Question 1

Find the limit, if it exists.
-$$\lim _ { h \rightarrow 8 } \frac { 2 } { \sqrt { 3 h + 4 } + 2 }$$

Accepted Answer

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

Question 2

Find the average rate of change of the function over the given interval.
-$$y = 4 x ^ { 2 } , \left[ 0 , \frac { 7 } { 4 } \right]$$

Accepted Answer

A)  $- \frac { 3 } { 10 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  2 
D)  7 
A)  $- \frac { 3 } { 10 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  2 
D)  7

Question 3

Find all points where the function is discontinuous.
-

Accepted Answer

A)  x = 1, x = 5 
B)  x = 1, x = 4, x = 5 
C)  x = 4 
D)  None 
A)  x = 1, x = 5 
B)  x = 1, x = 4, x = 5 
C)  x = 4 
D)  None

Question 4

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

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  12 
C)  $\frac { 1 } { 12 }$ 
D)  0 
A)  $\frac { 1 } { 6 }$ 
B)  12 
C)  $\frac { 1 } { 12 }$ 
D)  0

Question 5

A function $\mathrm { f } ( \mathrm { x } )$, a point $\mathrm { c }$, the limit of $\mathrm { f } ( \mathrm { x } )$ as $x$ approaches $\mathrm { c }$, and a positive number $\varepsilon$ is given. Find a number $\delta > 0$ such that for all $x , 0  0 , L = 4 m , c = 4$, and $\varepsilon = 0.05$

Accepted Answer

A)  $\delta = 4 + \frac { 0.05 } { m }$ 
B)  $\delta = 4 - \mathrm { m }$ 
C)  $\delta = 0.05$ 
D)  $\delta = \frac { 0.05 } { m }$ 
A)  $\delta = 4 + \frac { 0.05 } { m }$ 
B)  $\delta = 4 - \mathrm { m }$ 
C)  $\delta = 0.05$ 
D)  $\delta = \frac { 0.05 } { m }$

Question 6

$$\text { Use the graph to find a } \delta > 0 \text { such that for all } x , 0 < | x - c | < \delta \Rightarrow | f ( x ) - L | < \varepsilon \text {. }$$
-

Accepted Answer

A)  $\delta = 3$ 
B)  $\delta = 0.4$ 
C)  $\delta = 0.1$ 
D)  $\delta = 0.2$ 
A)  $\delta = 3$ 
B)  $\delta = 0.4$ 
C)  $\delta = 0.1$ 
D)  $\delta = 0.2$

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

Question 8

Solve the problem.
-You are asked to make some circular cylinders, each with a cross-sectional area of $1 \mathrm {~cm} ^ { 2 }$. To do this, you need to know how much deviation from the ideal cylinder diameter of $x _ { 0 } = 1.99 \mathrm {~cm}$ you can allow and still have the area come within $0.1 \mathrm {~cm} ^ { 2 }$ of the required $1 \mathrm {~cm} ^ { 2 }$. To find out, let $A = \pi \left( \frac { x } { 2 } \right) ^ { 2 }$ and look for the interval in which you must hold $x$ to make $| A - 1 | < 0.1$. What interval do you find?

Accepted Answer

A)  (1.8974, 2.0976) 
B)  (1.0705, 1.1835) 
C)  (0.7569, 0.8368) 
D)  (0.5642, 0.5642) 
A)  (1.8974, 2.0976) 
B)  (1.0705, 1.1835) 
C)  (0.7569, 0.8368) 
D)  (0.5642, 0.5642)

Question 9

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

Accepted Answer

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

Question 10

Find the limit.
-$$\lim _ { x \rightarrow 0 } \frac { - 1 + ( 6 / x ) } { 5 - \left( 1 / x ^ { 2 } \right) }$$

Accepted Answer

A)  $\infty$ 
B)  $- \frac { 1 } { 5 }$ 
C)  $\approx$ 
D)  $\frac { 1 } { 5 }$ 
A)  $\infty$ 
B)  $- \frac { 1 } { 5 }$ 
C)  $\approx$ 
D)  $\frac { 1 } { 5 }$

Question 11

$$\text { Use the graph to find a } \delta > 0 \text { such that for all } x , 0 < | x - c | < \delta \Rightarrow | f ( x ) - L | < \varepsilon \text {. }$$
-

Accepted Answer

The answer of \[\text { Use the graph to find...

Question 12

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit.
-$$\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } - 1 } { \sqrt { x ^ { 2 } + 3 } - 2 }$$

Accepted Answer

A)  2 
B)  4 
C)  $\frac { 1 } { 4 }$ 
D)  1 
A)  2 
B)  4 
C)  $\frac { 1 } { 4 }$ 
D)  1

Question 13

Divide numerator and denominator by the highest power of x in the denominator to find the limit.
-$$\lim _ { x \rightarrow \infty } \frac { \sqrt [ 3 ] { x } - 6 x - 4 } { - 7 x + x ^ { 2 / 3 } + 4 }$$

Accepted Answer

A)  $*$ 
B)  0 
C)  $\frac { 7 } { 6 }$ 
D)  $\frac { 6 } { 7 }$ 
A)  $*$ 
B)  0 
C)  $\frac { 7 } { 6 }$ 
D)  $\frac { 6 } { 7 }$

Question 14

Find the limit and determine if the function is continuous at the point being approached.
-$$\lim _ { x \rightarrow 4 } \sin \left( x \sin ^ { 2 } x + x \cos ^ { 2 } x + 3 \right)$$

