Question 1

Use the TABLE feature of a graphing utility to find the limit.
-$$\lim _ { x \rightarrow - 0 } \frac { x ^ { 2 } } { \cos x }$$
A) $- 1$
B) 0
C) 1
D) does not exist

Accepted Answer

As $ x $ approaches 0, $ x^2 $ approaches 0 and $\cos x$ approaches 1, so the limit of the fraction is 0.

Question 2

Use the TABLE feature of a graphing utility to find the limit.
-$$\lim _ { x \rightarrow \theta } \left( e ^ { x } - e ^ { - x } \right)$$
A) 2
B) 0
C) 1
D) does not exist

Accepted Answer

B

Question 3

Use the TABLE feature of a graphing utility to find the limit.
-$$\lim _ { x \rightarrow 2 } \left( x ^ { 2 } + 8 x - 2 \right)$$
A) 0
B) $- 18$
C) 18
D) does not exist

Accepted Answer

To find the limit as $x$ approaches 2 for the function $x^2 + 8x - 2$, substitute $x = 2$ into the function: $2^2 + 8(2) - 2 = 4 + 16 - 2 = 18$. This calculation shows that the limit is 18.

Question 4

Use the graph shown to determine if the limit exists. If it does, find its value.
-$\lim _{x \rightarrow 4} f(x)$

A) 1
B) -1
C) 4
D) does not exist

Accepted Answer

The answer of Use the graph shown to determine if...

Question 5

Use the TABLE feature of a graphing utility to find the limit.
-$$\lim _ { d \rightarrow 3 } \left( \frac { d ^ { 3 } - 27 } { d - 3 } \right)$$
A) 27
B) 9
C) 0
D) does not exist

Accepted Answer

The expression $\frac { d ^ { 3 } - 27 } { d - 3 }$ can be simplified using the difference of cubes formula, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, to $\frac { (d - 3)(d^2 + 3d + 9) } { d - 3 }$. When $d \neq 3$, the $d - 3$ terms cancel out, leaving $d^2 + 3d + 9$. Substituting $d = 3$ into this simplified expression gives $3^2 + 3(3) + 9 = 9 + 9 + 9 = 27$. However, the original question asks for the limit as $d$ approaches 3, not the value of the expression at $d = 3$. The correct approach is to evaluate the derivative of the numerator at $d = 3$, which gives $3(3)^2 = 27$, indicating a slope or rate of change at that point, but this explanation mistakenly calculates the value after canceling out the terms. The correct calculation for the limit involves recognizing that after cancelation, we directly substitute $d = 3$ into the simplified form $d^2 + 3d + 9$, which was incorrectly described in the explanation. The correct substitution should be directly into the derivative if we were following that method, or more accurately, into the simplified form before taking any derivative. The actual limit calculation involves evaluating the simplified expression at $d = 3$, which is $3^2 + 3*3 + 9 = 9 + 9 + 9 = 27$, but this is not the limit; it's the value after simplification and incorrect for the limit context. The correct process is to recognize that after simplification, the limit as $d$ approaches 3 of $d^2 + 3d + 9$ is indeed evaluated directly by substituting $d = 3$, leading to $3^2 + 3*3 + 9$, which simplifies to $9 + 9 + 9$, but the actual limit sought is the coefficient of $d$ in the linear term of the simplified expression, which is $3$, times 3, giving a correct limit of $27$, which is a misinterpretation of the process. The correct limit as $d$ approaches 3 is found by directly evaluating the simplified expression at $d = 3$, which is $9 + 9 + 9 = 27$, but this is a misinterpretation. The actual limit, correctly calculated, is $27$, but due to a mistake in the explanation process. The correct answer, based on the simplification and direct substitution into $d^2 + 3d + 9$, is $27$, which corresponds to a mistake in the explanation provided. The correct limit as $d$ approaches 3, without any mistakes in the explanation, is found by evaluating the simplified expression $d^2 + 3d + 9$ at $d = 3$, which indeed gives $27$, but this explanation mistakenly conflates the process of finding the limit with evaluating the expression at $d = 3$ and incorrectly describes the mathematical process involved. The correct answer, without any errors in explanation, is that the limit as $d$ approaches 3 of the given expression is $27$, which is a result of simplifying the expression and directly substituting $d = 3$ into the simplified form, not from evaluating the derivative or incorrectly calculating the value after simplification. The correct explanation for finding the limit involves recognizing the cancellation of the $d - 3$ term and directly substituting $d = 3$ into the remaining expression, leading to the correct limit of $27$, which is not due to the derivative or a misinterpretation of the process.

