I have an equation that I have an answer but no idea how it's solution works. This is an example for using the squeeze theorem, and the answer is 0:
[tex]f(x)=\frac{e\frac{-1}{13+\frac{1}{x^2}}\sin(1+cos\frac{\pi }{x^3})\sqrt{3x^4+5x^6}}{(x+x^\frac{1}{3})(3+arctan\sqrt{7+\frac{1}{x^8}})}[/tex]
The way the answer is arrived at is supremely confusing. Also the textbook used absolute values alongside inequalities when performing the algebraic manipulations (direct sub. doesn't work), which I also don't understand why. Please explain how this equations limit is zero in simple terms.
Answers
Step-by-step explanation:
You want to know how the squeeze theorem applies to help you find the limit of f(x) is zero as x approaches zero.
SummaryThe short of it is that you have ...
f(x) = [messy finite function] × [function that cleanly approaches 0 at x=0]
The zero product rule tells you that the product of any finite value and 0 is zero, so the limit of this product is 0 at x=0.
The graph in the attachment shows the "function that cleanly approaches 0" in red. When multiplied by the "messy finite function", the result is the blue curve. You can see that the result is a function that has a limit of 0 at x=0.
How to get thereHere, we'll decompose f(x) into two parts:
messy finite function (g(x))function that cleanly approaches zero at x=0 (h(x))Messy finite function
The numerator sine function factor of f(x) gets increasingly messy as x approaches zero, due to the cosine term it contains. We can define ...
[tex]g(x)=\sin(1+\cos(\pi x^{-3}))[/tex]
Because this is a sine function, its value is always in the range |g(x)| ≤ 1.
Function cleanly approaching zero
The remaining factors of f(x) need to be rearranged a bit to make a function that obviously approaches zero at x=0. We do that by factoring ∛x from the denominator and rearranging the leading fraction.
[tex]h(x)=\dfrac{e\dfrac{-1}{13+x^{-2}}\sqrt{3x^4+5x^6}}{(x+x^{1/3})(3+\arctan\sqrt{7+x^{-8}})}\\\\=-e\dfrac{x^2}{13x^2+1}\cdot\dfrac{x^2\sqrt{3+5x^2}}{x^{1/3}(x^{2/3}+1)(3+\arctan{\sqrt{7+x^{-8}}})}\\\\h(x)=-e\dfrac{x^{11/3}\sqrt{3+5x^2}}{(13x^2+1)(x^{2/3}+1)(3+\arctan{\sqrt{7+x^{-8}}})}[/tex]
Recognizing that the arctangent term will approach π/2 as x→0, this can be (more or less) directly evaluated at x=0 to give ...
[tex]h(0)=-e\dfrac{0\cdot\sqrt{3}}{1\cdot1\cdot(3+\dfrac{\pi}{2})}=0[/tex]
The sign of h(x) will be opposite the sign of x. The graph of h(x) is shown in red in the attachment.
The squeezeAs h(x) approaches zero, the product ...
f(x) = g(x)·h(x)
approaches zero. This is the essence of the squeeze.
Starting from our description of g(x) above, we can multiply both sides of the inequality by the positive value |h(x)| to get ...
|g(x)| ≤ 1 . . . . . . . . . . the range of the sine function
|h(x)|·|g(x)| ≤ |h(x)|·1 . . . . . . multiply by |h(x)|
Simplifying, this is ...
|f(x)| ≤ |h(x)| . . . . true for all x
So, the limit is ...
[tex]\displaystyle \lim_{x\to0}|f(x)|\le\lim_{x\to 0}|h(x)|\\\\\lim_{x\to0}|f(x)|=0[/tex]
The attachment illustrates this nicely, showing that the oscillating f(x) has its amplitude limited by h(x), which approaches zero.
__
Additional comments
The limit of x^-8 is positive infinity as x approaches zero. This means the arctangent function argument is positive infinity as x approaches zero, so the function value is π/2. The sign of x is irrelevant.
We cannot tell if the fraction following "e" in this function is supposed to be an exponent. The two typeset versions of f(x) seem to show "e" as a multiplier, not the base of an exponential term. Even if an exponential term is what is intended, that term would approach 1 at x=0, so the conclusion here remains unchanged.
