Show (3 steps) that the function is either continuous or discontinuous at the given value of x

1. f(x) = 1/x at x = 3
2. f(x) = |x+1|/x at x = 2
3. f(x)= [x] at x = 2

Answers

Answer 1

Using the continuity concept, the functions are classified as follows:

1) Continuous.

2) Continuous.

3) Discontinuous.

What is the continuity concept?

A function f(x) is continuous at x = a if it is defined at x = a, and:

[tex]\lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = f(a)[/tex]

For functions 1 and 2, the functions are given by simple definitions, hence they are continuous, the limits and the numeric value will al be equal.

In item 3, the function will be discontinuous. This is because the function [x] returns the greatest integer less than x, hence:

[1.999999999999] = 1.[2.000000000001] = 2.

Thus:

[tex]\lim_{x \rightarrow 2^-} f(x) = 1[/tex][tex]\lim_{x \rightarrow 2^+} f(x) = 2[/tex]

Due to different lateral limits, the function is not continuous at x = 2.

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Related Questions

Whats the constant term for y=3x + 5


1. 3
2. 5

Answers

i answered this one already on your other post for it

3 is the constant/slope

5 is the y intercept

The system of equations graphed below has how many solutions?
y = 2x + 2
y= 2x

Answers

The given system has zero solutions.

Lines that never cross one other are said to be parallel. Hence, a pair of parallel lines must have the same slope but distinct intercepts. (On the other hand, identical lines have same intercept).

In the general equation of a line, y = mx + b, m represents the slope of the line.

The general equations of parallel lines would be of the form:

(1) y = mx + b

(2) y = mx + c

where b and c are any constants.

Now, we observe that the given equations are of the above form with m=2, b=2 and c=0, therefore, they are parallel lines. Since, parallel lines never intersect each other, they have no solution.

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Find the rate of change of f(x) = x² - 2x + 3 on [-1,2].

Answers

Answer:

-1.

Step-by-step explanation:

Rate of change = [ f(2) - f(-1)] / [2 - (-1)]

                          = [(2)^2 - 2(2) + 3)  - ((-1)^2  - 2(-1) + 3] / 3

                           = ( 3 - 6)/3

                            = -1.

Kathi and Robert Hawn had a pottery stand at the annual Skippack Craft Fair. They sold some of their pottery at the original price of $10.50 ​each, but later decreased the price of each by ​$2.00. If they sold all 98 pieces and took in $893.00

find how many they sold at the original price and how many they sold at the reduced price.

Answers

Answer:

See work below.

30 sold for full price and 68 sold for a discount.

Step-by-step explanation:

a box of cereal has a volume of 1250 cubic cm with a length of 20cm and a width of 5cm. what is the height of the box?

Define in words variables for the unknown values
create an equation
solve using algebra

Answers

The height of the box that has a volume of 1250 cubic cm with a length of 20cm and a width of 5cm will be 12.5cm.

How to calculate the height?

It should be noted that the formula for the volume will be:

= Length × Width × Height

Therefore the height will be:

Volume = LWH

where

L = length = 20cm

W = width = 5cm

H = height = unknown

1250 = (20 × 5 × H)

1250 = 100H

Height = 1250/100

Height = 12.5 cm

Therefore, the height is 12.5cm.

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Use the diagram to find the value of the following. what is the value of x.
Angle A-
Angle B -
Angle C-

Answers

Answer:

x=29 angle a=63 Ang b=46 and c=71

Step-by-step explanation:

2x+5+x+17+3x-16=180 (sum of angles in triangles)

value of x is 29 substitute it in angle a b c

a=2*29+5

=63

b=29+17 = 46

c=3*29-16 = 71

i really need help! please

Answers

Solution: A

Explanation:

Factor the top and bottom

(x-2)(x-2) / (x+2)(x-2)

Cancel one (x-2) to find the “hole” or removable discontinuity.

Set equal to 0

x-2=0
x=2

Find the values of x and y. Write all radicals in simplest form.

Answers

Answer:

For the 60-30-90 degree triangle:

x=12

y=[tex]12\sqrt{3}[/tex]

For the 30-60-90 degree triangle:

x=16

y=[tex]8\sqrt{3}[/tex]

For the 45-45-90 degree triangle:

x=7

y=[tex]7\sqrt{2}[/tex]

Step-by-step explanation:

For 30-60-90 triangles, the side opposite the 30 degree angle is x, while the side opposite the hypotenuse is 2x, while the side opposite the 60 degree angle is [tex]x\sqrt{3}[/tex].

