Question 1

Provide an appropriate response.
-What conditions, when present, are sufficient to conclude that a function f(x) has a limit as x approaches some value of a?

Accepted Answer

A)  The limit of f(x) as x $\rightarrow$a from the left exists, the limit of f(x) as  $\rightarrow$a from the right exists, and at least one of these limits is the same as f(a). 
B)  The limit of f(x) as  $\rightarrow$a from the left exists, the limit of f(x) as  $\rightarrow$a from the right exists, and these two limits are the same. 
C)  Either the limit of f(x) as  $\rightarrow$a from the left exists or the limit of f(x) as  $\rightarrow$a from the right exists 
D)  f(a) exists, the limit of f(x) as  $\rightarrow$a from the left exists, and the limit of f(x) as  $\rightarrow$a from the right exists. 
A)  The limit of f(x) as x $\rightarrow$a from the left exists, the limit of f(x) as  $\rightarrow$a from the right exists, and at least one of these limits is the same as f(a). 
B)  The limit of f(x) as  $\rightarrow$a from the left exists, the limit of f(x) as  $\rightarrow$a from the right exists, and these two limits are the same. 
C)  Either the limit of f(x) as  $\rightarrow$a from the left exists or the limit of f(x) as  $\rightarrow$a from the right exists 
D)  f(a) exists, the limit of f(x) as  $\rightarrow$a from the left exists, and the limit of f(x) as  $\rightarrow$a from the right exists.

Question 2

Find the limit, if it exists.
-$$\lim _ { x \rightarrow 1 } \frac { 2 x - 7 } { 4 x + 5 }$$

Accepted Answer

A)  $- \frac { 5 } { 9 }$ 
B)  Does not exist 
C)  $- \frac { 1 } { 2 }$ 
D)  $- \frac { 7 } { 5 }$ 
A)  $- \frac { 5 } { 9 }$ 
B)  Does not exist 
C)  $- \frac { 1 } { 2 }$ 
D)  $- \frac { 7 } { 5 }$

Question 3

Determine the limit by sketching an appropriate graph.
-$$\lim _ { x \rightarrow 2 ^ { - } } f ( x ) , \text { where } f ( x ) = \left\{ \begin{array} { l l } 
\sqrt { 1 - x ^ { 2 } } & 0 \leq x < 1 \
1 & 1 \leq x < 2 \
2 & x = 2
\end{array} \right.$$

Accepted Answer

A)  2 
B)  0 
C)  Does not exist 
D)  1 
A)  2 
B)  0 
C)  Does not exist 
D)  1

Question 4

Find the limit, if it exists.
-$$\lim _ { x \rightarrow 7 } \frac { | 7 - x | } { 7 - x }$$

Accepted Answer

A)  0 
B)  $- 1$ 
C)  Does not exist 
D)  1 
A)  0 
B)  $- 1$ 
C)  Does not exist 
D)  1

Question 5

Find the limit.
-$\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)$

Accepted Answer

A)  0 
B)  does not exist 
C)  29 
D)  15 
A)  0 
B)  does not exist 
C)  29 
D)  15

Question 6

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

Accepted Answer

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

Question 7

Use the table of values of f to estimate the limit.
-$\text { Let } f(x)=\frac{x+5}{x^{2}+7 x+10} \text {, find } \lim _{x \rightarrow 5} f(x)$

$\begin{array}{c|l|l|l|l|l|l}
\mathrm{x} & -5.1 & -5.01 & -5.001 & -4.999 & -4.99 & -4.9 \
\hline \mathrm{f}(\mathrm{x}) & & & & & &
\end{array}$

