exercise 4.2.8. compute each limit or state that it does not exist. use the tools developed in this section to justify each conclusion. (a) limx→2 |x−2| x−2 (b) limx→7/4 |x−2| x−2 (c) limx→0(−1)[[1/x]] (d) limx→0 3√x(−1)[[1/x]]
Answers
(a) The limit limx→2 |x−2| / (x−2) does not exist. (b)The right-hand limit as x approaches 7/4 is 1. (c)the limit limx→0 (-1)[[1/x]] does not exist. (d) the limit limx→0 3√x(-1)[[1/x]] equals 0.
(a) To compute the limit limx→2 |x−2| / (x−2), we need to consider the left-hand limit and the right-hand limit separately. For x < 2, |x−2| / (x−2) = -1, as x−2 is negative.
Therefore, the left-hand limit as x approaches 2 is -1. For x > 2, |x−2| / (x−2) = 1, as x−2 is positive.
Therefore, the right-hand limit as x approaches 2 is 1.
Since the left-hand limit (-1) and the right-hand limit (1) are not equal, the limit limx→2 |x−2| / (x−2) does not exist.
(b) Similarly, for limx→7/4 |x−2| / (x−2), we evaluate the left-hand and right-hand limits. For x < 7/4, |x−2| / (x−2) = -1, as x−2 is negative. So, the left-hand limit as x approaches 7/4 is -1.
For x > 7/4, |x−2| / (x−2) = 1, as x−2 is positive. Thus, the right-hand limit as x approaches 7/4 is 1.
Since the left-hand limit (-1) and the right-hand limit (1) are not equal, the limit limx→7/4 |x−2| / (x−2) does not exist.
(c) For limx→0 (-1)[[1/x]], the expression [[1/x]] represents the greatest integer less than or equal to 1/x.
Since 1/x approaches infinity as x approaches 0 from both sides, [[1/x]] equals 0 for x > 0 and -1 for x < 0.
Therefore, the left-hand limit and right-hand limit are different, and the limit limx→0 (-1)[[1/x]] does not exist.
(d) In limx→0 3√x(-1)[[1/x]], [[1/x]] equals 0 for x > 0 and -1 for x < 0. As x approaches 0 from both sides, √x approaches 0.
Hence, the limit limx→0 3√x(-1)[[1/x]] equals 0.
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Related Questions
In ABC, find the value of x.
Answers
Answer:
x =62
Step-by-step explanation:
\((x+1)^{0} +55^{0} +x^{0} =180^{0}\)
\(x+1+55+x=180\)
\(2x+56=180\)
\(2x=180-56\)
\(2x=124\)
\(x=124/2\)
\(x=62\)
Hope this helps
the dimensions of a rectangular crate are 4 feet 6 feet and 8 feet. what is the volume?
Answers
Answer:
192
Step-by-step explanation:
Volume=l×w×h
V=4×6×8
V=192
As part of a science experiment, a student observes the growth of a population of bacteria in four different media. The table below lists the
observations the student made.
1 — Population increases by 10 every hour
11 — Population decreases by 10 every hour
111 — Population increases by 10% every hour
1111-Population decreases by 10% every hour
In which media can the change in the population
of bacteria be modeled by a linear function?