For 45-45-90 triangles, the sides opposite the 45 degree angles are x, while the hypotenuse is [tex]x\sqrt{2}[/tex].

Find the unit tangent vector for the following parameterized curves.
r(t)=cos(t)i+sin(t)j+sin(t)k, 0≤t<2π.

Answers

The unit tangent vector for the parametrized curve is [tex]\vec T(t) = - \frac{\sin t}{\sqrt{1 + \cos ^{2} t}}\,\hat{i} + \frac{\cos t}{\sqrt{1 + \cos ^{2} t}}\,\hat{j} + \frac{\cos t}{\sqrt{1 + \cos ^{2} t}}\,\hat{k}[/tex].

How to determine the unit tangent vector of a parametrized curve

In this question we know a three-dimension parametrized curve, of which we must derive its unit tangent vector in accordance with the following formula:

[tex]\vec T(t) = \frac{\vec R'(t)}{\|\vec R'(t)\|}[/tex]

Where:

[tex]\vec T(t)[/tex] - Tangent vector.[tex]\|\vec R'(t)\|[/tex] - Magnitude of the first derivative of the parametrized curve.

First, determine the first derivative of the parametrized curve:

[tex]\vec R'(t) = -\sin t \,\hat{i} + \cos t \,\hat{j} + \cos t \,\hat{k}[/tex]

Second, derive the magnitude of the parametrized curve:

[tex]\|\vec R'(t)\|[/tex] = √[(- sin t)² + (cos t)² + (cos t)²]

[tex]\|\vec R'(t)\|[/tex] = √(1 + cos² t)

Third, substitute every term in the unit tangent curve formula:

[tex]\vec T(t) = - \frac{\sin t}{\sqrt{1 + \cos ^{2} t}}\,\hat{i} + \frac{\cos t}{\sqrt{1 + \cos ^{2} t}}\,\hat{j} + \frac{\cos t}{\sqrt{1 + \cos ^{2} t}}\,\hat{k}[/tex]

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solve the equation for x (3x/4)+2=4x-1

Answers

Answer:

Step-by-step explanation:

The blueprint specifications for a machined part calls for its thickness to be 3.145 in.
with a tolerance of +-0.010 in. Find the limit dimensions of the part?

Answers

The limit dimensions of the part are 3.135 inches and 3.155 inches

How to find the limit dimensions of the part?

The given parameters are

Thickness = 3.145 inchesTolerance = 0.010 inches

The limit dimensions of the part are calculated as

Limit = Thickness +/- Tolerance

So, we have

Limit = 3.145 inches +/- 0.010 inches

Expand the above expression

So, we have

Limit = (3.145 inches - 0.010 inches, 3.145 inches + 0.010 inches)

Evaluate the sum

Limit = (3.135 inches, 3.155 inches)

Hence, the limit dimensions of the part are 3.135 inches and 3.155 inches

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solve the equation by performing two operations on both sides. State the operation in order of use

Answers

Answer:

  C.  Multiply by 5, then subtract 7.

Step-by-step explanation:

You want to know the two steps required to solve this 2-step linear equation.

Solution

The variable x has 7 added to it, and the sum is divided by 5. To find the value of x, we must undo these operations in reverse order.

To undo division by 5, we must multiply by 5.

Then, to undo the addition of 7, we must subtract 7.

Here is the "work":

  [tex]\dfrac{x+7}{5}=10\qquad\text{given}\\\\x+7 = 50\qquad\text{multiply both sides by 5}\\\\x=43\qquad\text{subtract 7 from both sides}[/tex]

The operations in order of use are ...

  Multiply both sides by 5 first, then subtract 7 from both sides.

__

Additional comment

The properties of equality tell you that you must do the same operation to both sides of the equation.

3y^3+20y^2=7y

Solve the polynomial by factoring. The answer is y(3y-1)(y+7)
I can't seem to figure out how they got the answer. I need an explanation before my test tomorrow.

Answers

Answer:

Hello,

Step-by-step explanation:

[tex]3y^3+20y^2-7y=y(3y^2+20y-7)\\\\=y(3y^2+21y-y-7)\\\\=y(3y(y+7)-(y+7))\\\\=y(y+7)(3y-1)\\[/tex]

QUESTION 94
Percentages
In January 2021, a study by the Bureau of Transportation statistics found that 12 out of 35 flights arrived
on time. What percent of the flights were on time?
Choose one 4 points
O 72%
O 34.3%
O 34%
O 27%

Answers

Answer: 34.3%

Step-by-step explanation:

[tex]Given:\ 12/35\ flights\ on\ time[/tex]

