Question 1

Find the limit.
-If $\lim _ { x \rightarrow0} \frac { f ( x ) } { x ^ { 2 } } = 1$, find $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x }$.

Accepted Answer

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

Question 2

Give an appropriate answer.
-$$f ( x ) = 4 \sqrt { x } \text { for } x _ { 0 } = 4$$

Accepted Answer

A)  Does not exist 
B)  8 
C)  4 
D)  1 
A)  Does not exist 
B)  8 
C)  4 
D)  1

Question 3

Use the graph to evaluate the limit.
-$\lim _ { x \rightarrow 0 } f ( x )$
S1U1B

Accepted Answer

A)  does not exist 
B)  $\infty$ 
C)  1 
D)  -1 
A)  does not exist 
B)  $\infty$ 
C)  1 
D)  -1

Question 4

Give an appropriate answer.
-Let $\lim _ { x \rightarrow 9 } f ( x ) = 4$. Find $\lim _ { x \rightarrow 9 } ( - 3 ) ^ { f ( x ) }$.

Accepted Answer

A)  81 
B)  $- 3$ 
C)  $- 19,683$ 
D)  4 
A)  81 
B)  $- 3$ 
C)  $- 19,683$ 
D)  4

Question 5

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow 1 ^ { + } } \left( \frac { x } { x + 9 } \right) \left( \frac { - 3 x + 8 } { x ^ { 2 } + 9 x } \right)$$

Accepted Answer

A)  Does not exist 
B)  $\frac { 11 } { 82 }$ 
C)  $\frac { 11 } { 64 }$ 
D)  $\frac { 1 } { 11 }$ 
A)  Does not exist 
B)  $\frac { 11 } { 82 }$ 
C)  $\frac { 11 } { 64 }$ 
D)  $\frac { 1 } { 11 }$

Question 6

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow 2 ^ { + } } ( x + 5 ) \left( \frac { | x + 2 | } { x + 2 } \right)$$

Accepted Answer

A)  Does not exist 
B)  3 
C)  $- 3$ 
D)  7 
A)  Does not exist 
B)  3 
C)  $- 3$ 
D)  7

Question 7

Give an appropriate answer.
-$\text { Let } \lim _ { x \rightarrow 5 } f ( x ) = 7 \text { and } \lim _ { x \rightarrow 5 } g ( x ) = 10 \text {. Find } \lim _ { x \rightarrow 5 } [ f ( x ) \cdot g ( x ) ] \text {. }$

Accepted Answer

A)  10 
B)  70 
C)  5 
D)  17 
A)  10 
B)  70 
C)  5 
D)  17

Question 8

Find the average rate of change of the function over the given interval.
-$$h ( t ) = \sin ( 4 t ) , \left[ 0 , \frac { \pi } { 8 } \right]$$

Accepted Answer

A)  $\frac { 4 } { \pi }$ 
B)  $- \frac { 8 } { \pi }$ 
C)  $\frac { \pi } { 8 }$ 
D)  $\frac { 8 } { \pi }$ 
A)  $\frac { 4 } { \pi }$ 
B)  $- \frac { 8 } { \pi }$ 
C)  $\frac { \pi } { 8 }$ 
D)  $\frac { 8 } { \pi }$

Question 9

Give an appropriate answer.
-$$f ( x ) = \frac { x } { 3 } + 4 \text { for } x _ { 0 } = 4$$

Accepted Answer

A)  $\frac { 16 } { 3 }$ 
B)  $\frac { 4 } { 3 }$ 
C)  Does not exist 
D)  $\frac { 1 } { 3 }$ 
A)  $\frac { 16 } { 3 }$ 
B)  $\frac { 4 } { 3 }$ 
C)  Does not exist 
D)  $\frac { 1 } { 3 }$

Question 10

Use the table of values of f to estimate the limit.
-$$\begin{array}{l}
\text { Let } f ( \theta ) = \frac { \cos ( 4 \theta ) } { \theta } \text {, find } \lim _ { \theta \rightarrow \theta } f ( \theta ) \text {. }\
\begin{array} { l | c | c | c | c | c | c } 
\mathrm { x } & - 0.1 & - 0.01 & - 0.001 & 0.001 & 0.01 & 0.1 \
\hline \mathrm { f } ( \theta ) & - 9.2106099 & & & & & 9.2106099
\end{array}
\end{array}$$