Accepted Answer

A)  does not exist; no 
B)  sin -1; yes 
C)  sin 7; no 
D)  sin 7; yes 
A)  does not exist; no 
B)  sin -1; yes 
C)  sin 7; no 
D)  sin 7; yes

Question 15

Provide an appropriate response.
-$$\text { Let } \lim _ { x \rightarrow 10 } f ( x ) = 4 \text {. Find } \lim _ { x \rightarrow 10 } ( - 5 ) ^ { f ( x ) }$$

Accepted Answer

A)  9,765,625 
B)  -5 
C)  4 
D)  625 
A)  9,765,625 
B)  -5 
C)  4 
D)  625

Question 16

Use the table to estimate the rate of change of y at the specified value of x.
-$x= 1$.
$\begin{array}{c|c}
x & y \
\hline .900 & -0.05263 \
1.990 & -0.00503 \
1.999 & -0.0005 \
.000 & 0.0000 \
.001 & 0.0005 \
.010 & 0.00498 \
.100 & 0.04762
\end{array}$

Accepted Answer

A)  $- 0.5$ 
B)  $0.5$ 
C)  0 
D)  1 
A)  $- 0.5$ 
B)  $0.5$ 
C)  0 
D)  1

Question 17

Provide an appropriate response.
-$\text { Let } \lim _ { x \rightarrow - 9 } f ( x ) = 81 \text {. Find } \lim _ { x \rightarrow 9 } \sqrt { f ( x ) } \text {. }$

Accepted Answer

A)  -9 
B)  3.0000 
C)  81 
D)  9 
A)  -9 
B)  3.0000 
C)  81 
D)  9

Question 18

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

Accepted Answer

A)  0 
B)  $\infty$ 
C)  $- \infty$ 
D)  Does not exist 
A)  0 
B)  $\infty$ 
C)  $- \infty$ 
D)  Does not exist

Question 19

Answer the question.
-$\text { Does } \lim _ { x - ( 1 ) ^ { + } } f ( x ) \text { exist? }$

Accepted Answer

A) True 
 B)False

Question 20

Provide an appropriate response.
-Use a calculator to graph the function $\mathrm { f }$ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function's value at $x = 0$. If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right $c$ the left? If so, what do you think the extended function's value(s) should be?
$f ( x ) = \frac { 6 ^ { x } - 1 } { x }$

Accepted Answer

A)  continuous extension exists from the right; $f ( 0 ) \approx 1.8145$ 
B)  continuous extension exists at origin; $f ( 0 ) = 0$ 
C)  continuous extension exists at origin; $f ( 0 ) \approx 1.8145$ 
D)  continuous extension exists from the left; $\mathrm { f } ( 0 ) \approx 1.8145$ 
A)  continuous extension exists from the right; $f ( 0 ) \approx 1.8145$ 
B)  continuous extension exists at origin; $f ( 0 ) = 0$ 
C)  continuous extension exists at origin; $f ( 0 ) \approx 1.8145$ 
D)  continuous extension exists from the left; $\mathrm { f } ( 0 ) \approx 1.8145$

Find the limit, if it exists. - $\lim _ { h \rightarrow 8 } \frac { 2 } { \sqrt { 3 h + 4 } + 2 }$

Find the average rate of change of the function over the given interval. - $y = 4 x ^ { 2 } , \left[ 0 , \frac { 7 } { 4 } \right]$

Find all points where the function is discontinuous. -

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

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

Find the limit. - $\lim _ { x \rightarrow 0 } \frac { - 1 + ( 6 / x ) } { 5 - \left( 1 / x ^ { 2 } \right) }$

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } - 1 } { \sqrt { x ^ { 2 } + 3 } - 2 }$

Divide numerator and denominator by the highest power of x in the denominator to find the limit. - $\lim _ { x \rightarrow \infty } \frac { \sqrt [ 3 ] { x } - 6 x - 4 } { - 7 x + x ^ { 2 / 3 } + 4 }$

Find the limit and determine if the function is continuous at the point being approached. - $\lim _ { x \rightarrow 4 } \sin \left( x \sin ^ { 2 } x + x \cos ^ { 2 } x + 3 \right)$

Provide an appropriate response. - $\text { Let } \lim _ { x \rightarrow 10 } f ( x ) = 4 \text {. Find } \lim _ { x \rightarrow 10 } ( - 5 ) ^ { f ( x ) }$

Use the table to estimate the rate of change of y at the specified value of x. - $x= 1$ . x y .900 -0.05263 1.990 -0.00503 1.999 -0.0005 .000 0.0000 .001 0.0005 .010 0.00498 .100 0.04762

Provide an appropriate response. - $\text { Let } \lim _ { x \rightarrow - 9 } f ( x ) = 81 \text {. Find } \lim _ { x \rightarrow 9 } \sqrt { f ( x ) } \text {. }$

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

Answer the question. - $\text { Does } \lim _ { x - ( 1 ) ^ { + } } f ( x ) \text { exist? }$ $Answer the question. - \text { Does } \lim _ { x - ( 1 ) ^ { + } } f ( x ) \text { exist? }$

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Exam 3: Limits and Continuity

Find the limit, if it exists. - lim⁡h→823h+4+2\lim _ { h \rightarrow 8 } \frac { 2 } { \sqrt { 3 h + 4 } + 2 }limh→8​3h+4​+22​

Find the average rate of change of the function over the given interval. - y=4x2,[0,74]y = 4 x ^ { 2 } , \left[ 0 , \frac { 7 } { 4 } \right]y=4x2,[0,47​]