Answers
Answer:
B
Step-by-step explanation:
Answer:A
Step-by-step explanation:it says linear so it has to increase the same every time so It can’t be very percentages, so A
En el mercado del satisfactor X hay 10,000 individuos idénticos, cada uno con una función de demanda definida por Qdx = 12 – Px y 1,000 productores idénticos del satisfactor X, cada uno con una función de oferta dada por Qsx = 20 Px
Answers
Answer:
Px = $4
Step-by-step explanation:
Given that:
En el mercado de satisfacción X hay 10,000 individuos idénticos, cada uno con una función de demanda definida por Qdx = 12 - Px y 1,000 productores idénticos de X satisfactorio, cada uno con una función de oferta dada por Qsx = 20 Px
El precio de equilibrio y la cantidad de equilibrio se pueden determinar de la siguiente manera;
Qdx = 10000(12-Px)
Qdx = 120000 - 10000Px
Qsx = 1000(20 Px)
Qsx = 20000 Px
El punto de equilibrio para completar el enunciado, cuando Qdx = Qsx es:
Qdx = Qsx
120000 - 10000 Px = 20000 Px
120000 = 20000 Px + 10000 Px
120000 = 30000 Px
Px = \(\mathbf{ \dfrac{120000}{30000}}\)
Px = $4
Problems 14 – 19 Translate from English into a math equation and then solve for X Answer the questions with a fraction or a mixed number. All fractions must be in lowest terms. 14. One eighth divided by one third times 125 equals X 15. Two and one third multiplied by five and one sixth equals X 16. One and a half plus two and a fifth equals X 17. four and one eighth minus two and three fourths equals X 18. Turn the decimal 0.004 into a fraction in lowest terms. INI 19. Turn the ratio 34:68 into a fraction in lowest terms
Answers
Let's solve the equations one by one:
One eighth divided by one third times 125 equals X:
Math equation: (1/8) ÷ (1/3) × 125 = X
Solution: (1/8) ÷ (1/3) × 125 = (1/8) × (3/1) × 125 = (3/8) × 125 = 375/8 = 46 7/8
Therefore, X = 46 7/8
Two and one third multiplied by five and one sixth equals X:
Math equation: (2 1/3) × (5 1/6) = X
Solution: (2 1/3) × (5 1/6) = (7/3) × (31/6) = 217/18 = 12 1/18
Therefore, X = 12 1/18
One and a half plus two and two-fifths equals X:
Math equation: (1 1/2) + (2 2/5) = X
Solution: (1 1/2) + (2 2/5) = (3/2) + (12/5) = (15/10) + (24/10) = 39/10 = 3 9/10
Therefore, X = 3 9/10
Four and one eighth minus two and three fourths equals X:
Math equation: (4 1/8) - (2 3/4) = X
Solution: (4 1/8) - (2 3/4) = (33/8) - (11/4) = (33/8) - (22/8) = 11/8 = 1 3/8
Therefore, X = 1 3/8
Turn the decimal 0.004 into a fraction in lowest terms:
To convert a decimal to a fraction, we can place the decimal value over a power of 10.
0.004 = 4/1000
To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 4.
4/1000 = 1/250
Therefore, 0.004 is equivalent to 1/250 in lowest terms.
Turn the ratio 34:68 into a fraction in lowest terms:
To convert a ratio to a fraction, we can simply place the first number as the numerator and the second number as the denominator.
34/68
To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 34.
34/68 = 1/2
Therefore, the ratio 34:68 is equivalent to 1/2 in lowest terms.
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Solve for x if AC =33
Answers
Answer:
6.2
Step-by-step explanation:
can i hae brailiezt plzz
on the first day it was posted online, a music video got 800 views. the number of views that the video got each day increased by 5% per day. how many total views did the video get over the course of the first 17 days, to the nearest whole number?
Answers
The total number of views the video got over the course of the first 17 days is approximately 17,536.
To solve this problem, we can use the formula for the sum of a geometric sequence, which is given by:
Sn = a(1 - r^n) / (1 - r)
where
Sn is the sum of the first n terms
a is the first term
r is the common ratio between the terms
In this case, a = 800 (the first day's views), r = 1.05 (the 5% increase in views per day), and n = 17 (the number of days).
Substituting these values into the formula, we get:
Sn = 800(1 - 1.05^17) / (1 - 1.05) ≈ 17,536
Therefore, the video got approximately 17,536 views over the course of the first 17 days.
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If we play a game where I start with 2 dollars and you start with one dollar, and I have a probability of 1/3 of winning a dollar from you and you have a 1/3 probability of winning a dollar from me, what is the probability that I win all of your money
Answers
Answer:
1/3 or 0.33
Step-by-step explanation:
Hope it helps
LMK if it does and weather it is wrong or right
If 45 out of 1,000 babies are born with a particular dominant trait, what is the frequency of the recessive allele
Answers
Answer:
0.976 or 97.6%
Step-by-step explanation:
To calculate the frequency of the recessive allele, we need to use the information provided about the frequency of the dominant trait.
Let's assume that the particular dominant trait is determined by a single gene with two alleles: the dominant allele (A) and the recessive allele (a).
Given that 45 out of 1,000 babies are born with the dominant trait, we can infer that the remaining babies (1,000 - 45 = 955) do not have the dominant trait and can be considered as the recessive trait carriers.