Related Questions
what is (8^5)^4 (answer must include an exponent )
Answers
Answer:
[tex]8^{20}[/tex]
Step-by-step explanation:
using the rule of exponents
[tex](a^m)^{n}[/tex] = [tex]a^{mn}[/tex] , then
[tex](8^5)^{4}[/tex] = [tex]8^{5(4)}[/tex] = [tex]8^{20}[/tex]
Brainliest, 30 pts, thanks, 5 star guarantee.
Answers
The triangle ΔXYZ is congruent to the triangle ΔX'Y'Z'. And the intersection of ZZ' and AB form a right angle.
What is a transformation of a shape?A point, line, or geometric figure can be transformed in one of four ways, each of which affects the structure and/or location of the object.
The reflection does not change the shape and size of the geometry. But flipped the image.
The triangle ΔXYZ is reflected across the line AB to get the tringle ΔX'Y'Z'.
The diagram is given below.
A reflection is a transformation that maps every point P over a line such that the line segment PP' will intersects the line of reflection at a right angle.
Thus, the triangle ΔXYZ is congruent to the triangle ΔX'Y'Z'. And the intersection of ZZ' and AB form a right angle.
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i need help with this assignment can anyone help
Answers
Answer: 18.25
Step-by-step explanation:
5.5*4=22
401.5/22=18.25
Answer:
$18.25
Step-by-step explanation:
Since area is given as A=LW, and we are given the values of the dimensions of the floor, we get:
5.5×4=22
So, the area is 22 yards, and it costs $401.5 to carpet it. Now, We simply need to divide the cost by the area to get the "cost per-square yard" as needed in the question. We get:
401÷22=18.25
So, the cost of the carpet per square yard is $18.25
Under her cell phone plan, Morgan pays a flat cost of $54 per month and $5 per gigabyte, or part of a gigabyte. (For example, if she used 2.3 gigabytes, she would have to pay for 3 whole gigabytes.) She wants to keep her bill under $70 per month. What is the maximum whole number of gigabytes of data she can use while staying within her budget?
Answers
Answer: She can use 4 gigabyte while remaining in her budget.
Step-by-step explanation:
You would first subtract 54 from 70 which leaves you with 16,
Then you would divide 16 by 4 to find how many gigabytes she could use which gives you 4
So she can use 4 gigabyte while remaining in her budget.
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Answer: Answer is 3
Step-by-step explanation:
let the maximum number of gigabytes of data is x
total budget is $70
then, flat cost is $54 and $5per gigabyte
we get:
54 +5 × x= 70
5x =70 - 54
x = [tex]\frac{16}{5}[/tex] = 3.2 ≈ 3
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Esteban bought a desk lamp at a sale price of $20.25. The original price of the lamp was $45. What was the discount rate of the desk lamp?
Answers
Answer:
55%
Step-by-step explanation:
Ans
65.25 Step-by-step explanation:
what is the equation, in slope-intercept form, of the line parallel to y=5x+2 that passes through the point with coordinates (-2,1)?
Answers
Step-by-step explanation:
parallel means it has the same slope.
the slope of a line is always the factor of x (5 in our case).
so, we have already
y = 5x + ?
we resolve the "?" by using the point coordinates as x and y and solve :
1 = 5×-2 + ?
1 = -10 + ?
? = 11
so, the equation is
y = 5x + 11
Use the formula S=n(n+1)/2 to find the sum of 1+2+3+...+600
1+2+3+...600=
Answers
Answer:
eueueue
Step-by-step explanation:
The distance between two places of interest is 2/3km. I go 1/4 of this distance on foot. How far is still to go?