To find the percent, divide then multiply by 100:

[tex]\frac{12}{35} = 0.343\times100=\large\boxed{34.3}[/tex]

Professor Cramer determines a final grade based on attendance, two papers, three major tests, and a final exam. Each of these activities has a total of 100 possible points. However, the activities carry different weights. Attendance is worth 7%, each paper is worth 9%, each test is worth 14%, and the final is worth 33%, (a) What is the average for a student with 64 on attendance, 82 on the first paper, 61 on the second paper, 72 on test 1, 64 on test 2, 93 on test 3, and 82 on the final exam? (Round your answer to one decimal place.) (b) Compute the average for a student with the above scores on the papers, tests, and final exam, but with a score of only 29 on attendance. (Round your answer to one decimal place.)​

Answers

The  average for a student with 64 on attendance, 82 on the first paper, 61 on the second paper, 72 on test 1, 64 on test 2, 93 on test 3, and 82 on the final exam is 77.31% and the average for a student with the above scores on the papers, tests, and final exam, but with a score of only 29 on attendance is 73.23%.

Average

a. Average:

Average: 82(0.07)+ 61(0.09)+ 72(0.14) +64(0.14) + 93(0.14)+ 82(0.33)÷0.07+0.09+3(0.14)+0.33

Average=5.74+5.49+10.08+8.96+13.02+27.06÷0.91

Average=70.35÷0.91

Average=77.31%

b. Average

Average= 29(0.07)+ 61(0.09)+ 72(0.14) +64(0.14) + 93(0.14)+ 82(0.33)÷0.07+0.09+3(0.14)+0.33

Average=2.03+5.49+10.08+8.96+13.02+27.06÷0.91

Average=66.64÷0.91

Average=73.23%

Therefore the average for both a and b is 77.31% and 73.23%.

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Please answer this screenshot please 10 points good idea yes.

Answers

Answer:

Yes it is a function

Step-by-step explanation:

There is a relationship between x and y. It is not linear but quadratic

Answer:

yes it is a function

Step-by-step explanation:

that is the answer bc I said I was

Urgent (I added 50 points)
The diagram below is drawn on a one-centimetre square grid


Work out the area of the shaded triangle:

Answers

The area of the shaded triangle will be 11.5 square centimeters.

What is the area of the shaded region?

It deals with the size of geometry, region, and density of the different forms both 2D and 3D.

The area of the square will be given as,

A₁ = 5 x 5

A₁ = 25

The area of the triangles will be given as,

A₂ = 1/2 (1 x 5 + 4 x 3 + 2 x 5)

A₂ = 1/2 (27)

A₂ = 27/2

Then the area of the shaded triangle will be given as,

A = A₁ - A₂

A = 25 - 27/2

A = (50 - 27) / 2

A = 23/2

A = 11.5 square cm

The area of the shaded triangle will be 11.5 square centimeters.

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Answer:

In the graph, excluding the shaded triangle, there are three right angled triangles surrounding the shaded triangle in a square

Area of shaded triangle = area of square - area of three triangles

[tex]{ \sf{area = (5 \times 5) - ( \frac{1}{2} \times 4 \times 3) - ( \frac{1}{2 } \times 5 \times 2) - ( \frac{1}{2} \times 5 \times 1)}} \\ \\ { \sf{area = 25 - 6 - 5 - 2.5}} \\ \\ { \sf{area = 11.5 \: {cm}^{2} }}[/tex]

What is an equation of the line that passes through the points (3, 6) and (-1, -6)?

Answers

point slope intercept form:y-6=3(x-3)
slope intercept form: y= 3x-3

A dressmaker needs to cut 18-inch pieces of ribbon from rolls of ribbon that are 3 feet in length. How many 18-inch pieces can the dressmaker cut from 15 of these rolls of ribbon?

Answers

You have 10 rolls and each is 6 feet so therefore you have 60 feet of ribbon. Convert that to inches you need to multiply by 12 so 60 x 12 = 720 inches of ribbon.

If the dress maker needs 18 inch pieces you divide to 720 by 18.
720 divided by 18 will give you 40 pieces.

Final answer:

The dressmaker will get 40, 18-inch pieces of ribbon from 10, 6 foot rolls.

If you spend 3.5 hours a week studying for English and 7.5 hours studying for math what is the ratio of time
spent studying in math to studying for English?