The frequency of the recessive allele (q) can be calculated using the Hardy-Weinberg equation:
q = sqrt((Recessive individuals) / (Total individuals))
In this case, the total number of individuals is 1,000, and the number of recessive individuals is 955.
q = sqrt(955 / 1,000)
Using a calculator, we can find the value:
q ≈ 0.976
Therefore, the frequency of the recessive allele is approximately 0.976 or 97.6%.
The zeros of the quadratic x^2+4x=-2x+16 are x=-8 and x=2. What does this tell you about the graph of this quadratic function?
a. The parabola crosses the x-axis at x=-8 and x=2.
b. The parabola touches the x-axis only at and x=2.
c. The parabola does not touch or cross the x-axis.
d. The parabola crosses the x-axis at x=-8 and x=2 but you can ignore the zero x=-8 because it is negative.
Answers
The graph of the quadratic function x²+4x=-2x+16 is a parabola that crosses the x-axis at x=-8 and x=2. Correct option is (a).
What are zeros and graph of an quadratic equation?Let ax² +bx +c = 0 is an quadratic equation its zeroes are the values of x for which the Left hand side becomes zero.
To find zeroes we can use quadratic formula that is
x = (-b ±√(b²-4ac))/2
Graph of the quadratic equation are points ( x , y(x) )
where y(x) = ax² +bx +c
when x= t is zero of quadratic equation ax² +bx +c = 0
then y(t) = 0
(t, 0) points are points on the x-axis.
Given an quadratic equation
x²+4x=-2x+16 has roots x =-8 and x =2
This can we written as x² +6x -16 =0
The graph of this quadratic equation is given by (x, y(x))
where y = x² +6x -16 which is a parabola
for x= -8 and x= 2 , y(-8) =y(2) = 0
Then (-8,0) and (2,0) are points on this parabola which are at x-axis
Therefore, the parabola y = x² +6x -16 crosses the x-axis at x=-8 and x=2.Thus (a) is the correct option.
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Solve for
�
cc.
Give an exact answer.
0.2
(
10
−
5
�
)
=
5
�
−
16
0.2(10−5c)=5c−16
Answers
The solution to the equation 0.2(10 - 5c) = 5c - 16 is c = 3.
To solve the equation 0.2(10 - 5c) = 5c - 16, we will first distribute the 0.2 on the left side of the equation:
0.2 * 10 - 0.2 * 5c = 5c - 16
Simplifying further:
2 - 1c = 5c - 16
Next, we will group the variables on one side and the constants on the other side by adding c to both sides:
2 - 1c + c = 5c + c - 16
Simplifying:
2 = 6c - 16
To isolate the variable term, we will add 16 to both sides:
2 + 16 = 6c - 16 + 16
Simplifying:
18 = 6c
Finally, we will divide both sides by 6 to solve for c:
18/6 = 6c/6
Simplifying:
3 = c
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Rai compares the fee structure for two different art classes. which equation can be used to solve for the number of classes, c, for which class a and class b cost the same?
Answers
Answer:The answer is 40c=25c+60
Step-by-step explanation:
(c) explain what would happen to the length of the interval if the confidence level were... (i) increased to 99% (ii) decreased to 80%
Answers
If the confidence level were increased to 99%, the width of the confidence interval increases, when decreased to 80%, the width of the confidence interval decreases.
What is confidence interval?A confidence interval provides a range of estimated values for a population parameter's real value. A confidence level is connected to confidence intervals. The width of the confidence interval grows as the confidence level increases. The width of the confidence interval reduces as the confidence level goes down.
Since the margin of error grows as the confidence level does as well, the confidence interval will be bigger. Hence it increases for 99% and decreases for 80%.
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Each day, the cafeteria serves more than 200 apples with lunch. Write an inequality that represents this situation. Let a represent the number of apples.
Answers
Here is the answer
a > 200
The expression for the inequality is written as a > 200.
What is inequality?When two expressions are connected by a sign like "not equal to," "greater than," or "less than," it is said to be inequitable. The inequality shows the greater than and less than relationship between variables and the numbers.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that each day, the cafeteria serves more than 200 apples with lunch.