Accepted Answer

A) 
$\begin{array}{c|cccccc}
\mathrm{x} & -5.1 & -5.01 & -5.001 & -4.999 & -4.99 & -4.9 \
\hline \mathrm{f}(\mathrm{x}) & -0.4226 & -0.4322 & -0.4332 & -0.4334 & -0.4344 & -0.4448
\end{array} \text {; limit }=-0.4333$ 
B) 
$\begin{array}{c|cccccc}
\mathrm{x} & -5.1 & -5.01 & -5.001 & -4.999 & -4.99 & -4.9 \
\hline \mathrm{f}(\mathrm{x}) & -0.2226 & -0.2322 & -0.2332 & -0.2334 & -0.2344 & -0.2448
\end{array} \text {; limit }=-0.2333$ 
C) 
$\begin{array}{c|cccccc}
\mathrm{x} & -5.1 & -5.01 & -5.001 & -4.999 & -4.99 & -4.9 \
\hline \mathrm{f}(\mathrm{x}) & 0.3226 & 0.3322 & 0.3332 & 0.3334 & 0.3344 & 0.3448
\end{array} \text {; limit }=0.3333$ 
D) 
$\begin{array}{c|cccccc}
\mathrm{x} & -5.1 & -5.01 & -5.001 & -4.999 & -4.99 & -4.9 \
\hline \mathrm{f}(\mathrm{x}) & -0.3226 & -0.3322 & -0.3332 & -0.3334 & -0.3344 & -0.3448
\end{array} \text { limit }=-0.3333$ 
A) 
$\begin{array}{c|cccccc}
\mathrm{x} & -5.1 & -5.01 & -5.001 & -4.999 & -4.99 & -4.9 \
\hline \mathrm{f}(\mathrm{x}) & -0.4226 & -0.4322 & -0.4332 & -0.4334 & -0.4344 & -0.4448
\end{array} \text {; limit }=-0.4333$ 
B) 
$\begin{array}{c|cccccc}
\mathrm{x} & -5.1 & -5.01 & -5.001 & -4.999 & -4.99 & -4.9 \
\hline \mathrm{f}(\mathrm{x}) & -0.2226 & -0.2322 & -0.2332 & -0.2334 & -0.2344 & -0.2448
\end{array} \text {; limit }=-0.2333$ 
C) 
$\begin{array}{c|cccccc}
\mathrm{x} & -5.1 & -5.01 & -5.001 & -4.999 & -4.99 & -4.9 \
\hline \mathrm{f}(\mathrm{x}) & 0.3226 & 0.3322 & 0.3332 & 0.3334 & 0.3344 & 0.3448
\end{array} \text {; limit }=0.3333$ 
D) 
$\begin{array}{c|cccccc}
\mathrm{x} & -5.1 & -5.01 & -5.001 & -4.999 & -4.99 & -4.9 \
\hline \mathrm{f}(\mathrm{x}) & -0.3226 & -0.3322 & -0.3332 & -0.3334 & -0.3344 & -0.3448
\end{array} \text { limit }=-0.3333$

Question 8

Use the slopes of UQ, UR, US, and UT to estimate the rate of change of y at the specified value of x.
-x = 5

Accepted Answer

A)  $\frac{25}{4}$ 
B)  0 
C)  52 
D)  54 
A)  $\frac{25}{4}$ 
B)  0 
C)  52 
D)  54

Question 9

Find the limit, if it exists.
-$$\lim _ { x\rightarrow 0 } \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$$

Accepted Answer

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

Question 10

Give an appropriate answer.
-Let $\lim _ { x \rightarrow 7 } f ( x ) = - 7$ and $\lim _ { x \rightarrow 7 } g ( x ) = - 5$. Find $\lim _ { x \rightarrow 7 } [ f ( x ) + g ( x ) ] ^ { 2 }$.

Accepted Answer

A)  144 
B)  $- 2$ 
C)  $- 12$ 
D)  74 
A)  144 
B)  $- 2$ 
C)  $- 12$ 
D)  74

Question 11

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$\lim _ { h \rightarrow 0^ { + } } \frac { \sqrt { h ^ { 2 } + 13 h + 14 } - \sqrt { 14 } } { h }$

Accepted Answer

A)  $\frac { 13 } { \sqrt { 28 } }$ 
B)  $\frac { 13 } { 2 \sqrt { 14 } }$ 
C)  $\frac { 13 } { 28 }$ 
D)  Does not exist 
A)  $\frac { 13 } { \sqrt { 28 } }$ 
B)  $\frac { 13 } { 2 \sqrt { 14 } }$ 
C)  $\frac { 13 } { 28 }$ 
D)  Does not exist

Question 12

Provide an appropriate response.
-If $x ^ { 3 } \leq f ( x ) \leq x$ for $x$ in $[ - 1,1 ]$, find $\lim _ { x \rightarrow \theta } f ( x )$ if it exists.

Accepted Answer

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

Question 13

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

Accepted Answer

A)  6 
B)  -1 
C)  does not exist 
D)  0 
A)  6 
B)  -1 
C)  does not exist 
D)  0

Question 14

$\lim _ { x \rightarrow 0 } f ( x )$

Accepted Answer

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

Question 15

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 - 4, L = -24, x0 = 2, and $\varepsilon$ = 0.01

Accepted Answer

A)  0.002 
B)  0.0005 
C)  -0.005 
D)  0.001 
A)  0.002 
B)  0.0005 
C)  -0.005 
D)  0.001

Question 16

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$
-$$\lim _ { x \rightarrow 8 ^ { + } } \frac { \sqrt { 5 x } ( x - 3 ) } { | x - 3 | }$$

Accepted Answer

A)  0 
B)  $- \sqrt { 15 }$ 
C)  $\sqrt { 15 }$ 
D)  Does not exist 
A)  0 
B)  $- \sqrt { 15 }$ 
C)  $\sqrt { 15 }$ 
D)  Does not exist

Question 17

Provide an appropriate response.
-Given $\varepsilon > 0$, find an interval $\mathrm { I } = ( 1,1 + \delta ) , \delta > 0$, such that if $x$ lies in $\mathrm { I }$, then $\sqrt { x - 1 } < \varepsilon$. What limit is being verified and what is its value?

