Question 1

Find the limit.
-$$\lim _ { x \rightarrow 1 ^ { - } } \frac { 3 } { x ^ { 2 } - 1 }$$

Accepted Answer

A)  $\infty$ 
B)  1 
C)  0 
D)  $- \infty$ 
A)  $\infty$ 
B)  1 
C)  0 
D)  $- \infty$

Question 2

Find the limit.
-$\lim _ { x \rightarrow 1 ^ { - } } \frac { x ^ { 2 } - 5 x + 4 } { x ^ { 3 } - x }$

Accepted Answer

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

Question 3

Find the limit, if it exists.
-$$\lim _ { x \rightarrow 5 } \frac { x ^ { 2 } - 25 } { x ^ { 2 } - 9 x + 20 }$$

Accepted Answer

A)  Does not exist 
B)  0 
C)  5 
D)  10 
A)  Does not exist 
B)  0 
C)  5 
D)  10

Question 4

For the function f whose graph is given, determine the limit.
-$\text { Find } \lim _{x \rightarrow 0} f(x) \text {. }$

Accepted Answer

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

Question 5

Provide an appropriate response.
-Suppose $\lim _ { x \rightarrow \theta } f ( x ) = 1$ and $\lim _ { x \rightarrow \theta } g ( x ) = - 3$. Name the limit rules that are used to accomplish steps (a), (b), and (c) of the following calculation.

Accepted Answer

A)  (a) Quotient Rule 
B)  (a) Quotient Rule
(b) Difference Rule, Sum Rule
(b) Difference Rule
(c) Constant Multiple Rule and Power Rule
(c) Constant Multiple Rule 
C)  (a) Difference Rule 
D)  (a) Quotient Rule
(b) Power Rule
(b) Difference Rule, Power Rule
(c) Sum Rule
(c) Constant Multiple Rule and Sum Rule 
A)  (a) Quotient Rule 
B)  (a) Quotient Rule
(b) Difference Rule, Sum Rule
(b) Difference Rule
(c) Constant Multiple Rule and Power Rule
(c) Constant Multiple Rule 
C)  (a) Difference Rule 
D)  (a) Quotient Rule
(b) Power Rule
(b) Difference Rule, Power Rule
(c) Sum Rule
(c) Constant Multiple Rule and Sum Rule

Question 6

Provide an appropriate response.
-It can be shown that the inequalities $- x \leq x \cos \left( \frac { 1 } { x } \right) \leq x$ hold for all values of $x \geq 0$. Find $\lim _ { x \rightarrow 0 } x \cos \left( \frac { 1 } { x } \right)$ if it exists.

Accepted Answer

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

Question 7

Solve the problem.
-The cross-sectional area of a cylinder is given by $\mathrm { A } = \pi \mathrm { D } ^ { 2 } / 4$, where $\mathrm { D }$ is the cylinder diameter. Find the tolerance range of $\mathrm { D }$ such that $| \mathrm { A } - 10 | < 0.01$ as long as $\mathrm { D } _ { \min } < \mathrm { D } < \mathrm { D } _ { \max }$.

Accepted Answer

A)  $\mathrm { D } _ { \min } = 3.567 , \mathrm { D } _ { \max } = 3.570$ 
B)  $\mathrm { D } _ { \min } = 3.567 , \mathrm { D } _ { \max } = 3.578$ 
C)  $\mathrm { D } _ { \min } = 3.558 , \mathrm { D } _ { \max } = 3.570$ 
D)  $\mathrm { D } _ { \min } = 3.558 , \mathrm { D } _ { \max } = 3.578$ 
A)  $\mathrm { D } _ { \min } = 3.567 , \mathrm { D } _ { \max } = 3.570$ 
B)  $\mathrm { D } _ { \min } = 3.567 , \mathrm { D } _ { \max } = 3.578$ 
C)  $\mathrm { D } _ { \min } = 3.558 , \mathrm { D } _ { \max } = 3.570$ 
D)  $\mathrm { D } _ { \min } = 3.558 , \mathrm { D } _ { \max } = 3.578$

