Question 1

Find the limit and determine if the function is continuous at the point being approached.
-$$\lim _ { x \rightarrow \pi / 2 } \cos ( 3 x - \cos 3 x )$$

Accepted Answer

A)  0; no 
B)  does not exist; yes 
C)  0; yes 
D)  does not exist; no 
A)  0; no 
B)  does not exist; yes 
C)  0; yes 
D)  does not exist; no

Question 2

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ occur frequently in calculus. Evaluate this limit for the given value of $x$ and function $f$.
-$( x ) = 4 \sqrt { x } , x = 4$

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

Provide an appropriate response.
-Let $\lim _ { x \rightarrow 3 } f ( x ) = 16$. Find $\lim _ { x \rightarrow 3 } \log _ { 4 } f ( x )$.

Accepted Answer

A)  2 
B)  $\frac { 1 } { 2 }$ 
C)  16 
D)  $- 3$ 
A)  2 
B)  $\frac { 1 } { 2 }$ 
C)  16 
D)  $- 3$

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 $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$.
-$f ( x ) = \sqrt { x + 2 } , L = 3 , c = 7$, and $\varepsilon = 1$

Accepted Answer

A)  $\delta = 16$ 
B)  $\delta = 4$ 
C)  $\delta = 5$ 
D)  $\delta = 7$ 
A)  $\delta = 16$ 
B)  $\delta = 4$ 
C)  $\delta = 5$ 
D)  $\delta = 7$

Question 6

Find the limit.
-$$\lim _ { x \rightarrow - ^ { - } } \frac { \sqrt { 3 x } ( x - 3 ) } { | x - 3 | }$$

Accepted Answer

A)  -3 
B)  Does not exist 
C)  0 
D)  3 
A)  -3 
B)  Does not exist 
C)  0 
D)  3

Question 7

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

Accepted Answer

A)  $\frac { 4 } { \pi }$ 
B)  $- \frac { 4 } { \pi }$ 
C)  $- \frac { 3 } { 2 }$ 
D)  0 
A)  $\frac { 4 } { \pi }$ 
B)  $- \frac { 4 } { \pi }$ 
C)  $- \frac { 3 } { 2 }$ 
D)  0

Question 8

Sketch the graph of a function y = f(x) that satisfies the given conditions.
-$$\lim _ { x \rightarrow \infty } f ( x ) = 1 , \lim _ { x \rightarrow \theta ^ { + } } f ( x ) = - 1 , \lim _ { x \rightarrow } f ( x ) = - 1 , \lim _ { x \rightarrow \theta ^ { - } } f ( x ) = 1$$

Accepted Answer

\[\text { (Answers may vary.)

Question 9

Provide an appropriate response.
-Give an example of a function f(x) that is continuous at all values of x except at x = 7, where it has a removable
discontinuity. Explain how you know that f is discontinuous at x = 7 and how you know the discontinuity is
removable.

Accepted Answer

Let @#LAT-DLM& be defined for all @#LAT-DLM&. The function @#LAT-DLM&

Question 10

Find the limit, if it exists.
-$$\lim _ { x \rightarrow 1 } \frac { x ^ { 4 } - 1 } { x - 1 }$$

Accepted Answer

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

Question 11

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit.
-The statement "the limit of a constant times a function is the constant times the limit" follows from a combination of two fundamental limit principles. What are they?

Accepted Answer

A)  The limit of a constant is the constant, and the limit of a product is the product of the limits. 
B)  The limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits. 
C)  The limit of a function is a constant times a limit, and the limit of a constant is the constant. 
D)  The limit of a product is the product of the limits, and a constant is continuous. 
A)  The limit of a constant is the constant, and the limit of a product is the product of the limits. 
B)  The limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits. 
C)  The limit of a function is a constant times a limit, and the limit of a constant is the constant. 
D)  The limit of a product is the product of the limits, and a constant is continuous.

Question 12

Use the table of values of f to estimate the limit.
-$\text { Let } f(x)=\frac{x^{2}+7 x+12}{x^{2}+5 x+4} \text {, find } \lim _{x \rightarrow 4} f(x)$
$\begin{array}{c|l|l|l|l|l|l}
\mathrm{x} & -4.1 & -4.01 & -4.001 & -3.999 & -3.99 & -3.9 \
\hline \mathrm{f}(\mathrm{x}) & & & & & &
\end{array}$