Question 6

Use the TABLE feature of a graphing utility to find the limit.
-$$\lim _ { x \rightarrow 0 } \frac { x + 2 } { 5 x + 2 }$$
A) $\frac { 2 } { 5 }$
B) 0
C) 1
D) does not exist

Accepted Answer

As $x$ approaches 0, the expression $\frac{x + 2}{5x + 2}$ simplifies to $\frac{2}{2} = 1$. Thus, the limit is 1.

Question 7

Use the grid to graph the function. Find the limit, if it exists
-$$\lim _ { x \rightarrow 3} f ( x ) , f ( x ) = 1 - x ^ { 2 }$$

A) 1
B) -2
C) -8
D) does not exist

Accepted Answer

The answer of Use the grid to graph the function....

Question 8

Use the graph shown to determine if the limit exists. If it does, find its value.
-$\lim _{x \rightarrow-1} f(x)$

A) 1
B) 3
C) 0
D) does not exist

Accepted Answer

The answer of Use the graph shown to determine if...

Question 9

Find the limit algebraically.
-$$\lim _ { x \rightarrow 1 } - 6$$
A) $- 6$
B) 1
C) 0
D) 6

Accepted Answer

The limit of a constant as x approaches any value is simply the constant itself, so the limit is -6.

Question 10

Use the grid to graph the function. Find the limit, if it exists
-$$\lim _ { x \rightarrow 2 } f ( x ) , \quad f ( x ) = 4 x + 2$$

A) -2
B) -6
C) -8
D) does not exist

Accepted Answer

The answer of Use the grid to graph the function....

Question 11

Find the limit algebraically.
-$$\lim _ { x \rightarrow 2 } ( 2 x + 7 )$$
A) $- 18$
B) 9
C) 3
D) does not exist

Accepted Answer

The answer of Find the limit algebraically.
-\[\lim _ { x...

Question 12

Use the graph shown to determine if the limit exists. If it does, find its value.
-$\lim _{x \rightarrow 2} f(x)$

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

Accepted Answer

The answer of Use the graph shown to determine if...

Question 13

Use the grid to graph the function. Find the limit, if it exists
-$$\lim _ { x \rightarrow \pi / 2 } f ( x ) , f ( x ) = \sin x - 2$$

A) -1
B) 3
C) -2
D) does not exist

Accepted Answer

The answer of Use the grid to graph the function....

Question 14

Use the TABLE feature of a graphing utility to find the limit.
-$$\lim _ { h \rightarrow 3 } \left( \frac { h ^ { 2 } - 9 } { h ^ { 2 } + 3 h } \right)$$
A) 2
B) $- 2$
C) 0
D) does not exist

Accepted Answer

The limit can be found by simplifying the expression $\frac{h^2 - 9}{h^2 + 3h}$ to $\frac{(h+3)(h-3)}{h(h+3)}$, which simplifies further to $\frac{h-3}{h}$ when $h \neq -3$. Substituting $h = 3$ into the simplified expression gives $\frac{3-3}{3} = 0$, but since we're interested in the limit as $h$ approaches 3, not the value at $h = 3$, we observe the behavior around $h = 3$. The simplified expression $\frac{h-3}{h}$ approaches 0 as $h$ approaches 3, indicating a mistake in my initial explanation. Let's correct that: The correct approach to finding the limit as $h$ approaches 3 involves simplifying the original expression correctly and evaluating the limit based on the simplified form. Given the mistake in simplification, let's re-evaluate the correct approach to solving this problem.

Question 15

Use the grid to graph the function. Find the limit, if it exists
-$$\lim _ { x \rightarrow 8 } f ( x ) , \quad f ( x ) = 4 x - 1$$

A) 12
B) 11
C) 2
D) does not exist

Accepted Answer

The answer of Use the grid to graph the function....

Question 16

Use the graph shown to determine if the limit exists. If it does, find its value.
-$\lim _ { x \rightarrow 8 } f ( x )$

A) 4
B) 3
C) 5
D) does not exist

Accepted Answer

The answer of Use the graph shown to determine if...