Accepted Answer

A)  limit = 9.2106099 
B)  limit does not exist 
C)  limit = 4 
D)  limit = 0 
A)  limit = 9.2106099 
B)  limit does not exist 
C)  limit = 4 
D)  limit = 0

Question 11

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow  0 } \frac { \sin 4 x } { \sin 5 x }$$

Accepted Answer

A)  does not exist 
B)  $\frac { 5 } { 4 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  0 
A)  does not exist 
B)  $\frac { 5 } { 4 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  0

Question 12

Determine the limit by sketching an appropriate graph.
-$$\lim _ { x \rightarrow 4 ^ { + } } f ( x ) , \text { where } f ( x ) = \left\{ \begin{array} { l l } 
x ^ { 2 } + 3 & \text { for } x \neq 4 \
0 & \text { for } x = 4
\end{array} \right.$$

Accepted Answer

A)  13 
B)  19 
C)  0 
D)  16 
A)  13 
B)  19 
C)  0 
D)  16

Question 13

Find the limit.
-$\lim _ { x \rightarrow 2 } ( 8 x + 3 )$

Accepted Answer

A)  19 
B)  -13 
C)  11 
D)  3 
A)  19 
B)  -13 
C)  11 
D)  3

Question 14

Find the limit.
-$\lim _ { x \rightarrow 2 } \left( 3 x ^ { 5 } - 3 x ^ { 4 } - 4 x ^ { 3 } + x ^ { 2 } + 5 \right)$

Accepted Answer

A)  -23 
B)  25 
C)  121 
D)  89 
A)  -23 
B)  25 
C)  121 
D)  89

Question 15

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$ \lim _{x\rightarrow\ 6^{-}}(x-\lfloor x\rfloor) $

Accepted Answer

A)  12 
B)  6 
C)  0 
D)  -12 
A)  12 
B)  6 
C)  0 
D)  -12

Question 16

Prove the limit statement
-$$\lim _ { x \rightarrow 5 } ( 5 x - 3 ) = 22$$

Accepted Answer

Let @#LAT-DLM& be given. Choose @#LAT-DLM&. Then @#LAT-DLM& implies th

Question 17

Provide an appropriate response.
-$$a = \frac { 3 } { 9 } , b = \frac { 9 } { 9 } , x _ { 0 } = \frac { 4 } { 9 }$$

Accepted Answer

A)  $\delta = \frac { 5 } { 9 }$ 
B)  $\delta = \frac { 1 } { 9 }$ 
C)  $\delta = 0$ 
D)  $\delta = 1$ 
A)  $\delta = \frac { 5 } { 9 }$ 
B)  $\delta = \frac { 1 } { 9 }$ 
C)  $\delta = 0$ 
D)  $\delta = 1$

Question 18

Provide an appropriate response.
-The inequality $1 - \frac { x ^ { 2 } } { 2 } < \frac { \sin x } { x } < 1$ holds when $x$ is measured in radians and $| x | < 1$. Find $\lim _ { x \rightarrow \theta } \frac { \sin x } { x }$ if it exists.

Accepted Answer

A)  0 
B)  1 
C)  does not exist 
D)  $0.0007$ 
A)  0 
B)  1 
C)  does not exist 
D)  $0.0007$

Question 19

$\lim _ { x \rightarrow 0 } f ( x )$

Accepted Answer

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

Question 20

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit.
-$\lim _ { x \rightarrow 0 } \frac { \sqrt { 36 + 2 x } - 6 } { x }$

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  36 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 12 }$ 
A)  $\frac { 1 } { 6 }$ 
B)  36 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 12 }$

Find the limit. -If $\lim _ { x \rightarrow0} \frac { f ( x ) } { x ^ { 2 } } = 1$ , find $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x }$ .

