Question 1

Find the limit if it exists.
-$$\lim _ { x \rightarrow 3 } ( 4 x - 4 )$$

Accepted Answer

A)  8 
B)  $- 8$ 
C)  16 
D)  $- 16$ 
A)  8 
B)  $- 8$ 
C)  16 
D)  $- 16$ D

Question 2

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

Accepted Answer

A) 
$\begin{array}{r|llllll}
\mathrm{x} & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \
\hline \mathrm{f}(\mathrm{x}) & 3.97484 & 3.99750 & 3.99975 & 4.00025 & 4.00250 & 4.02485
\end{array} \text {; limit = 4.0 }$ 
B) 
$\begin{array}{c|llllll}
\mathrm{x} & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \
\hline \mathrm{f}(\mathrm{x}) & 1.19245 & 1.19925 & 1.19993 & 1.20007 & 1.20075 & 1.20745
\end{array} \text {; limit }=1.20$ 
C) 
$\begin{array}{c|llllll}
\mathrm{x} & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \
\hline \mathrm{f}(\mathrm{x}) & 1.19245 & 1.19925 & 1.19993 & 1.20007 & 1.20075 & 1.20745
\end{array} ; \text { limit }=\infty$ 
D) 
$\begin{array}{c|llllll}
\mathrm{x} & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \
\hline \mathrm{f}(\mathrm{x}) & 5.07736 & 5.09775 & 5.09978 & 5.10022 & 5.10225 & 5.12236
\end{array} \text {; limit }=5.10$ 
A) 
$\begin{array}{r|llllll}
\mathrm{x} & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \
\hline \mathrm{f}(\mathrm{x}) & 3.97484 & 3.99750 & 3.99975 & 4.00025 & 4.00250 & 4.02485
\end{array} \text {; limit = 4.0 }$ 
B) 
$\begin{array}{c|llllll}
\mathrm{x} & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \
\hline \mathrm{f}(\mathrm{x}) & 1.19245 & 1.19925 & 1.19993 & 1.20007 & 1.20075 & 1.20745
\end{array} \text {; limit }=1.20$ 
C) 
$\begin{array}{c|llllll}
\mathrm{x} & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \
\hline \mathrm{f}(\mathrm{x}) & 1.19245 & 1.19925 & 1.19993 & 1.20007 & 1.20075 & 1.20745
\end{array} ; \text { limit }=\infty$ 
D) 
$\begin{array}{c|llllll}
\mathrm{x} & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \
\hline \mathrm{f}(\mathrm{x}) & 5.07736 & 5.09775 & 5.09978 & 5.10022 & 5.10225 & 5.12236
\end{array} \text {; limit }=5.10$ A

Question 3

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

Accepted Answer

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

Question 4

Use the table to estimate the rate of change of y at the specified value of x.
-x = 1. 
$\begin{array} { c | c } 
\mathrm { x } & \mathrm { y } \
\hline 0 & 0 \
0.2 & 0.12 \
0.4 & 0.48 \
0.6 & 1.08 \
0.8 & 1.92 \
1.0 & 3 \
1.2 & 4.32 \
1.4 & 5.88
\end{array}$

Accepted Answer

A)  8 
B)  6 
C)  2 
D)  4 
A)  8 
B)  6 
C)  2 
D)  4

Question 5

Provide an appropriate response.
-$$\mathrm { a } = 1.317 , \mathrm {~b} = 2.713 , \mathrm { x } _ { 0 } = 1.976$$

Accepted Answer

A)  $\delta = 1$ 
B)  $\delta = 0.659$ 
C)  $\delta = 1.396$ 
D)  $\delta = 0.737$ 
A)  $\delta = 1$ 
B)  $\delta = 0.659$ 
C)  $\delta = 1.396$ 
D)  $\delta = 0.737$

Question 6

Give an appropriate answer.
-Let $\lim _ { x \rightarrow 8 } f ( x ) = 32$. Find $\lim _ { x \rightarrow 8 } \sqrt [ 5 ] { f ( x ) }$.

