Question 1

Use the graph to estimate the specified limit.
-Find lim x $\rightarrow$0 f(x)

Accepted Answer

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

Question 2

Find the limit.
-$$\lim _ { x \rightarrow 0 } ( 3 \sin x - 1 )$$

Accepted Answer

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

Question 3

Find the limit if it exists.
-$$\lim _ { x \rightarrow \frac { 1 } { 6 } } 6 x \left( x - \frac { 2 } { 5 } \right)$$

Accepted Answer

A)  $\frac { 17 } { 30 }$ 
B)  $- \frac { 7 } { 30 }$ 
C)  $- \frac { 7 } { 5 }$ 
D)  $- \frac { 7 } { 180 }$ 
A)  $\frac { 17 } { 30 }$ 
B)  $- \frac { 7 } { 30 }$ 
C)  $- \frac { 7 } { 5 }$ 
D)  $- \frac { 7 } { 180 }$

Question 4

Give an appropriate answer.
-$\text { Let } \lim _ { x \rightarrow 7 } f ( x ) = - 8 \text { and } \lim _ { x \rightarrow 7 } g ( x ) = 5 \text {. Find } \lim _ { x \rightarrow 7 } [ f ( x ) - g ( x ) ] \text {. }$

Accepted Answer

A)  -3 
B)  -13 
C)  -8 
D)  7 
A)  -3 
B)  -13 
C)  -8 
D)  7

Question 5

Use the graph to estimate the specified limit.
-$\text { Find } \lim _ { x \rightarrow 0 ^ { - } } f ( x ) \text { and } \lim _ { x \rightarrow 0^ { + } } f ( x )$

Accepted Answer

A)  2; 1 
B)  -2; -1 
C)  2; -1 
D)  -1; 2 
A)  2; 1 
B)  -2; -1 
C)  2; -1 
D)  -1; 2

Question 6

Find the limit $L$ for the given function $f$, the point $x _ { 0 }$, and the positive number $\varepsilon$. Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$.
-$$\mathrm { f } ( \mathrm { x } ) = \frac { 10 } { \mathrm { x } } , \mathrm { x } _ { 0 } = 5 , \varepsilon = 0.1$$

Accepted Answer

A)  $\mathrm { L } = 2 ; \delta = 0.53$ 
B)  $\mathrm { L } = 2 ; \delta = 0.24$ 
C)  $L = 2 ; \delta = 0.26$ 
D)  $L = 2 ; \delta = 2.63$ 
A)  $\mathrm { L } = 2 ; \delta = 0.53$ 
B)  $\mathrm { L } = 2 ; \delta = 0.24$ 
C)  $L = 2 ; \delta = 0.26$ 
D)  $L = 2 ; \delta = 2.63$

Question 7

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow 0 } 6 x ^ { 2 } ( \cot 3 x ) ( \csc 2 x )$$

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

Find the limit $L$ for the given function $f$, the point $x _ { 0 }$, and the positive number $\varepsilon$. Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$.
-f(x) = -10x + 5, L = -15, x0 = 2, and $\varepsilon$ = 0.01

Accepted Answer

A)  0.004 
B)  0.001 
C)  0.002 
D)  -0.005 
A)  0.004 
B)  0.001 
C)  0.002 
D)  -0.005

Question 10

Find the limit $L$ for the given function $f$, the point $x _ { 0 }$, and the positive number $\varepsilon$. Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$.
-$$\mathrm { f } ( \mathrm { x } ) = 6 \mathrm { x } + 3 , \mathrm { x } _ { 0 } = - 5 , \varepsilon = 0.06$$

Accepted Answer

A)  $\mathrm { L } = - 27 ; \delta = 0.01$ 
B)  $L = - 33 ; \delta = 0.01$ 
C)  $L = - 27 ; \delta = 0.02$ 
D)  $\mathrm { L } = 33 ; \delta = 0.02$ 
A)  $\mathrm { L } = - 27 ; \delta = 0.01$ 
B)  $L = - 33 ; \delta = 0.01$ 
C)  $L = - 27 ; \delta = 0.02$ 
D)  $\mathrm { L } = 33 ; \delta = 0.02$

Question 11

Give an appropriate answer.
-$$f ( x ) = 3 \sqrt { x } + 8 \text { for } x _ { 0 } = 9$$

Accepted Answer

A)  $\frac { 27 } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  Does not exist 
D)  $\frac { 9 } { 2 }$ 
A)  $\frac { 27 } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  Does not exist 
D)  $\frac { 9 } { 2 }$

Question 12

Find the limit $L$ for the given function $f$, the point $x _ { 0 }$, and the positive number $\varepsilon$. Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$.
-f(x) = 3x - 2, L = 1, x0 = 1, and $\varepsilon$ = 0.01