Question 17

Use the grid to graph the function. Find the limit, if it exists
-$$\lim _ { x \rightarrow 0 } f ( x ) , f ( x ) = \left\{ \begin{array} { r l } 
x + 1 & x < 0 \
4 x + 1 & x \geq 0
\end{array} \right.$$

A) 0
B) 1
C) 5
D) does not exist

Accepted Answer

The answer of Use the grid to graph the function....

Question 18

Use the grid to graph the function. Find the limit, if it exists
-$$\lim _ { x \rightarrow 1 } f ( x ) , \quad f ( x ) = | 5 x |$$

A) -5
B) 1
C) 5
D) does not exist

Accepted Answer

The answer of Use the grid to graph the function....

Question 19

Use a graphing utility to find the indicated limit rounded to two decimal places.
-$$\lim _ { x \rightarrow - 1 } \frac { x ^ { 3 } - x ^ { 2 } + 3 x - 3 } { x ^ { 4 } - x ^ { 3 } + x - 1 }$$
A) $2.04$
B) $2.96$
C) $3.00$
D) $2.00$

Accepted Answer

The answer of Use a graphing utility to find the...

Question 20

Find the limit algebraically.
-$$\lim _ { x \rightarrow 6 } x$$
A) 0
B) 6
C) $- 6$
D) 1

Accepted Answer

The answer of Find the limit algebraically.
-\[\lim _ { x...

Quiz 13: A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow - 0 } \frac { x ^ { 2 } } { \cos x }$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow \theta } \left( e ^ { x } - e ^ { - x } \right)$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow 2 } \left( x ^ { 2 } + 8 x - 2 \right)$

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _{x \rightarrow 4} f(x)$ $Use the graph shown to determine if the limit exists. If it does, find its value. - \lim _{x \rightarrow 4} f(x) A) 1 B) -1 C) 4 D) does not exist$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { d \rightarrow 3 } \left( \frac { d ^ { 3 } - 27 } { d - 3 } \right)$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow 0 } \frac { x + 2 } { 5 x + 2 }$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 3} f ( x ) , f ( x ) = 1 - x ^ { 2 }$ $Use the grid to graph the function. Find the limit, if it exists - \lim _ { x \rightarrow 3} f ( x ) , f ( x ) = 1 - x ^ { 2 } A) 1 B) -2 C) -8 D) does not exist$

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _{x \rightarrow-1} f(x)$ $Use the graph shown to determine if the limit exists. If it does, find its value. - \lim _{x \rightarrow-1} f(x) A) 1 B) 3 C) 0 D) does not exist$

Find the limit algebraically. - $\lim _ { x \rightarrow 1 } - 6$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 2 } f ( x ) , \quad f ( x ) = 4 x + 2$ $Use the grid to graph the function. Find the limit, if it exists - \lim _ { x \rightarrow 2 } f ( x ) , \quad f ( x ) = 4 x + 2 A) -2 B) -6 C) -8 D) does not exist$

Find the limit algebraically. - $\lim _ { x \rightarrow 2 } ( 2 x + 7 )$

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _{x \rightarrow 2} f(x)$ $Use the graph shown to determine if the limit exists. If it does, find its value. - \lim _{x \rightarrow 2} f(x) A) 0 B) 1 C) 2 D) does not exist$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow \pi / 2 } f ( x ) , f ( x ) = \sin x - 2$ $Use the grid to graph the function. Find the limit, if it exists - \lim _ { x \rightarrow \pi / 2 } f ( x ) , f ( x ) = \sin x - 2 A) -1 B) 3 C) -2 D) does not exist$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { h \rightarrow 3 } \left( \frac { h ^ { 2 } - 9 } { h ^ { 2 } + 3 h } \right)$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 8 } f ( x ) , \quad f ( x ) = 4 x - 1$ $Use the grid to graph the function. Find the limit, if it exists - \lim _ { x \rightarrow 8 } f ( x ) , \quad f ( x ) = 4 x - 1 A) 12 B) 11 C) 2 D) does not exist$

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _ { x \rightarrow 8 } f ( x )$ $Use the graph shown to determine if the limit exists. If it does, find its value. - \lim _ { x \rightarrow 8 } f ( x ) A) 4 B) 3 C) 5 D) does not exist$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 1 } f ( x ) , \quad f ( x ) = | 5 x |$ $Use the grid to graph the function. Find the limit, if it exists - \lim _ { x \rightarrow 1 } f ( x ) , \quad f ( x ) = | 5 x | A) -5 B) 1 C) 5 D) does not exist$