Accepted Answer

A)  2 
B)  32 
C)  8 
D)  5 
A)  2 
B)  32 
C)  8 
D)  5

Question 7

Provide an appropriate response.
-Given $\lim _ { x \rightarrow 0^  { - } } \mathrm { f } ( \mathrm { x } ) = \mathrm { L } _ { \mathrm { l } } , \lim _ { \mathrm { x } \rightarrow 0 ^ { + } } \mathrm { f } ( \mathrm { x } ) = \mathrm { L } _ { \mathrm { r } }$, and $\mathrm { L } _ { \mathrm { l } } = \mathrm { L } _ { \mathrm { r } }$, which of the following statements is false?
I. $\lim _ { x \rightarrow 0 } f ( x ) = L _ { 1 }$
II. $\lim _ { x \rightarrow0 } f ( x ) = L _ { r }$
III. $\lim _ { x \rightarrow 0} f ( x )$ does not exist.

Accepted Answer

A)  II 
B)  none 
C)  III 
D)  I 
A)  II 
B)  none 
C)  III 
D)  I

Question 8

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

Accepted Answer

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

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 $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$.
-

Accepted Answer

A)  0.2 
B)  0.1 
C)  3.5 
D)  -0.2 
A)  0.2 
B)  0.1 
C)  3.5 
D)  -0.2

Question 10

Give an appropriate answer.
-Let $\lim _ { x \rightarrow -9 } f ( x ) = 49$. Find $\lim _ { x \rightarrow -9 } \sqrt { f ( x ) }$.

Accepted Answer

A)  49 
B)  $- 9$ 
C)  7 
D)  $2.6458$ 
A)  49 
B)  $- 9$ 
C)  7 
D)  $2.6458$

Question 11

Find all points where the function is discontinuous.
-Does $\lim f ( x )$ exist?
$$f ( x ) = \left\{ \begin{array} { l l } 
- x ^ { 2 } + 1 , & - 1 \leq x < 0 \
2 x , & 0 < x < 1 \
- 4 , & x = 1 \
- 2 x + 4 & 1 < x < 3 \
5 , & 3 < x < 5
\end{array} \right.$$

Accepted Answer

A) True 
 B)False

Question 12

Use the table to estimate the rate of change of y at the specified value of x.
-x = 1. 
$\begin{array} { c | c } 
x & y \
\hline 0 & 0 \
0.2 & 0.01 \
0.4 & 0.04 \
0.6 & 0.09 \
0.8 & 0.16 \
1.0 & 0.25 \
1.2 & 0.36 \
1.4 & 0.49
\end{array}$

Accepted Answer

A)  2 
B)  1 
C)  0.5 
D)  1.5 
A)  2 
B)  1 
C)  0.5 
D)  1.5

Question 13

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 $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$.
-

Accepted Answer

A)  0.5625 
B)  2 
C)  1 
D)  0.4375 
A)  0.5625 
B)  2 
C)  1 
D)  0.4375

Question 14

Find the slope of the curve at the given point P and an equation of the tangent line at P.
-$y = 4 - x ^ { 3 } , ( - 1,5 )$

Accepted Answer

A)  slope is $- 1 ; y = - x + 2$ 
B)  slope is $0 ; y = 2$ 
C)  slope is $- 3 ; y = - 3 x + 2$ 
D)  slope is $3 ; y = 3 x + 2$ 
A)  slope is $- 1 ; y = - x + 2$ 
B)  slope is $0 ; y = 2$ 
C)  slope is $- 3 ; y = - 3 x + 2$ 
D)  slope is $3 ; y = 3 x + 2$

Question 15

Find all points where the function is discontinuous.
-

Accepted Answer

A)  x = 1 
B)  x = 0 
C)  None 
D)  x = 0, x = 1 
A)  x = 1 
B)  x = 0 
C)  None 
D)  x = 0, x = 1

Question 16

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow0 } \frac { x } { \sin 3 x }$$