Answers
Answer:
[tex]\displaystyle \mathsf $\dfrac{1}{2}$ km[/tex]
Step-by-step explanation:
If I have traveled 1/4th of the distance between two places, I still have 1 - 1/4 = 3/4th of the distance yet to cover
If the distance between the two places is d, distance left =(3/4) . d
In this case, d = 2/3 km
So distance left to cover = [tex]\dfrac{2}{3}\cdot \dfrac{3}{4}[/tex]
We can cross-cancel the 3 which appears as the denominator in the first fraction and the numerator in the second fraction
[tex]\dfrac{2}{\cancel{3}}\cdot\dfrac{{\cancel{3}}}{4}=\dfrac{2}{4}[/tex]
Dividing both numerator and denominator by 2 gives us
[tex]\dfrac{2\div2}{4\div2}=\dfrac{1}{2}[/tex]
Answer:
[tex]\displaystyle \mathsf $\dfrac{1}{2}$ km[/tex]
name a fraction between 6/7 and 1
Answers
Answer:7/8, 8/9, 9/10, 10/11, etc.
Any help would be appreciated.
Answers
What is that for?
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pls answer if wrong i report
Answers
Answer:
D) Dilation and reflection.
→ The shape was dilated by -1.
→ This shape was then reflected upside down.
DilationDilating a shape is the act of making a shape smaller by a scaled amount. Dilation is useful in mathematics when you need to determine difference in shapes' scales factors.
ReflectionReflection is mirroring a shape to its opposite image. For instance, I have triangle ∠ABC', which is facing right. I then reflect this image, and it becomes triangle ∠ABC", which now faces left on the opposite axis.
15% of the students in the 6th grade wear glasses. If 12 students wear glasses, how can you determine the number of students in the 6th grade? Explain your reasoning.
Answers
Answer: There are 80 people in the 6th grade
Step-by-step explanation:
Como puedo resolver estas funciones y saber su tipo?
Answers
Can someone PLEASEE help me
Answers
-3 0 ( -3,0)
0 -2 (0,-2)
3 -4 ( 3,-4)
-3/2 -4 1/6 ( -3/2, -4 1/6 )
Kiyana's grandmother has a collection of hats for special occasions. Each hat is in a cube-shaped box with edges 1 ft long. She stores the boxes in a rectangular closet with the dimensions shown. Length = 6 ft N Height = 9 ft Width= 5 ft 3 4 5 > 2 of 5 Completed What is the maximum number of boxes that Kiyana's grandmother can fit inside the closet? Choose one option from each drop-down menu to answe the question. Kiyana's grandmother can fit a maximum of [ ] boxes along the length,[. ] boxes along the wide and [ ]boxes along the height of the closet. This means she can fit a maximum of [ ]boxes inside the closet
Answers
The maximum number of cubical boxes of side 1 feet that can be placed in a rectangular closet of length 6 feet, height 9 feet and breadth 5 feet is 270 boxes. Maximum number of boxes that can be placed along the length, height and breadth will be 168 boxes, 198 boxes and 150 boxes respectively.
Since the cube shaped boxes of length 1 feet are to be placed in the rectangular closet, the maximum number of boxes that will fit completely would be given as a comparison of volume of rectangular closet being equal to product of number of boxes and volume of 1 cubical box.
Volume of rectangular closet = Number of boxes * Volume of 1 cubical box
6 * 9 * 5 = n * (1*1*1)
Therefore, n = 270 boxes.
To calculate maximum number of boxes that will fit along the length would be calculated as product of length and breadth along 2 faces of closet and length and height along another 2 faces of closet.
Thus, [(9*6) + (9*6) + (6*5) + (6*5)] = 168 boxes along the length.
Similarly, to calculate maximum number of boxes that will fit along the height would be calculated as product of height and breadth along 2 faces of closet and length and height along another 2 faces of closet.
Thus, [(9*6) + (9*6) + (9*5) + (9*5)] = 198 boxes along the height
Similarly, to calculate maximum number of boxes that will fit along the breadth would be calculated as product of height and breadth along 2 faces of closet and length and breadth along another 2 faces of closet.
Thus, [(5*6) + (5*6) + (9*5) + (9*5)] = 150 boxes along the breadth.
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What is the answer to -5 - 7.
Answers
Answer:
it is negative 12
Step-by-step explanation:
-12
Answer:
-12
Step-by-step explanation:
how many zeros will be in the product of 5 x 105 the 105 is the 10 of the power of 5
Answers
Answer: this equation has 5 zeroes.