Answers

Answer:

7.5 : 3.5

Step-by-step explanation:

ratio of time spent studying in math to studying for English is ;

time spent studying in math. / time spent in studying English

[tex] \frac{7.5}{3.5} [/tex]

= 7.5 : 3.5

[tex] \rm \int_{ \infty }^{ - \infty } \frac{ { {e}^{ { - x}^{2} } }(5 {x}^{2} + 2 {x}^{4} )}{ {x}^{2}( {x}^{2} + 1)} dx \\ [/tex]​

Answers

Consider the integral

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx[/tex]

which is the negative of yours. Bit strange to integrate over [tex](\infty,-\infty)[/tex], but if that's what you actually intended, just multiply the final result by -1. Of course, I've already canceled the superfluous factors of [tex]x^2[/tex].

Expand the integrand into partial fractions.

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx = \int_{-\infty}^\infty \left(2 + \frac3{1+x^2}\right) e^{-x^2} \, dx[/tex]

Recall that for [tex]\alpha>0[/tex],

[tex]\displaystyle \int_{-\infty}^\infty e^{-\alpha x^2} \, dx = \sqrt{\frac\pi\alpha}[/tex]

Now let

[tex]\displaystyle I(a) = \int_{-\infty}^\infty \frac{e^{-ax^2}}{1+x^2} \, dx[/tex]

Together, these give

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx = 2\sqrt\pi + 3I(1)[/tex]

Differentiate [tex]I(a)[/tex] under the integral sign with respect to [tex]a[/tex] to obtain a simple linear differential equation.

[tex]\displaystyle \frac{dI}{da} = -\int_{-\infty}^\infty \frac{x^2 e^{-ax^2}}{1+x^2} \, dx \\\\ ~~~~~~~~ = - \int_{-\infty}^\infty \left(1 - \frac1{1+x^2}\right) e^{-ax^2} \, dx \\\\ ~~~~~~~~ = -\sqrt{\frac\pi a} + I(a)[/tex]

Solve for [tex]I(a)[/tex] with the initial value [tex]I(1) = \sqrt\pi[/tex]. Using an integrating factor,

[tex]\displaystyle \frac{dI}{da} - I(a) = -\sqrt{\frac\pi a} \\\\ e^{-a} \frac{dI}{da} - e^{-a} I(a) = -\sqrt{\frac\pi a}\,e^{-a} \\\\ \frac{d}{da}\left[e^{-a} I(a)\right] = -\sqrt{\frac\pi a}\,e^{-a}[/tex]

By the fundamental theorem of calculus,

[tex]\displaystyle e^{-a} I(a) = e^{-a}I(a)\bigg|_{a=0} - \sqrt\pi \int_0^a \frac{e^{-\xi}}{\sqrt\xi} \, d\xi \\\\ I(a) = \pi e^a - \sqrt\pi \, e^a \int_0^a \frac{e^{-\xi}}{\sqrt\xi} \, d\xi[/tex]

so that

[tex]\displaystyle I(1) = \pi e - \sqrt\pi\,e \int_0^1 \frac{e^{-\xi}}{\sqrt\xi} \, d\xi[/tex]

Substitute [tex]t=\sqrt\xi[/tex].

[tex]\displaystyle I(1) = \pi e - 2\sqrt\pi\,e \int_0^1 e^{-t^2} \, dt[/tex]

Recall the error function,

[tex]\mathrm{erf}(x) = \displaystyle \frac2{\sqrt\pi} \int_0^x e^{-t^2} \, dt[/tex]

which we can use to write

[tex]I(1) = \pi e - 2\sqrt\pi e \cdot \dfrac{\sqrt\pi}2\,\mathrm{erf}(1) = \pi e - \pi e \,\mathrm{erf}(1)[/tex]

Finally, we arrive at

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx = \boxed{2\sqrt\pi + 3\pi e - 3\pi e \, \mathrm{erf}(1)}[/tex]

if 2/5 kilogram of soil fills 1/3 of a container,can1 a kilogram of soil fit in the container? Explain or show your reasoning

Answers

Answer:

Yes

Step-by-step explanation:

1 kilogram of soil will fill:

1/3 ÷ 2/5 = 5/6 (of a container)

Because 5/6 < 1

=> 1 kilogram of soil will fit in the container

Question 7 of 20:
Select the best answer for the question.
7. Percent means "per
O A. 100
OB. 1000
O C. 1
O D. 10
11
K
OMark for review (Will be highlighted on the review p

Answers

parts per hundred
Percent means parts per hundred. The word comes from the Latin phrase per centum, which means per hundred. In mathematics, we use the symbol % for percent.
The answer is Percentage means "per 100" and denotes a piece of a total amount. For example, 45% represents 45 out of 100, or 45% of the total amount. The answer is 100-A I hope this helps:)

Draw the image of

ΔDEF

∆DEF


under the dilation with scale factor

−1/2


and the origin as the center of dilation. Label the image

ΔD'E'F'

∆D′E′F′


. Make sure to include the coordinates of D’, E’, and F’.