Accepted Answer

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

Question 17

Determine the limit by sketching an appropriate graph.
-$$\lim _ { x \rightarrow 3 ^ { - } } f ( x ) \text {, where } f ( x ) = \left\{ \begin{array} { l l } 
- 4 x + 1 & \text { for } x < 3 \
2 x + 2 & \text { for } x \geq 3
\end{array} \right.$$

Accepted Answer

A)  2 
B)  3 
C)  -11 
D)  8 
A)  2 
B)  3 
C)  -11 
D)  8

Question 18

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow 0 } \frac { \sin 3 x \cot 4 x } { \cot 5 x }$$

Accepted Answer

A)  0 
B)  $\frac { 15 } { 4 }$ 
C)  does not exist 
D)  $\frac { 12 } { 5 }$ 
A)  0 
B)  $\frac { 15 } { 4 }$ 
C)  does not exist 
D)  $\frac { 12 } { 5 }$

Question 19

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 $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$.
-$\mathrm { f } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 2 } , \mathrm {~L} = 243 , \mathrm { x } _ { 0 } = 9 \text {, and } \varepsilon = 0.5$

Accepted Answer

A)  $0.00926$ 
B)  $9.00925$ 
C)  $0.00925$ 
D)  $8.99074$ 
A)  $0.00926$ 
B)  $9.00925$ 
C)  $0.00925$ 
D)  $8.99074$

Question 20

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

Accepted Answer

A)  0 
B)  $\frac { \sqrt { 6 } } { 2 }$ 
C)  $\sqrt { 6 }$ 
D)  $\frac { 1 } { 2 }$ 
A)  0 
B)  $\frac { \sqrt { 6 } } { 2 }$ 
C)  $\sqrt { 6 }$ 
D)  $\frac { 1 } { 2 }$

Find the limit if it exists. - $\lim _ { x \rightarrow 3 } ( 4 x - 4 )$

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

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

Use the table to estimate the rate of change of y at the specified value of x. -x = 1. 0 0 0.2 0.12 0.4 0.48 0.6 1.08 0.8 1.92 1.0 3 1.2 4.32 1.4 5.88

Provide an appropriate response. - $\mathrm { a } = 1.317 , \mathrm {~b} = 2.713 , \mathrm { x } _ { 0 } = 1.976$

Give an appropriate answer. -Let $\lim _ { x \rightarrow 8 } f ( x ) = 32$ . Find $\lim _ { x \rightarrow 8 } \sqrt [ 5 ] { f ( x ) }$ .

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

Give an appropriate answer. -Let $\lim _ { x \rightarrow -9 } f ( x ) = 49$ . Find $\lim _ { x \rightarrow -9 } \sqrt { f ( x ) }$ .

Use the table to estimate the rate of change of y at the specified value of x. -x = 1. x y 0 0 0.2 0.01 0.4 0.04 0.6 0.09 0.8 0.16 1.0 0.25 1.2 0.36 1.4 0.49

Find the slope of the curve at the given point P and an equation of the tangent line at P. - $y = 4 - x ^ { 3 } , ( - 1,5 )$

Find all points where the function is discontinuous. -

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow0 } \frac { x } { \sin 3 x }$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { \sin 3 x \cot 4 x } { \cot 5 x }$

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

Functions

Differentiation

Applications of Derivatives

Integration

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Filters

Exam 2: Limits and Derivatives

Find the limit if it exists. - lim⁡x→3(4x−4)\lim _ { x \rightarrow 3 } ( 4 x - 4 )limx→3​(4x−4)

Use the table of values of f to estimate the limit. - Let f(x)=x−4x−2, find lim⁡x→4f(x)\text { Let } f(x)=\frac{x-4}{\sqrt{x}-2} \text {, find } \lim _{x \rightarrow 4} f(x) Let f(x)=x​−2x−4​, find limx→4​f(x) 3.9 3.99 3.999 4.001 4.01 4.1 ()

Find the limit, if it exists. - lim⁡h→θ23h+4+2\lim _ { h \rightarrow \theta } \frac { 2 } { \sqrt { 3 h + 4 } + 2 }limh→θ​3h+4​+22​