Accepted Answer

A) 
$\begin{array}{c|cccccc}
\mathrm{x} & -4.1 & -4.01 & -4.001 & -3.999 & -3.99 & -3.9 \
\hline \mathrm{f}(\mathrm{x}) & 0.2548 & 0.2355 & 0.2336 & 0.2331 & 0.2311 & 0.2103
\end{array} \text { limit }=0.2333$ 
B) 
$\begin{array}{c|cccccc}
\mathrm{x} & -4.1 & -4.01 & -4.001 & -3.999 & -3.99 & -3.9 \
\hline \mathrm{f}(\mathrm{x}) & 0.4548 & 0.4355 & 0.4336 & 0.4331 & 0.4311 & 0.4103
\end{array} \text {; limit }=0.4333$ 
C) 
$\begin{array}{c|cccccc}
\mathrm{x} & -4.1 & -4.01 & -4.001 & -3.999 & -3.99 & -3.9 \
\hline \mathrm{f}(\mathrm{x}) & 0.3548 & 0.3355 & 0.3336 & 0.3331 & 0.3311 & 0.3103
\end{array} \text {; limit }=0.3333$ 
D) 
$\begin{array}{c|cccccc}
\mathrm{x} & -4.1 & -4.01 & -4.001 & -3.999 & -3.99 & -3.9 \
\hline \mathrm{f}(\mathrm{x}) & 1.4082 & 1.4008 & 1.4001 & 1.3999 & 1.3992 & 1.3922
\end{array} \text {; limit }=1.4$ 
A) 
$\begin{array}{c|cccccc}
\mathrm{x} & -4.1 & -4.01 & -4.001 & -3.999 & -3.99 & -3.9 \
\hline \mathrm{f}(\mathrm{x}) & 0.2548 & 0.2355 & 0.2336 & 0.2331 & 0.2311 & 0.2103
\end{array} \text { limit }=0.2333$ 
B) 
$\begin{array}{c|cccccc}
\mathrm{x} & -4.1 & -4.01 & -4.001 & -3.999 & -3.99 & -3.9 \
\hline \mathrm{f}(\mathrm{x}) & 0.4548 & 0.4355 & 0.4336 & 0.4331 & 0.4311 & 0.4103
\end{array} \text {; limit }=0.4333$ 
C) 
$\begin{array}{c|cccccc}
\mathrm{x} & -4.1 & -4.01 & -4.001 & -3.999 & -3.99 & -3.9 \
\hline \mathrm{f}(\mathrm{x}) & 0.3548 & 0.3355 & 0.3336 & 0.3331 & 0.3311 & 0.3103
\end{array} \text {; limit }=0.3333$ 
D) 
$\begin{array}{c|cccccc}
\mathrm{x} & -4.1 & -4.01 & -4.001 & -3.999 & -3.99 & -3.9 \
\hline \mathrm{f}(\mathrm{x}) & 1.4082 & 1.4008 & 1.4001 & 1.3999 & 1.3992 & 1.3922
\end{array} \text {; limit }=1.4$

Question 13

Find the limit, if it exists.
-$\lim _ { - \theta } \frac { \sqrt { 1 + x } - 1 } { x }$

Accepted Answer

A)  $1 / 2$ 
B)  0 
C)  Does not exist 
D)  $1 / 4$ 
A)  $1 / 2$ 
B)  0 
C)  Does not exist 
D)  $1 / 4$

Question 14

Use the graph to estimate the specified limit.
-Find $\lim _ { x \rightarrow 8 ^ { - } } f ( x )$

Accepted Answer

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

Question 15

Find the limit.
-f $\lim _ { x \rightarrow } \frac { f ( x ) - 4 } { x - 2 } = 3$, find $\lim _ { x \rightarrow 3 } f ( x )$.

Accepted Answer

A)  11 
B)  7 
C)  3 
D)  Does not exist 
A)  11 
B)  7 
C)  3 
D)  Does not exist

Question 16

Provide an appropriate response.
-$\text { Use the Intermediate Value Theorem to prove that } 3 x ^ { 3 } + 9 x ^ { 2 } - 3 x + 5 = 0 \text { has a solution between } - 4 \text { and } - 3 \text {. }$

Accepted Answer

Let @#LAT-DLM& and let @#LAT-DLM& and @#LAT-DLM&. Since @#LAT-DLM& is continuous

Question 17

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 $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$.
-$$f ( x ) = \frac { 14 } { x } , c = 7 , \varepsilon = 0.4$$

Accepted Answer

A)  $L = 2 ; \delta = 3.5$ 
B)  $L = 2 ; \delta = 1.75$ 
C)  $\mathrm { L } = 2 ; \delta = 1.17$ 
D)  $L = 2 ; \delta = 17.5$ 
A)  $L = 2 ; \delta = 3.5$ 
B)  $L = 2 ; \delta = 1.75$ 
C)  $\mathrm { L } = 2 ; \delta = 1.17$ 
D)  $L = 2 ; \delta = 17.5$