The inequality will be written as,
a > 200
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Please answer this in two minutes
Answers
Answer:
\( s = 14.1 \)
Step-by-step explanation:
Given:
<S = 31°
SQ = r = 9
SR = q = 21
s = ?
To find s, use the Law of Cosines:
s² = r² + q² - 2rs*cos(S)
\( s^2 = 9^2 + 21^2 - 2*9*21*cos(31) \)
\( s^2 = 81 + 441 - 378*0.8572 \)
\( s^2 = 522 - 324.02 \)
\( s^2 = 197.98 \)
\( s = 14.1 \) (nearest tenth)
Use >, <, or = to compare the expressions.
Answers
Answer:
-(∛2)³ = (∛-2)³
(-√2)² > -(√2)²
2³ > -2³
-2² < (-2)²
∛64 = √16
I hope this is the answer you're looking for. this took me a while so brainliest plz...
Answer:
=
>
>
<
=
Step-by-step explanation:
evaluate 3|-3|
a. -6
b. 6
c. 9
d. -9
Answers
Answer:is that 3i or what?
-
Step-by-step explanation
Answer:
It's answer is d) 9
.........
A rectangular storage container without a lid is to have a volume of 10 m3. the length of its base is twice the width. material for the base costs $15 per square meter. material for the sides costs $9 per square meter. let w denote the width of the base. Find a function in the variable w giving the cost C in dollars) of constructing the box.
Answers
The function in variable w giving the cost C (in dollars) of constructing the box is C(w) = 30w² + 270/w. The result is obtained by using the formula of volume and area of the box.
How to determine the function?We have a rectangular storage container without a lid.
Volume, V = 10 m³Length, l = 2wWidth, w = wBase costs $15/m²Sides costs $9/m²The formula of volume of the box is
V = l × w × h
Where
l = lengthw = widthh = heightSo, the height is
10 = 2w × w × h
10 = 2w² × h
h = 10/2w²
h = 5/w²
To find the total cost, calculate the area of base and sides of the box!
See the picture in the attachment!
The base area is
A₁ = 2w × w = 2w² m²
The sides area is
A₂ = 2(2wh + wh)
A₂ = 2(3wh)
A₂ = 6wh
A₂ = 6w(5/w²)
A₂ = 30/w m²
The total cost is
C = $15(2w²) + $9(30/w)
C = $30w² + $270/w
The function of the total cost is
C(w) = 30w² + 270/w
Hence, the function of constructing the box is C(w) = 30w² + 270/w.
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multiplying rational numbers (-7) (-2) SHOW WORK PLEASE
Answers
Suppose you have 15 months in which to save $1800 for a vacation cruise. If you can earn an APR of 3. 7%, compounded monthly, how much should you deposit each month. (Hint: use monthly payment formula) 4. Calculate the monthly payments for a shack mortgage of $127,000 with a fixed APR of 9. 1% for 30 years
Answers
For the first problem, using the monthly payment formula monthly deposit needed is $118.69. For the second problem, using the same formula the monthly payment is $1029.73.
We have the following variables
P = Monthly payment
r = Annual interest rate = 3.7% = 0.037/12 per month
n = Number of payments = 15 months
A = Amount to be saved = $1800
Using the monthly payment formula
P = (r * A) / (1 - (1 + r)⁻ⁿ)
Substituting the given values
P = (0.003083 * 1800) / (1 - (1 + 0.003083)⁻¹⁵)
P ≈ $118.69
Therefore, you should deposit approximately $118.69 each month to save $1800 in 15 months, assuming an APR of 3.7%, compounded monthly.
We have the following variables
P = Monthly payment
r = Annual interest rate = 9.1% = 0.091/12 per month
n = Number of payments = 30 years * 12 months = 360 months
A = Mortgage amount = $127,000
Using the monthly payment formula
P = (r * A) / (1 - (1 + r)⁻ⁿ)
Substituting the given values
P = (0.007583 * 127000) / (1 - (1 + 0.007583)⁻³⁶⁰)
P ≈ $1029.73
Therefore, the monthly payment for a shack mortgage of $127,000 with a fixed APR of 9.1% for 30 years would be approximately $1029.73.
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which of the following statements is true of pie charts? group of answer choices they are ineffective in showing quantitative totals. they are ineffective in showing percentages. they present data in columns and rows. they make it easier to understand processes.