Accepted Answer

A)  0.003333 
B)  0.001667 
C)  0.006667 
D)  0.01 
A)  0.003333 
B)  0.001667 
C)  0.006667 
D)  0.01

Question 13

Use the table of values of f to estimate the limit.
-$\text { Let } f(x)=x^{2}+8 x-2, \text { find } \lim _{x \rightarrow 2} f(x)$
$\begin{array}{c|l|l|l|l|l|l}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & & & & & &
\end{array}$

Accepted Answer

A) 
$\begin{array}{c|ccrrcc}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & 16.692 & 17.592 & 17.689 & 17.710 & 17.808 & 18.789
\end{array} \text {; limit }=17.70$ 
B)  
$\begin{array}{c|ccrrcc}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & 16.810 & 17.880 & 17.988 & 18.012 & 18.120 & 19.210
\end{array} \text {; limit }=18.0$ 
C) 
$\begin{array}{c|llllll}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & 5.043 & 5.364 & 5.396 & 5.404 & 5.436 & 5.763
\end{array} \text {; limit }=\infty$ 
D)  
$\begin{array}{c|llllll}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & 5.043 & 5.364 & 5.396 & 5.404 & 5.436 & 5.763
\end{array} \text {; limit }=5.40$ 
A) 
$\begin{array}{c|ccrrcc}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & 16.692 & 17.592 & 17.689 & 17.710 & 17.808 & 18.789
\end{array} \text {; limit }=17.70$ 
B)  
$\begin{array}{c|ccrrcc}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & 16.810 & 17.880 & 17.988 & 18.012 & 18.120 & 19.210
\end{array} \text {; limit }=18.0$ 
C) 
$\begin{array}{c|llllll}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & 5.043 & 5.364 & 5.396 & 5.404 & 5.436 & 5.763
\end{array} \text {; limit }=\infty$ 
D)  
$\begin{array}{c|llllll}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & 5.043 & 5.364 & 5.396 & 5.404 & 5.436 & 5.763
\end{array} \text {; limit }=5.40$

Question 14

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow 2 ^ { - } } \frac { \lfloor x \rfloor } { x }$$

Accepted Answer

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

Question 15

Use the graph to estimate the specified limit.
-$\text { Find } \lim _ { x \rightarrow 2 ^ { - } } f ( x ) \text { and } \lim _ { x \rightarrow 2^ { + } } f ( x )$

Accepted Answer

A)  5; -3 
B)  1; 1 
C)  -3; 5 
D)  does not exist; does not exist 
A)  5; -3 
B)  1; 1 
C)  -3; 5 
D)  does not exist; does not exist

Question 16

Provide an appropriate response.
-It can be shown that the inequalities $1 - \frac { x ^ { 2 } } { 6 } < \frac { x \sin ( x ) } { 2 - 2 \cos ( x ) } < 1$ hold for all values of $x$ close to zero. What, if anything, does this tell you about $\frac { x \sin ( x ) } { 2 - 2 \cos ( x ) }$ ? Explain.

Accepted Answer

Answers may vary. One possibil

Question 17

Provide an appropriate response.
-If $\lim f ( x ) = 1$ and $f ( x )$ is an odd function, which of the following statements are true?
I. $\lim _ { x \rightarrow 0 } f ( x ) = 1$
II. $\lim _ { x \rightarrow 0^ { + } } f ( x ) = - 1$
III. $\lim _ { x \rightarrow - 0 } f ( x )$ does not exist.

Accepted Answer

A)  II and III only 
B)  I and III only 
C)  I, II, and III 
D)  I and II only 
A)  II and III only 
B)  I and III only 
C)  I, II, and III 
D)  I and II only

Question 18

Use the table to estimate the rate of change of y at the specified value of x.
-x = 2.
$\begin{array} { c | c } 
\mathrm { x } & \mathrm { y } \
\hline 0 & 10 \
0.5 & 38 \
1.0 & 58 \
1.5 & 70 \
2.0 & 74 \
2.5 & 70 \
3.0 & 58 \
3.5 & 38 \
4.0 & 10
\end{array}$

Accepted Answer

A)  0 
B)  8 
C)  -8 
D)  4 
A)  0 
B)  8 
C)  -8 
D)  4

Question 19

Use the table to estimate the rate of change of y at the specified value of x.
-When exposed to ethylene gas, green bananas will ripen at an accelerated rate. The number of days for ripening becomes shorter for longer exposure times. Assume that the table below gives average ripening times of
Bananas for several different ethylene exposure times.   Plot the data and then find a line approximating the data. With the aid of this line, determine the rate of change of Ripening time with respect to exposure time. Round your answer to two significant digits.