Question 8

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

Accepted Answer

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

Question 9

Graph the rational function. Include the graphs and equations of the asymptotes.
-$y = \frac { x } { x ^ { 2 } + x + 4 }$

Accepted Answer

A)  asymptote: $\mathrm { y } = 0$
  
B)  asymptotes: $x = 4 , x = - 4$
  
C)  asymptote: $y = 0$
  
D)  asymptote: $y = 1$
  
A)  asymptote: $\mathrm { y } = 0$
  
B)  asymptotes: $x = 4 , x = - 4$
  
C)  asymptote: $y = 0$
  
D)  asymptote: $y = 1$

Question 10

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

Accepted Answer

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

Question 11

Find the limit.
-$$\lim _ { x \rightarrow } \frac { 5 x + 1 } { 16 x - 7 }$$

Accepted Answer

A)  $- \frac { 1 } { 7 }$ 
B)  $\frac { 5 } { 16 }$ 
C)  0 
D)  $\infty$ 
A)  $- \frac { 1 } { 7 }$ 
B)  $\frac { 5 } { 16 }$ 
C)  0 
D)  $\infty$

Question 12

Find the limit and determine if the function is continuous at the point being approached.
-$$\lim _ { x \rightarrow 0 } \sin ^ { - 1 } \left( e ^ { x ^ { 5 } } \right)$$

Accepted Answer

A)  $\frac { \pi } { 2 }$; yes 
B)  $\frac { \pi } { 4 }$; no 
C)  does not exist; no 
D)  $\frac { \pi } { 4 }$; yes 
A)  $\frac { \pi } { 2 }$; yes 
B)  $\frac { \pi } { 4 }$; no 
C)  does not exist; no 
D)  $\frac { \pi } { 4 }$; yes

Question 13

Solve the problem.
-$$\text { The graph below shows the number of tuberculosis deaths in the United States from } 1989 \text { to } 1998 .$$
 
 Estimate the average rate of change in tuberculosis deaths from 1996 to 1998.

Accepted Answer

A)  About -0.5 deaths per year 
B)  About -90 deaths per year 
C)  About -50 deaths per year 
D)  About -20 deaths per year 
A)  About -0.5 deaths per year 
B)  About -90 deaths per year 
C)  About -50 deaths per year 
D)  About -20 deaths per year

Question 14

Find the limit.
-$$\lim _ { x \rightarrow 0 } ( 5 \sin x - 1 )$$

Accepted Answer

A)  5 
B)  0 
C)  5 - 1 
D)  -1 
A)  5 
B)  0 
C)  5 - 1 
D)  -1

Question 15

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

Accepted Answer

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

Question 16

Use the table of values of f to estimate the limit.
-$\text { Let } f(x)=\frac{x-2}{x^{2}-6 x+8} \text {, find } \lim _{x \rightarrow 2} f(x)$
$\begin{array}{c|l|l|l|l|l|l}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & & & & & &
\end{array}$

Accepted Answer

A) 
$\begin{array}{c|cccccc}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & -0.3762 & -0.3975 & -0.3998 & -0.4003 & -0.4025 & -0.4263
\end{array} \text {; limit }=-0.4$ 
B) 
$\begin{array}{c|cccccc}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & 0.4762 & 0.4975 & 0.4998 & 0.5003 & 0.5025 & 0.5263
\end{array} \text {; limit }=0.5$ 
C) 
$\begin{array}{c|cccccc}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & -0.4762 & -0.4975 & -0.4998 & -0.5003 & -0.5025 & -0.5263
\end{array} \text { limit }=-0.5$ 
D)

$\begin{array}{c|cccccc}\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & -0.5762 & -0.5975 & -0.5998 & -0.6003 & -0.6025 & -0.6263
\end{array} ; \text { limit }=-0.6$ 
A) 
$\begin{array}{c|cccccc}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & -0.3762 & -0.3975 & -0.3998 & -0.4003 & -0.4025 & -0.4263
\end{array} \text {; limit }=-0.4$ 
B) 
$\begin{array}{c|cccccc}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & 0.4762 & 0.4975 & 0.4998 & 0.5003 & 0.5025 & 0.5263
\end{array} \text {; limit }=0.5$ 
C) 
$\begin{array}{c|cccccc}
\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & -0.4762 & -0.4975 & -0.4998 & -0.5003 & -0.5025 & -0.5263
\end{array} \text { limit }=-0.5$ 
D)