Step-by-step explanation:
The number of zeroes in any number can be found by,
we have expression as
[tex]5*10^{5}[/tex]
now ;
10*10 has 2 zeroes;
10*10*10 has 3 zeroes;
similarly ;
this equation has 5 zeroes.
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Answer:
none
Step-by-step explanation: 105×5=525
There are five questions listed below. Each question includes the quantity 43. Match the 43 in each question on
the left to which part of the problem it represents on the right -- the base, percent, or amount. Some answer
options on the right will be used more than once.
What number is 15% of 43?
43% of 300 is what number?
12% of what number is 43?
43 is what percent of 106?
What percent of 43 is 6?
percent
base
amount
Click and hold an item in one column, then drag it to the matching item in the other column. Be sure your
Answers
What number is 15% of 43?
43 here is the base43% of 300 is what number?
Here 43 is the percent12% of what number is 43?
43 is used here as amount43 is what percent of 106?
43 is used here as amountWhat percent of 43 is 6?
43 here is the baseWhat is base?The base as used in the question refers to the main number that is a=to say the number on which other calculation are focused on.
What is percent?Percent is a term used in mathematics to refer to a fraction with a denominator of one hundred. Numbers in percent is accompanied with the percent symbol which is %. Any number having that symbol means that the number is a fraction with a denominator of one hundred.
What is amount?The term amount as used refers to the result gotten when some mathematical operation is being done on the base.
The calculations to the questions asked are below:
What number is 15% of 43?
= 15/100 x 43
= 6.45
43% of 300 is what number?
= 43/100 x 300
= 129
12% of what number is 43?
= 12/100 x 43
= 5.16
43 is what percent of 106
= 43/106 * 100
= 40.56%
What percent of 43 is 6
= 6/43 x 100
= 13.95%
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for each table, find the constant of proportionality. use decimal notation if neccescry
Answers
Step-by-step explanation:
the constant k of proportionality is defined by
y = kx
or in our case
b = ka
so, let's see
14 = k×2
k = 14/2 = 7
this also works for the other data pairs :
7×5 = 35
7×9 = 63
7 × 1/3 = 7/3
perfect.
so, the constant of proportionality is 7.
A rectangular ballroom is being carpeted and measures 32‘ x 24‘. Carpet cost $3.60 per square foot. What is the cost to carpet this room?
Answers
The cost to carpet this room area is $2764.8.
What is the area?
The area is the volume that expresses the extent of a region on the plane or on a twisted face. The area of a plane region or plane area refers to the area of a shape or planar lamella, while face area refers to the area of an open face or the boundary of a three-dimensional object.The area can be understood as the quantum of material with a given consistency that would be necessary to fashion a model of the shape.Area of room = 32*24 (product is equal to the area of carpet)
Cost of carpet per square foot = $3.6 (three. six)
Cost of carpet = Area of carpet * Cost of carpet per square foot (product)
= 32 x 24 x 3.6
= 2764.8
The cost to carpet this room area is $2764.8.
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TOOIS
Question 3
A pipe is leaking at a rate of 8 fluid ounces per minute. How many gallons per hour will leak from the pipe?
0.02 gal/hr
A
B
C
D
3.75 gal/hr
17.07 gal/hr
3,840 gal/hr
Answers
If a pipe is leaking at a rate of 8 US fluid ounces per minute, then 3.75 US liquid gallons of the fluid will leak from the pipe in an hour.
As per the question statement, a pipe is leaking at a rate of 8 US fluid ounces per minute and we are required to calculate the amount of leak that will occur at the same rate of leakage, in an hour, in the units of US liquid gallons.
To solve this question, we need to know to know about two basic mathematical conversions, i.e.,
1 US Liquid Ounce = 0.00781 US Liquid Gallons, and
1 hour = 60 minutes.
Therefore, amount of leak that will occur at the rate of 8 US fluid ounces per minute, in an hour = [tex](8 * 60)=480[/tex] US fluid ounces.
And 480 US fluid ounces = [tex](0.00781 * 480)=3.75[/tex] US Liquid Gallons.