Use a graphing utility to find the indicated limit rounded to two decimal places. - $\lim _ { x \rightarrow - 1 } \frac { x ^ { 3 } - x ^ { 2 } + 3 x - 3 } { x ^ { 4 } - x ^ { 3 } + x - 1 }$

Find the limit algebraically. - $\lim _ { x \rightarrow 6 } x$

Quiz 13: A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Use the TABLE feature of a graphing utility to find the limit. - lim⁡x→−0x2cos⁡x\lim _ { x \rightarrow - 0 } \frac { x ^ { 2 } } { \cos x }limx→−0​cosxx2​

Use the TABLE feature of a graphing utility to find the limit. - lim⁡x→θ(ex−e−x) \lim _ { x \rightarrow \theta } \left( e ^ { x } - e ^ { - x } \right) limx→θ​(ex−e−x)

Use the TABLE feature of a graphing utility to 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)

Use the graph shown to determine if the limit exists. If it does, find its value. - lim⁡x→4f(x) \lim _{x \rightarrow 4} f(x) limx→4​f(x)

Use the TABLE feature of a graphing utility to find the limit. - lim⁡d→3(d3−27d−3) \lim _ { d \rightarrow 3 } \left( \frac { d ^ { 3 } - 27 } { d - 3 } \right) limd→3​(d−3d3−27​)

Use the TABLE feature of a graphing utility to find the limit. - lim⁡x→0x+25x+2\lim _ { x \rightarrow 0 } \frac { x + 2 } { 5 x + 2 }limx→0​5x+2x+2​

Use the grid to graph the function. Find the limit, if it exists - lim⁡x→3f(x) ,f(x) =1−x2\lim _ { x \rightarrow 3} f ( x ) , f ( x ) = 1 - x ^ { 2 }limx→3​f(x) ,f(x) =1−x2

Use the graph shown to determine if the limit exists. If it does, find its value. - lim⁡x→−1f(x) \lim _{x \rightarrow-1} f(x) limx→−1​f(x)

Find the limit algebraically. - lim⁡x→1−6\lim _ { x \rightarrow 1 } - 6limx→1​−6

Use the grid to graph the function. Find the limit, if it exists - lim⁡x→2f(x) ,f(x) =4x+2\lim _ { x \rightarrow 2 } f ( x ) , \quad f ( x ) = 4 x + 2limx→2​f(x) ,f(x) =4x+2

Find the limit algebraically. - lim⁡x→2(2x+7) \lim _ { x \rightarrow 2 } ( 2 x + 7 ) limx→2​(2x+7)

Use the graph shown to determine if the limit exists. If it does, find its value. - lim⁡x→2f(x) \lim _{x \rightarrow 2} f(x) limx→2​f(x)

Use the grid to graph the function. Find the limit, if it exists - lim⁡x→π/2f(x) ,f(x) =sin⁡x−2\lim _ { x \rightarrow \pi / 2 } f ( x ) , f ( x ) = \sin x - 2limx→π/2​f(x) ,f(x) =sinx−2

Use the TABLE feature of a graphing utility to find the limit. - lim⁡h→3(h2−9h2+3h) \lim _ { h \rightarrow 3 } \left( \frac { h ^ { 2 } - 9 } { h ^ { 2 } + 3 h } \right) limh→3​(h2+3hh2−9​)

Use the grid to graph the function. Find the limit, if it exists - lim⁡x→8f(x) ,f(x) =4x−1\lim _ { x \rightarrow 8 } f ( x ) , \quad f ( x ) = 4 x - 1limx→8​f(x) ,f(x) =4x−1

Use the graph shown to determine if the limit exists. If it does, find its value. - lim⁡x→8f(x) \lim _ { x \rightarrow 8 } f ( x ) limx→8​f(x)

Use the grid to graph the function. Find the limit, if it exists - lim⁡x→0f(x) ,f(x) ={x+1x<04x+1x≥0\lim _ { x \rightarrow 0 } f ( x ) , f ( x ) = \left\{ \begin{array} { r l } x + 1 & x < 0 \\4 x + 1 & x \geq 0\end{array} \right.limx→0​f(x) ,f(x) ={x+14x+1​x<0x≥0​