Answers
Pie charts are effective in conveying percentages and are commonly used in data analysis and presentations.
The true statement about pie charts is that they are effective in showing percentages. Pie charts are a common data visualization tool used to represent proportions or percentages of a whole.
The chart is divided into slices, with each slice representing a specific category or data point. The size of each slice is proportional to the percentage it represents within the whole.
This visual representation makes it easier for viewers to understand the distribution and relative proportions of different categories or variables. Pie charts are therefore frequently used in data analysis and presentations because they are good at communicating percentages.
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Question 15 of 26
AABC has side lengths that measure 10 centimeters, 10 centimeters, and 15
centimeters. Which of the following best describes this type of triangle?
Answers
Answer:
isosceles
Step-by-step explanation:
▪︎ Isosceles triangle is a triangle that has two sides of equal length.
a factory worker productivity is normally distributed. one worker produces an average of 75 units per day with a standard deviation of 20. another worker produces at an average rate of 65 per day with a standard deviation of 21. what is the probability that in 1 week (5 working days), worker 1 will outproduce worker 2
Answers
By using the concept of Probability, 0.771 is the probability by which worker 1 will outproduce worker 2
Let Xi be the random variable representing the number of units the first worker produces in day i.
Define X = X₁+ X₂ + X₃ + X₄ + X₅ as the random variable representing the number of units the first worker produces during the entire week.
We know that Mean =(Sum of all quantities)/(Number of quantities)
Mean=75 and number of quantities =5(given)
Therefore, from the formula
=>Sum of all quantities=75×5
or We can Say that X₁+ X₂ + X₃ + X₄ + X₅=375--------------------------------(eq1)
Now, talking about the standard deviation of first worker
We know that standard deviation = \(\frac{\sqrt{(Each quantity - Mean)^{2} } }{\sqrt{Total Number of quantities} }\)
We are given standard deviation of first worker as 20,
Therefore 20×\(\sqrt{Total Number of quantities}\) = \(\sqrt{(Eachquantity -Mean)^{2} }\)
20√5 = √[(X₁ - Mean)²+(X₂ - Mean)²+(X₃ - Mean)²+(X₄ - Mean)²+(X₅ - Mean)²]-(eq2)
Therefore, from eq1 and eq2,
we get Mean(µx) =375 and standard deviation(σ\(x\)) =20√5
Similarly, define random variables Y₁, Y₂, . . . , Y₅ representing the number of units produces by the second worker during each of the five days and define Y = Y₁ + Y₂ + Y₃ + Y₄ + Y₅.
From the Mean formula,
we get Y₁ + Y₂ + Y₃ + Y₄ + Y₅=(65×5)--------------------------------(eq3)
Standard deviation of second worker = 21(given),
So using the standard deviation formula, we get
Therefore 21×\(\sqrt{Total Number of quantities}\)=\(\sqrt{(Eachquantity -Mean)^{2} }\)
21√5=√[(Y₁ - Mean)²+(Y₂ - Mean)²+(Y₃ - Mean)²+(Y₄ - Mean)²+(Y₅ - Mean)²]-(eq4)
Therefore, from eq3 and eq4,
we get Mean(µy) =325 and standard deviation(σy) =21√5
Of course, we assume that X and Y are independent. The problem asks for P(X > Y ) or in other words for P(X − Y > 0).
It is a quite surprising fact that the random variable U = X −Y , the difference between X and Y ,is also normally distributed with mean µU = µx−µy = 375−325 = 50 and standard deviation σu ,where σ\(u^{2}\) = σ\(x^{2}\)+σ\(y^{2}\) =400·5+441·5 = 841·5 = 4205
It follows that σu=√4205.
Now probability of first worker(P₁)=375/√4205
probability of second worker(P₂) =325/√4205
We can clearly see P₁>P₂
Difference in Probability of both workers(P)=P₁-P₂
=>P=[(375/√4205)-(325/√4205)]
=>P=50/√4205
=>P=50/64.84
=>P=0.771
Hence, probability by which worker 1 will outproduce worker 2 is 0.771.
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5x = 10 can someone help me
Answers
Answer:
X=2
Step-by-step explanation: To get a single
x
, divide
5
x
by
5
.