Question 18

Find the limit and determine if the function is continuous at the point being approached.
-$$\lim _ { \theta - 0 } \tan ( \sin ( 0 \cos ( \sin \theta ) ) )$$

Accepted Answer

A)  does not exist; no 
B)  0; no 
C)  1; yes 
D)  0; yes 
A)  does not exist; no 
B)  0; no 
C)  1; yes 
D)  0; yes

Question 19

Find the limit if it exists.
-$$\lim _ { x \rightarrow 9 } ( 7 x - 3 )$$

Accepted Answer

A)  -66 
B)  -60 
C)  66 
D)  60 
A)  -66 
B)  -60 
C)  66 
D)  60

Question 20

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

Accepted Answer

A)  8 
B)  2 
C)  Does not exist 
D)  -2 
A)  8 
B)  2 
C)  Does not exist 
D)  -2

Find the limit and determine if the function is continuous at the point being approached. - $\lim _ { x \rightarrow \pi / 2 } \cos ( 3 x - \cos 3 x )$

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ occur frequently in calculus. Evaluate this limit for the given value of $x$ and function $f$ . - $( x ) = 4 \sqrt { x } , x = 4$

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

Provide an appropriate response. -Let $\lim _ { x \rightarrow 3 } f ( x ) = 16$ . Find $\lim _ { x \rightarrow 3 } \log _ { 4 } f ( x )$ .

Find the limit. - $\lim _ { x \rightarrow - ^ { - } } \frac { \sqrt { 3 x } ( x - 3 ) } { | x - 3 | }$

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

Provide an appropriate response. -Give an example of a function f(x) that is continuous at all values of x except at x = 7, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 7 and how you know the discontinuity is removable.

Find the limit, if it exists. - $\lim _ { x \rightarrow 1 } \frac { x ^ { 4 } - 1 } { x - 1 }$

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. -The statement "the limit of a constant times a function is the constant times the limit" follows from a combination of two fundamental limit principles. What are they?

Use the table of values of f to estimate the limit. - $\text { Let } f(x)=\frac{x^{2}+7 x+12}{x^{2}+5 x+4} \text {, find } \lim _{x \rightarrow 4} f(x)$ -4.1 -4.01 -4.001 -3.999 -3.99 -3.9 ()

Find the limit, if it exists. - $\lim _ { - \theta } \frac { \sqrt { 1 + x } - 1 } { x }$

Use the graph to estimate the specified limit. -Find $\lim _ { x \rightarrow 8 ^ { - } } f ( x )$ $Use the graph to estimate the specified limit. -Find \lim _ { x \rightarrow 8 ^ { - } } f ( x )$

Find the limit. -f $\lim _ { x \rightarrow } \frac { f ( x ) - 4 } { x - 2 } = 3$ , find $\lim _ { x \rightarrow 3 } f ( x )$ .

Provide an appropriate response. - $\text { Use the Intermediate Value Theorem to prove that } 3 x ^ { 3 } + 9 x ^ { 2 } - 3 x + 5 = 0 \text { has a solution between } - 4 \text { and } - 3 \text {. }$

Find the limit and determine if the function is continuous at the point being approached. - $\lim _ { \theta - 0 } \tan ( \sin ( 0 \cos ( \sin \theta ) ) )$

Find the limit if it exists. - $\lim _ { x \rightarrow 9 } ( 7 x - 3 )$

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

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Exam 3: Limits and Continuity

Find the limit and determine if the function is continuous at the point being approached. - lim⁡x→π/2cos⁡(3x−cos⁡3x)\lim _ { x \rightarrow \pi / 2 } \cos ( 3 x - \cos 3 x )limx→π/2​cos(3x−cos3x)

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→θsin⁡4xsin⁡5x\lim _ { x \rightarrow \theta } \frac { \sin 4 x } { \sin 5 x }limx→θ​sin5xsin4x​

Provide an appropriate response. -Let lim⁡x→3f(x)=16\lim _ { x \rightarrow 3 } f ( x ) = 16limx→3​f(x)=16 . Find lim⁡x→3log⁡4f(x)\lim _ { x \rightarrow 3 } \log _ { 4 } f ( x )limx→3​log4​f(x) .

Find the limit. - lim⁡x→−−3x(x−3)∣x−3∣\lim _ { x \rightarrow - ^ { - } } \frac { \sqrt { 3 x } ( x - 3 ) } { | x - 3 | }limx→−−​∣x−3∣3x​(x−3)​

Find the average rate of change of the function over the given interval. - g(t)=4+tan⁡t,[−π4,π4]g ( t ) = 4 + \tan t , \left[ - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right]g(t)=4+tant,[−4π​,4π​]