$\begin{array}{c|cccccc}\mathrm{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \
\hline \mathrm{f}(\mathrm{x}) & -0.5762 & -0.5975 & -0.5998 & -0.6003 & -0.6025 & -0.6263
\end{array} ; \text { limit }=-0.6$

Question 17

Graph the rational function. Include the graphs and equations of the asymptotes.
-$y = \frac { 2 - 2 x - x ^ { 2 } } { x }$

Accepted Answer

A)  asymptotes: $x = 0 , y = - x - 4$
  
B)  asymptotes: $x = 0 , y = - x$
  
C)  asymptotes: $x = 0 , y = - x + 2$
  
D)  asymptotes: $x = 0 , y = - x - 2$
  
A)  asymptotes: $x = 0 , y = - x - 4$
  
B)  asymptotes: $x = 0 , y = - x$
  
C)  asymptotes: $x = 0 , y = - x + 2$
  
D)  asymptotes: $x = 0 , y = - x - 2$

Question 18

Solve the problem.
-The current in a simple electrical circuit is given by $\mathrm { I } = \mathrm { V } / \mathrm { R }$, where $\mathrm { I }$ is the current in amperes, $\mathrm { V }$ is the voltage in volts, and $\mathrm { R }$ is the resistance in ohms. When $\mathrm { V } = 12$ volts, what is a $12 \Omega$ resistor's tolerance for the current to be within $1 \pm 0.01$ amp?

Accepted Answer

A)  $0.01 \%$ 
B)  $10 \%$ 
C)  $1 \%$ 
D)  $0.1 \%$ 
A)  $0.01 \%$ 
B)  $10 \%$ 
C)  $1 \%$ 
D)  $0.1 \%$

Question 19

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

Accepted Answer

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

Question 20

Solve the problem.
-To what new value should $\mathrm { f } ( 2 )$ be changed to remove the discontinuity?
$f ( x ) = \left\{ \begin{array} { l l } 
2 x - 1 , & x  2
\end{array} \right.$

Accepted Answer

A)  2 
B)  -4 
C)  3 
D)  -5 
A)  2 
B)  -4 
C)  3 
D)  -5

Find the limit. - $\lim _ { x \rightarrow 1 ^ { - } } \frac { 3 } { x ^ { 2 } - 1 }$

Find the limit. - $\lim _ { x \rightarrow 1 ^ { - } } \frac { x ^ { 2 } - 5 x + 4 } { x ^ { 3 } - x }$

Find the limit, if it exists. - $\lim _ { x \rightarrow 5 } \frac { x ^ { 2 } - 25 } { x ^ { 2 } - 9 x + 20 }$

For the function f whose graph is given, determine the limit. - $\text { Find } \lim _{x \rightarrow 0} f(x) \text {. }$ $For the function f whose graph is given, determine the limit. - \text { Find } \lim _{x \rightarrow 0} f(x) \text {. }$

Provide an appropriate response. -It can be shown that the inequalities $- x \leq x \cos \left( \frac { 1 } { x } \right) \leq x$ hold for all values of $x \geq 0$ . Find $\lim _ { x \rightarrow 0 } x \cos \left( \frac { 1 } { x } \right)$ if it exists.

Graph the rational function. Include the graphs and equations of the asymptotes. - $y = \frac { x } { x ^ { 2 } + x + 4 }$ $Graph the rational function. Include the graphs and equations of the asymptotes. - y = \frac { x } { x ^ { 2 } + x + 4 }$

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

Find the limit. - $\lim _ { x \rightarrow } \frac { 5 x + 1 } { 16 x - 7 }$

Find the limit and determine if the function is continuous at the point being approached. - $\lim _ { x \rightarrow 0 } \sin ^ { - 1 } \left( e ^ { x ^ { 5 } } \right)$