Accepted Answer

A) 
 
$- 6.7$ days per minute 
B) 
 
$5.6$ days 
C) 
 
 38 minutes 
D) 
 
-0.14 day per minute 
A) 
 
$- 6.7$ days per minute 
B) 
 
$5.6$ days 
C) 
 
 38 minutes 
D) 
 
-0.14 day per minute

Question 20

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

Accepted Answer

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

Use the graph to estimate the specified limit. -Find lim x $\rightarrow$ 0 f(x) $Use the graph to estimate the specified limit. -Find lim x \rightarrow 0 f(x)$

Find the limit. - $\lim _ { x \rightarrow 0 } ( 3 \sin x - 1 )$

Find the limit if it exists. - $\lim _ { x \rightarrow \frac { 1 } { 6 } } 6 x \left( x - \frac { 2 } { 5 } \right)$

Give an appropriate answer. - $\text { Let } \lim _ { x \rightarrow 7 } f ( x ) = - 8 \text { and } \lim _ { x \rightarrow 7 } g ( x ) = 5 \text {. Find } \lim _ { x \rightarrow 7 } [ f ( x ) - g ( x ) ] \text {. }$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } 6 x ^ { 2 } ( \cot 3 x ) ( \csc 2 x )$

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

Give an appropriate answer. - $f ( x ) = 3 \sqrt { x } + 8 \text { for } x _ { 0 } = 9$

Use the table of values of f to estimate the limit. - $\text { Let } f(x)=x^{2}+8 x-2, \text { find } \lim _{x \rightarrow 2} f(x)$ 1.9 1.99 1.999 2.001 2.01 2.1 ()

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 2 ^ { - } } \frac { \lfloor x \rfloor } { x }$

Provide an appropriate response. -It can be shown that the inequalities $1 - \frac { x ^ { 2 } } { 6 } < \frac { x \sin ( x ) } { 2 - 2 \cos ( x ) } < 1$ hold for all values of $x$ close to zero. What, if anything, does this tell you about $\frac { x \sin ( x ) } { 2 - 2 \cos ( x ) }$ ? Explain.

Provide an appropriate response. -If $\lim f ( x ) = 1$ and $f ( x )$ is an odd function, which of the following statements are true? I. $\lim _ { x \rightarrow 0 } f ( x ) = 1$ II. $\lim _ { x \rightarrow 0^ { + } } f ( x ) = - 1$ III. $\lim _ { x \rightarrow - 0 } f ( x )$ does not exist.

Use the table to estimate the rate of change of y at the specified value of x. -x = 2. 0 10 0.5 38 1.0 58 1.5 70 2.0 74 2.5 70 3.0 58 3.5 38 4.0 10

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

Functions

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Exam 2: Limits and Derivatives

Use the graph to estimate the specified limit. -Find lim x →\rightarrow→ 0 f(x)

Find the limit. - lim⁡x→0(3sin⁡x−1)\lim _ { x \rightarrow 0 } ( 3 \sin x - 1 )limx→0​(3sinx−1)

Find the limit if it exists. - lim⁡x→166x(x−25)\lim _ { x \rightarrow \frac { 1 } { 6 } } 6 x \left( x - \frac { 2 } { 5 } \right)limx→61​​6x(x−52​)

Use the graph to estimate the specified limit. - Find lim⁡x→0−f(x) and lim⁡x→0+f(x)\text { Find } \lim _ { x \rightarrow 0 ^ { - } } f ( x ) \text { and } \lim _ { x \rightarrow 0^ { + } } f ( x ) Find limx→0−​f(x) and limx→0+​f(x)

Find the limit using lim⁡x=0sin⁡xx=1. \text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. } Find the limit using limx=0​xsinx​=1. - lim⁡x→06x2(cot⁡3x)(csc⁡2x)\lim _ { x \rightarrow 0 } 6 x ^ { 2 } ( \cot 3 x ) ( \csc 2 x )limx→0​6x2(cot3x)(csc2x)

Give an appropriate answer. - f(x)=3x+8 for x0=9f ( x ) = 3 \sqrt { x } + 8 \text { for } x _ { 0 } = 9f(x)=3x​+8 for x0​=9

Use the table of values of f to estimate the limit. - Let f(x)=x2+8x−2, find lim⁡x→2f(x)\text { Let } f(x)=x^{2}+8 x-2, \text { find } \lim _{x \rightarrow 2} f(x) Let f(x)=x2+8x−2, find limx→2​f(x) 1.9 1.99 1.999 2.001 2.01 2.1 ()

Find the limit using lim⁡x=0sin⁡xx=1. \text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. } Find the limit using limx=0​xsinx​=1. - lim⁡x→2−⌊x⌋x\lim _ { x \rightarrow 2 ^ { - } } \frac { \lfloor x \rfloor } { x }limx→2−​x⌊x⌋​

Use the graph to estimate the specified limit. - Find lim⁡x→2−f(x) and lim⁡x→2+f(x)\text { Find } \lim _ { x \rightarrow 2 ^ { - } } f ( x ) \text { and } \lim _ { x \rightarrow 2^ { + } } f ( x ) Find limx→2−​f(x) and limx→2+​f(x)