Whatever you do to one side of the equals sign, you must do to the other side. This means, to get the right answer, you must do
5
x
5
=
10
5
The
5
's on the left side of the equals sign cancel out so it would make
5x5=105
and then
x=105105
is equal to
2
so...
x=2
Answer: Hey user! x = 2.
Step-by-step explanation: Step 1: Divide both sides by 5.
5x /5 = 10 /5
x=2
Consider marking this answer as brainliest if it helped you out.
the sum of 4 and the second power of b is equal to 104. write as an equation
Answers
Answer:
4 + b² = 104
Hope this helps. :)
Answer:
b^{2} + 4 = 104.
Step-by-step explanation:
The sum of 4 and the second power of b basically means 4 + b^2.
b^2 + 4 = 104
\(b^{2} + 4 = 104\)
b^2 = 100
b = plus or minus 10.
Hope this helps!
answer this in 40 seconds for brainiest:)
Answers
Answer:
15
Step-by-step explanation: multiply 3x5 lol
Will make as the brainliest
you are using a taxi cab company that charges a flat fee of $5.00 and an additional $0.50 per mile. Let x represent the number of miles you drive in the taxi and y represent the amount you paid for the ride. Write an equation that can be used to find the amount of money you pay for the taxi ride.
a- standard form
b-slope intercept form
Answers
Answer:
In standard form: x - 2y = -10
In slope intercept form: y = (1/2)x + 5
Step-by-step explanation:
y = total cost
y = flat cost + mileage cost
The flat cost is $5, so you get
y = 5 + mileage cost
The mileage cost depends on the number of miles, x.
1 mile costs $0.50
2 miles cost $0.50 * 2
3 miles cost $0.50 * 3
x miles cost $0.50 * x, or simply 0.5x
Now we add the mileage cost to our expression.
y = 5 + 0.5x
y = 0.5x + 5
In standard form:
0.5x - y = -5
x - 2y = -10
In slope intercept form:
y = (1/2)x + 5
THE AREA OF A SECTOR IS A = 384 in².
A. IF THE RADIUS IS 24 IN. , WHAT IS THE MEASURE OF THE CENTRAL ANGLE, IN RADIANS?
B. IF THE CENTRAL ANGLE MEASURES RADIANS, WHAT IS THE RADIUS OF THE CIRCLE? ROUND TO THE NEAREST HUNDREDTH.
Answers
A) The measure of the central angle is \(\frac{4}{3} \pi\) .
B) The radius of the circle is 55.43 in.
A) We know,
The area of a sector is \(\frac{n}{2\pi } \pi R^{2}\)
where n = the degree of the central angle of the sector
R = radius of the circle
Suppose, the measure of the central angle is x.
\(\frac{x}{2\pi } .\pi .24^{2} = 384\pi\)
⇒ \(\frac{x}{2\pi } = \frac{384\pi }{576\pi }\)
⇒ \(x = \frac{384\pi }{576\pi } * 2\pi\)
⇒ \(x = \frac{4}{3} \pi\)
So, the measure of the central angle is \(\frac{4}{3} \pi\).
B) Suppose the radius of the circle is r in.
\(\frac{\pi /4}{2\pi } .\pi .r^{2} =384\pi\)
⇒\(\frac{1}{8} .\pi .r^{2} =384\pi\)
⇒ \(r^{2} = 384\pi / \frac{1}{8} \pi\)
⇒ \(r^{2} = 3072\)
As r > 0
So, r = 55.43
Hence, The radius of the circle is 55.43 in.
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IM NOT SURE I UNDERSTAND HELP
Answers
Answer:
Mixed Numbers
Step-by-step explanation:
It would be this because they contain decimals.
Answer: Irrational numbers
Step-by-step explanation: Not integer or whole numbers. They have e the symbols of irrational numbers, they are not simple fraction either
Solve the system of equations.
Answers
Answer:
x = 2, y = 1
Step-by-step explanation:
x + 3y = 5
3y = -x + 5
y = -1/3x + 5/3
-x + 6y = 4
-x + 6(-1/3x + 5/3) = 4
-x - 2x + 10 = 4
-3x = -6
x = 2
x + 3y = 5
2 + 3y = 5
3y = 3
y = 1