Find the limit. - $\lim _ { x \rightarrow 0 } ( 5 \sin x - 1 )$

Use the table of values of f to estimate the limit. - $\text { Let } f(x)=\frac{x-2}{x^{2}-6 x+8} \text {, find } \lim _{x \rightarrow 2} f(x)$ 1.9 1.99 1.999 2.001 2.01 2.1 ()

Graph the rational function. Include the graphs and equations of the asymptotes. - $y = \frac { 2 - 2 x - x ^ { 2 } } { x }$ $Graph the rational function. Include the graphs and equations of the asymptotes. - y = \frac { 2 - 2 x - x ^ { 2 } } { x }$

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

Solve the problem. -To what new value should $\mathrm { f } ( 2 )$ be changed to remove the discontinuity? $f ( x ) = \left\{ \begin{array} { l l } 2 x - 1 , & x < 2 \\5 & x = 2 \\x + 1 , & x > 2\end{array} \right.$

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Exam 3: Limits and Continuity

Find the limit. - lim⁡x→1−3x2−1\lim _ { x \rightarrow 1 ^ { - } } \frac { 3 } { x ^ { 2 } - 1 }limx→1−​x2−13​

Find the limit. - lim⁡x→1−x2−5x+4x3−x\lim _ { x \rightarrow 1 ^ { - } } \frac { x ^ { 2 } - 5 x + 4 } { x ^ { 3 } - x }limx→1−​x3−xx2−5x+4​

Find the limit, if it exists. - lim⁡x→5x2−25x2−9x+20\lim _ { x \rightarrow 5 } \frac { x ^ { 2 } - 25 } { x ^ { 2 } - 9 x + 20 }limx→5​x2−9x+20x2−25​

For the function f whose graph is given, determine the limit. - Find lim⁡x→0f(x). \text { Find } \lim _{x \rightarrow 0} f(x) \text {. } Find limx→0​f(x).

Graph the rational function. Include the graphs and equations of the asymptotes. - y=xx2+x+4y = \frac { x } { x ^ { 2 } + x + 4 }y=x2+x+4x​

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - lim⁡x→036+x−36−xx\lim _ { x \rightarrow 0 } \frac { \sqrt { 36 + x } - \sqrt { 36 - x } } { x }limx→0​x36+x​−36−x​​

Find the limit. - lim⁡x→5x+116x−7\lim _ { x \rightarrow } \frac { 5 x + 1 } { 16 x - 7 }limx→​16x−75x+1​

Find the limit and determine if the function is continuous at the point being approached. - lim⁡x→0sin⁡−1(ex5)\lim _ { x \rightarrow 0 } \sin ^ { - 1 } \left( e ^ { x ^ { 5 } } \right)limx→0​sin−1(ex5)

Find the limit. - lim⁡x→0(5sin⁡x−1)\lim _ { x \rightarrow 0 } ( 5 \sin x - 1 )limx→0​(5sinx−1)

Use the table of values of f to estimate the limit. - Let f(x)=x−2x2−6x+8, find lim⁡x→2f(x)\text { Let } f(x)=\frac{x-2}{x^{2}-6 x+8} \text {, find } \lim _{x \rightarrow 2} f(x) Let f(x)=x2−6x+8x−2​, find limx→2​f(x) 1.9 1.99 1.999 2.001 2.01 2.1 ()

Graph the rational function. Include the graphs and equations of the asymptotes. - y=2−2x−x2xy = \frac { 2 - 2 x - x ^ { 2 } } { x }y=x2−2x−x2​

Find the limit. - lim⁡x→2(x2+8x−2)\lim _ { x \rightarrow 2 } \left( x ^ { 2 } + 8 x - 2 \right)limx→2​(x2+8x−2)

Solve the problem. -To what new value should f(2)\mathrm { f } ( 2 )f(2) be changed to remove the discontinuity? f(x)={2x−1,x<25x=2x+1,x>2f ( x ) = \left\{ \begin{array} { l l } 2 x - 1 , & x < 2 \\5 & x = 2 \\x + 1 , & x > 2\end{array} \right.f(x)=⎩⎨⎧​2x−1,5x+1,​x<2x=2x>2​