches c of the average rate of change of the function from c to x.

-c = 9; $$f ( x ) = \frac { x } { 3 } + 10$$

A) 3
B) Does not exist
C) 13
D) $$\frac { 1 } { 3 }$$

Question 56

Find the limit algebraically.

- $$\lim _ { x \rightarrow0 } \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$$

A) 4
B) 0
C) -4
D) does not exist

Question 57

Find the limit algebraically.

- $$\lim _ { x \rightarrow 0 } \frac { 3 \tan x } { 7 x }$$

A) 1
B) Does not exist
C) $\frac { 3 } { 7 }$
D) 0

Question 58

Find the limit as x approaches c of the average rate of change of the function from c to x.

-c = -4; $$f ( x ) = x ^ { 3 }$$

A) 0
B) 16
C) 48
D) 32

Question 59

Use the graph of y = g(x) to answer the question.

-What is the range of g?

A) all real numbers
B) $\{ y \mid 2 \leq y \leq 4 \}$
C) $\{ y \mid - 2 \leq y \leq 4 \}$
D) $\{ y \mid - 2 \leq y \leq 5 \}$

Question 60

Find the one-sided limit.

- $\lim _ { x \rightarrow 0 ^ { - } } ( 4 - 5 x )$

A) 0
B) -1
C) 4
D) does not exist

Question 61

Determine whether f is continuous at c.

- $f ( x ) = \frac { - 1 } { x ^ { 2 } + 8 x } ; \quad c = - 8$

A) not continuous
B) continuous

Question 62

Determine whether f is continuous at c.

- $f ( x ) = \frac { 8 } { x ^ { 2 } + 3 x } ; \quad c = - 8$

A) continuous
B) not continuous

Question 63

Determine whether f is continuous at c.

- $f ( x ) = \frac { x + 2 } { x - 7 } ; c = 0$

A) continuous
B) not continuous

Question 64

Determine whether f is continuous at c.

- $f ( x ) = \frac { x + 6 } { ( x - 2 ) ( x - 3 ) } ; c = - 6$

A) not continuous
B) continuous

Question 65

Determine whether f is continuous at c.

- $f ( x ) = \frac { 5 } { x ^ { 2 } - 2 x } ; \quad c = 0$

A) continuous
B) not continuous

Question 66

Determine whether f is continuous at c.

- $f ( x ) = \frac { x ^ { 2 } - 36 } { x - 6 } ; \quad c = - 6$

A) continuous
B) not continuous

Question 67

Determine whether f is continuous at c.

- $f ( x ) = \frac { x + 4 } { ( x - 3 ) ( x + 2 ) } ; \quad c = - 2$

A) not continuous
B) continuous

Question 68

Find the one-sided limit.

- $$\lim _ { x \rightarrow ( 3 \pi / 4 ) ^ { - } } \sin x$$

A) $\frac { 1 } { 2 }$
B) 0
C) $\frac { \sqrt { 2 } } { 2 }$
D) 1

Question 69

Find the one-sided limit.

- $$\lim _ { x \rightarrow3 ^ { - } } \frac { x ^ { 2 } - 9 } { x - 3 }$$

A) 3
B) -6
C) 6
D) -3

Question 70

Determine whether f is continuous at c.

- $f ( x ) = \frac { x - 2 } { x + 9 } ; c = - 9$

A) not continuous
B) continuous

Question 71

Determine whether f is continuous at c.

- $f ( x ) = \frac { x - 2 } { ( x + 9 ) ( x + 8 ) } ; c = - 9$

A) not continuous
B) continuous

Question 72

Determine whether f is continuous at c.

- $f ( x ) = \frac { x - 7 } { x + 9 } ; c = 7$

A) continuous
B) not continuous

Question 73

Find the one-sided limit.

- $\lim _ { x \rightarrow 4 ^ { + } } \frac { 16 - x ^ { 2 } } { 4 - x }$

A) 8
B) 0
C) 4
D) does not exist

Question 74

Determine whether f is continuous at c.

- $f ( x ) = \frac { 5 } { x + 2 } ; c = - 2$

A) not continuous
B) continuous

Question 75

Determine whether f is continuous at c.

- $f ( x ) = \frac { 5 } { x + 7 } ; c = 0$

A) continuous
B) not continuous

Question 76

Find the one-sided limit.

- $\lim _ { x \rightarrow \theta ^ { + } } ( 2 \cos x )$

A) -2
B) 2
C) 0
D) does not exist

Question 77

Determine whether f is continuous at c.

- $$f ( x ) = \frac { x - 6 } { ( x - 1 ) ( x - 9 ) } ; \quad c = 0$$

A) continuous
B) not continuous

Question 78

Find the one-sided limit.

- $$\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 2 x - 15 } { x + 3 }$$

A) Does not exist
B) 5
C) -8
D) 0

Question 79

Determine whether f is continuous at c.

- $f ( x ) = \frac { x ^ { 2 } - 25 } { x - 5 } ; \quad c = 0$

A) not continuous
B) continuous

Question 80

Find the one-sided limit.

- $\lim _ { x \rightarrow 2 ^ { + } } \left( x ^ { 2 } - 3 x - 3 \right)$

A) 13
B) -5
C) 1
D) 7