Accepted Answer

A)  $\lim _ { x \rightarrow 1 ^ { + } } \sqrt { x } = 1$ 
B)  $\lim _ { x \rightarrow0 ^ { - } } \sqrt { x - 1 } = 0$ 
C)  $\lim _ { x \rightarrow \mathbf { 1 } ^ { + } } \sqrt { x - 1 } = 0$ 
D)  $\lim _ { x \rightarrow 1 ^ { - } } \sqrt { x - 1 } = 0$ 
A)  $\lim _ { x \rightarrow 1 ^ { + } } \sqrt { x } = 1$ 
B)  $\lim _ { x \rightarrow0 ^ { - } } \sqrt { x - 1 } = 0$ 
C)  $\lim _ { x \rightarrow \mathbf { 1 } ^ { + } } \sqrt { x - 1 } = 0$ 
D)  $\lim _ { x \rightarrow 1 ^ { - } } \sqrt { x - 1 } = 0$

Question 18

Give an appropriate answer.
- $\text { Evaluate } \lim _ { h \rightarrow \theta } \frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h } \text { for the given } x _ { 0 } \text { and function } f \text {. }$
$$f ( x ) = 3 x ^ { 2 } \text { for } x _ { 0 } = 9$$

Accepted Answer

A)  54 
B)  243 
C)  27 
D)  Does not exist 
A)  54 
B)  243 
C)  27 
D)  Does not exist

Question 19

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

Accepted Answer

A)  2 
B)  3 
C)  6 
D)  Does not exist 
A)  2 
B)  3 
C)  6 
D)  Does not exist

Question 20

Find the average rate of change of the function over the given interval.
-$y = \frac { 3 } { x - 2 } , [ 4,7 ]$

Accepted Answer

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

Provide an appropriate response. -What conditions, when present, are sufficient to conclude that a function f(x) has a limit as x approaches some value of a?

Find the limit, if it exists. - $\lim _ { x \rightarrow 1 } \frac { 2 x - 7 } { 4 x + 5 }$

Find the limit, if it exists. - $\lim _ { x \rightarrow 7 } \frac { | 7 - x | } { 7 - x }$

Find the limit. - $\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)$

Use the table of values of f to estimate the limit. - $\text { Let } f(x)=\frac{x+5}{x^{2}+7 x+10} \text {, find } \lim _{x \rightarrow 5} f(x)$ -5.1 -5.01 -5.001 -4.999 -4.99 -4.9 ()

Use the slopes of UQ, UR, US, and UT to estimate the rate of change of y at the specified value of x. -x = 5

Find the limit, if it exists. - $\lim _ { x\rightarrow 0 } \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$

Give an appropriate answer. -Let $\lim _ { x \rightarrow 7 } f ( x ) = - 7$ and $\lim _ { x \rightarrow 7 } g ( x ) = - 5$ . Find $\lim _ { x \rightarrow 7 } [ f ( x ) + g ( x ) ] ^ { 2 }$ .

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { h \rightarrow 0^ { + } } \frac { \sqrt { h ^ { 2 } + 13 h + 14 } - \sqrt { 14 } } { h }$

Provide an appropriate response. -If $x ^ { 3 } \leq f ( x ) \leq x$ for $x$ in $[ - 1,1 ]$ , find $\lim _ { x \rightarrow \theta } f ( x )$ if it exists.

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

$\lim _ { x \rightarrow 0 } f ( x )$ $\lim _ { x \rightarrow 0 } f ( x )$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 8 ^ { + } } \frac { \sqrt { 5 x } ( x - 3 ) } { | x - 3 | }$

Provide an appropriate response. -Given $\varepsilon > 0$ , find an interval $\mathrm { I } = ( 1,1 + \delta ) , \delta > 0$ , such that if $x$ lies in $\mathrm { I }$ , then $\sqrt { x - 1 } < \varepsilon$ . What limit is being verified and what is its value?

Give an appropriate answer. - $\text { Evaluate } \lim _ { h \rightarrow \theta } \frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h } \text { for the given } x _ { 0 } \text { and function } f \text {. }$ $f ( x ) = 3 x ^ { 2 } \text { for } x _ { 0 } = 9$

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

Find the average rate of change of the function over the given interval. - $y = \frac { 3 } { x - 2 } , [ 4,7 ]$

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

Provide an appropriate response. -What conditions, when present, are sufficient to conclude that a function f(x) has a limit as x approaches some value of a?

Find the limit, if it exists. - lim⁡x→12x−74x+5\lim _ { x \rightarrow 1 } \frac { 2 x - 7 } { 4 x + 5 }limx→1​4x+52x−7​

Find the limit, if it exists. - lim⁡x→7∣7−x∣7−x\lim _ { x \rightarrow 7 } \frac { | 7 - x | } { 7 - x }limx→7​7−x∣7−x∣​

Find the limit. - lim⁡x→2(x3+5x2−7x+1)\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)limx→2​(x3+5x2−7x+1)

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

Use the table of values of f to estimate the limit. - Let f(x)=x+5x2+7x+10, find lim⁡x→5f(x)\text { Let } f(x)=\frac{x+5}{x^{2}+7 x+10} \text {, find } \lim _{x \rightarrow 5} f(x) Let f(x)=x2+7x+10x+5​, find limx→5​f(x) -5.1 -5.01 -5.001 -4.999 -4.99 -4.9 ()