Give an appropriate answer. - $f ( x ) = 4 \sqrt { x } \text { for } x _ { 0 } = 4$

Use the graph to evaluate the limit. - $\lim _ { x \rightarrow 0 } f ( x )$ $Use the graph to evaluate the limit. - \lim _ { x \rightarrow 0 } f ( x ) </sub>$

Give an appropriate answer. -Let $\lim _ { x \rightarrow 9 } f ( x ) = 4$ . Find $\lim _ { x \rightarrow 9 } ( - 3 ) ^ { f ( x ) }$ .

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 1 ^ { + } } \left( \frac { x } { x + 9 } \right) \left( \frac { - 3 x + 8 } { x ^ { 2 } + 9 x } \right)$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 2 ^ { + } } ( x + 5 ) \left( \frac { | x + 2 | } { x + 2 } \right)$

Give an appropriate answer. - $\text { Let } \lim _ { x \rightarrow 5 } f ( x ) = 7 \text { and } \lim _ { x \rightarrow 5 } g ( x ) = 10 \text {. Find } \lim _ { x \rightarrow 5 } [ f ( x ) \cdot g ( x ) ] \text {. }$

Find the average rate of change of the function over the given interval. - $h ( t ) = \sin ( 4 t ) , \left[ 0 , \frac { \pi } { 8 } \right]$

Give an appropriate answer. - $f ( x ) = \frac { x } { 3 } + 4 \text { for } x _ { 0 } = 4$

Use the table of values of f to estimate the limit. - Let f(\theta)= , find f(\theta) . -0.1 -0.01 -0.001 0.001 0.01 0.1 (\theta) -9.2106099 9.2106099

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { \sin 4 x } { \sin 5 x }$

Find the limit. - $\lim _ { x \rightarrow 2 } ( 8 x + 3 )$

Find the limit. - $\lim _ { x \rightarrow 2 } \left( 3 x ^ { 5 } - 3 x ^ { 4 } - 4 x ^ { 3 } + x ^ { 2 } + 5 \right)$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _{x\rightarrow\ 6^{-}}(x-\lfloor x\rfloor)$

Prove the limit statement - $\lim _ { x \rightarrow 5 } ( 5 x - 3 ) = 22$

Provide an appropriate response. - $a = \frac { 3 } { 9 } , b = \frac { 9 } { 9 } , x _ { 0 } = \frac { 4 } { 9 }$

Provide an appropriate response. -The inequality $1 - \frac { x ^ { 2 } } { 2 } < \frac { \sin x } { x } < 1$ holds when $x$ is measured in radians and $| x | < 1$ . Find $\lim _ { x \rightarrow \theta } \frac { \sin x } { x }$ if it exists.

$\lim _ { x \rightarrow 0 } f ( x )$ $\lim _ { x \rightarrow 0 } f ( x )$

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 0 } \frac { \sqrt { 36 + 2 x } - 6 } { x }$

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Exam 2: Limits and Derivatives

Find the limit. -If lim⁡x→0f(x)x2=1\lim _ { x \rightarrow0} \frac { f ( x ) } { x ^ { 2 } } = 1limx→0​x2f(x)​=1 , find lim⁡x→0f(x)x\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x }limx→0​xf(x)​ .

Give an appropriate answer. - f(x)=4x for x0=4f ( x ) = 4 \sqrt { x } \text { for } x _ { 0 } = 4f(x)=4x​ for x0​=4

Use the graph to evaluate the limit. - lim⁡x→0f(x)\lim _ { x \rightarrow 0 } f ( x )limx→0​f(x)

Give an appropriate answer. -Let lim⁡x→9f(x)=4\lim _ { x \rightarrow 9 } f ( x ) = 4limx→9​f(x)=4 . Find lim⁡x→9(−3)f(x)\lim _ { x \rightarrow 9 } ( - 3 ) ^ { f ( x ) }limx→9​(−3)f(x) .

Find the average rate of change of the function over the given interval. - h(t)=sin⁡(4t),[0,π8]h ( t ) = \sin ( 4 t ) , \left[ 0 , \frac { \pi } { 8 } \right]h(t)=sin(4t),[0,8π​]

Give an appropriate answer. - f(x)=x3+4 for x0=4f ( x ) = \frac { x } { 3 } + 4 \text { for } x _ { 0 } = 4f(x)=3x​+4 for x0​=4