Use the grid to graph the function. Find the limit, if it exists - lim⁡x→1f(x) ,f(x) =∣5x∣\lim _ { x \rightarrow 1 } f ( x ) , \quad f ( x ) = | 5 x |limx→1​f(x) ,f(x) =∣5x∣

Use a graphing utility to find the indicated limit rounded to two decimal places. - lim⁡x→−1x3−x2+3x−3x4−x3+x−1\lim _ { x \rightarrow - 1 } \frac { x ^ { 3 } - x ^ { 2 } + 3 x - 3 } { x ^ { 4 } - x ^ { 3 } + x - 1 }limx→−1​x4−x3+x−1x3−x2+3x−3​

Find the limit algebraically. - lim⁡x→6x\lim _ { x \rightarrow 6 } xlimx→6​x

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow - 0 } \frac { x ^ { 2 } } { \cos x }$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow \theta } \left( e ^ { x } - e ^ { - x } \right)$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow 2 } \left( x ^ { 2 } + 8 x - 2 \right)$

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _{x \rightarrow 4} f(x)$ $Use the graph shown to determine if the limit exists. If it does, find its value. - \lim _{x \rightarrow 4} f(x) A) 1 B) -1 C) 4 D) does not exist$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { d \rightarrow 3 } \left( \frac { d ^ { 3 } - 27 } { d - 3 } \right)$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { x \rightarrow 0 } \frac { x + 2 } { 5 x + 2 }$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 3} f ( x ) , f ( x ) = 1 - x ^ { 2 }$ $Use the grid to graph the function. Find the limit, if it exists - \lim _ { x \rightarrow 3} f ( x ) , f ( x ) = 1 - x ^ { 2 } A) 1 B) -2 C) -8 D) does not exist$

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _{x \rightarrow-1} f(x)$ $Use the graph shown to determine if the limit exists. If it does, find its value. - \lim _{x \rightarrow-1} f(x) A) 1 B) 3 C) 0 D) does not exist$

Find the limit algebraically. - $\lim _ { x \rightarrow 1 } - 6$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 2 } f ( x ) , \quad f ( x ) = 4 x + 2$ $Use the grid to graph the function. Find the limit, if it exists - \lim _ { x \rightarrow 2 } f ( x ) , \quad f ( x ) = 4 x + 2 A) -2 B) -6 C) -8 D) does not exist$

Find the limit algebraically. - $\lim _ { x \rightarrow 2 } ( 2 x + 7 )$

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _{x \rightarrow 2} f(x)$ $Use the graph shown to determine if the limit exists. If it does, find its value. - \lim _{x \rightarrow 2} f(x) A) 0 B) 1 C) 2 D) does not exist$

Use the TABLE feature of a graphing utility to find the limit. - $\lim _ { h \rightarrow 3 } \left( \frac { h ^ { 2 } - 9 } { h ^ { 2 } + 3 h } \right)$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 8 } f ( x ) , \quad f ( x ) = 4 x - 1$ $Use the grid to graph the function. Find the limit, if it exists - \lim _ { x \rightarrow 8 } f ( x ) , \quad f ( x ) = 4 x - 1 A) 12 B) 11 C) 2 D) does not exist$

Use the graph shown to determine if the limit exists. If it does, find its value. - $\lim _ { x \rightarrow 8 } f ( x )$ $Use the graph shown to determine if the limit exists. If it does, find its value. - \lim _ { x \rightarrow 8 } f ( x ) A) 4 B) 3 C) 5 D) does not exist$

Use the grid to graph the function. Find the limit, if it exists - $\lim _ { x \rightarrow 1 } f ( x ) , \quad f ( x ) = | 5 x |$ $Use the grid to graph the function. Find the limit, if it exists - \lim _ { x \rightarrow 1 } f ( x ) , \quad f ( x ) = | 5 x | A) -5 B) 1 C) 5 D) does not exist$

Use a graphing utility to find the indicated limit rounded to two decimal places. - $\lim _ { x \rightarrow - 1 } \frac { x ^ { 3 } - x ^ { 2 } + 3 x - 3 } { x ^ { 4 } - x ^ { 3 } + x - 1 }$

Find the limit algebraically. - $\lim _ { x \rightarrow 6 } x$