Answers

The image of triangle DEF after the dilation with a scale factor of -1/2  is given at the end of the answer.

What are transformations on the graph of a function?

Transformations in the graph of a function involve operations in the vertices of the image, such as addition, subtraction, division or multiplication.

The transformation used in this problem is a dilation, in which the coordinates of the vertices of the original figure are multiplied by the scale factor.

For this problem, the vertices of the triangle are given as follows:

D(3,6), E(3,2) and F(5,4).

The dilation has a scale factor of -0.5 = -1/2, hence the coordinates of the image are given as follows:

D': (3 x -0.5, 6 x -0.5) = (-1.5, -3).E': (-1.5, -1).F': (-2.5, -2).

The dilated triangle is given by the graph at the end of this answer.

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At the beginning of each of her four years in college, Miranda took out a new Stafford loan. Each loan had a principal of $5,500, an interest rate of 7.5% compounded monthly, and a duration of ten years. Miranda paid off each loan by making constant monthly payments, starting with when she graduated. All of the loans were subsidized. What is the total lifetime cost for Miranda to pay off her 4 loans? Round each loan's calculation to the nearest cent.
a.
$23,650.00
b.
$29,481.08
c.
$7,834.32
d.
$31,337.27

Answers

The total lifetime cost for Miranda to pay off her 4 loans is: $31,337.27.

What is interest?

Interest is the sum of money paid for using someone else's funds. You must pay interest when you borrow money from lenders. You receive interest when you lend money to borrowers. It could be stated in terms of money or the rate of payment. You will learn more about interest in this article, including the different sorts of interest, what they are, and how to determine how much interest will be charged on a loan or a loan amount.

Given that,

At the beginning of each of her four years in college

Miranda took out a new Stafford loan.

Each loan had a principal of $5,500,

An interest rate of 7.5% compounded monthly

A duration of ten years.

Miranda paid off each loan by making constant monthly payments, starting with when she graduated.

All of the loans were subsidized.

The total lifetime cost for Miranda to pay off her 4 loans is: $31,337.27

Therefore, the total lifetime cost for Miranda to pay off her 4 loans is: $31,337.27

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Answer: D 31,337.27

Step-by-step explanation:

Which number below is NOT equal to the others?

1/8

18/10

1.8

1 4/5

Answers

Answer:

1/8

Step-by-step explanation:

When worked out, all numbers are equal to 1.8 except 1/8. (1/8 is equal to .125)

1/8) 1 divided by 8 is .125

18/10) 18 divided by 10 is 1.8

1.8) 1.8=1.8

1 4/5) 4 divided by 5 is .8, and when you add the 1 it ends up as 1.8

13x + 5 + 17x -4.5 + x

Answers

Answer:

31x+0.5

Step-by-step explanation:

In this problem, we must combine like terms to simplify the expression. We can begin by writing out the expression:


13x + 5 + 17x - 4.5 + x


Now that we have rewritten the expression, we can combine like terms to simplify the expression. This would look like:


13x + 17x + x = 31x


5 - 4.5 = 0.5

The answer would be “31x + 0.5” when simplified.

Need help with D pls will give brainly

Answers

The length of the pen is 2√3 feet

Area of a triangle

The formula for calculating the area of a rectangle is expressed according to the formula below:

Area of a right triangle = 1/2 * base * height

A = 1/2 bh

Given the following parameters

Area = 20 square feet

If the height is five times the length, then;

h = 5b

Substitute

A = 1/2 bh

A = 1/2(b)5b

30 = 1/2(5b²)

60 =5b²

b² = 12

b = 2√3

Hence the length of the pen is 2√3 feet

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Approximate √10 to the nearest tenth

Answers

√10 = 3.1622….. to the nearest tenth it would be 3.2

television and DVD player cost a total of $1242. The cost of the television is two times the cost of the DVD player. Find the cost of each item.

Answers

Answer: TV costs $828 and DVD player costs $414.

Step-by-step explanation: let x represent the cost of tv while y represents the cost of the player. x+y = 1242. since the tv costs twice as much as the dvd player, we can say that 2y=x. now replace x with 2y in our equation and we get 2y+y= 1242,  3y = 1242. now divide both sides by 3 and y = 414. Now that we now the value of y, just sub it into our original equation, x+y = 1242. x + 414 = 1242. x = 828.

Other Questions
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