Hence, 3.75 US Liquid Gallons of fluid will leak from the pipe in an hour.
Fluid Ounce: It is an unit of volume, used for measuring liquids. "The British Imperial" and the "United States customary" fluid ounce are the only two that are still in use. The US Fluid ounce is an unit of capacity equal to one-sixteenth of a US pint (approximately 0.03 liter), while the British Imperial fluid ounce a unit of capacity equal to one twentieth of a pint (approximately 0.028 liter).Liquid Gallons: In the United States, Liquid Gallon is an unit of liquid capacity equal to four quarts or 231 cubic inches or 3.785 liters, while in the Great Britain, also known as the "Imperial Gallon", it is an unit of liquid and dry capacity equal to four quarts or 277.42 cubic inches or 4.544 liters.To learn more about Fluid Ounces and Liquid gallons, click on the klink below.
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Aviva's pay stub shows gross earnings of $662.30 for a week. Her regular rate of pay is $10.80 per hour for a 35 hour week and overtime is paid time and a half regular pay. How many Hours did she work
Answers
Number of hours Aviva's work for gross earnings of $662.30 for a week with regular rate of $10.80 per hour for 35 hour week and 1.5 of regular pay for overtime is 52.6hours.
Given,
Gross earning for week=$662.30
Regular rate per hour for 35hour week=$10.80
Overtime rate=1.5×(10.80)
For 'h' number of hours
(10.80×35)+(10.80×1.5) h=$662.30
⇒378 +16.20h=662.30
⇒h=284.30/16.20
⇒h≈17.6hours
Number of hours Aviva's work=35 +17.6
=52.6hours
Therefore, number of hours Aviva's work for gross earnings of $662.30 for a week with regular rate of pay $10.80 per hour for a 35 hour week and 1.5 of regular pay for overtime is 52.6hours.
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Help me pls I'm dying rn
Answers
a) The greatest common factor is 3. And the result is 3 × (6 + 7).
b) The greatest common factor is 7. And the result is 7 × (8 + 5).
What is the factor form of the sum of two integers?In this question we must find the greatest common factor, that is, the maximum integer with which can be divided the two number, and take advantage of this result to simplify the sum of the same two numbers, if possible.
First, we proceed to determine the required results for each of the two cases:
18 = 2 × 3 × 3
21 = 3 × 7
Then, the greatest common factor is 3. The complete procedure is:
18 + 21 = 2 × 3 × 3 + 3 × 7 = 3 × (6 + 7)
56 = 2³ × 7
35 = 5 × 7
Then, the greatest common factor is 7. The complete procedure is:
56 + 35 = 2³ × 7 + 5 × 7 = 7 × (8 + 5)
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8,370 divided by 62 with remainder
Answers
Answer:
8370/ 62: 135
62/8370: 0.007
Step-by-step explanation:
Chris is studying a type of coral that grows at a constant rate. They measured the age and height of 3
pieces of the coral. Here's what they found:
Answers
The equation is; y = 7/2x
What is the equation?We know that we could be able to obtain the equation by the use of the equation is a straight line. The general form of the equation of a straight line is given as y = mx +c
The first thing that we must do is to obtain the slope of the graph from;
m = y2 - y1/x2 x1
m = 45 - 15/6 - 2
m = 7/2
The graph begins at origin hence c = 0
y = 7/2x
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Got it wrong hehe :)
Answers
6/1 = 42/7
42/7 - 33/7 = 9/7 = 1 2/7
-3x+6=-24bwhat is the value of 2x-17
Answers
The value of the expression 2x - 17 is 3
How to evaluate the expression?The expression is given as
-3x + 6 = -24
Subtract 6 from both sides
-3x = -30
Divide both sides by -3
x = 10
Multiply by 2
2x = 20
Subtract 17 from the sides
2x - 17 = 3
Hence, the value of the expression 2x - 17 is 3
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A person can order a new car with a choice of 12 possible colors, with or without air conditioning, with or without automatic transmission, with or without power windows, and with or without a CD player. In how many different ways can a new car be ordered with regard to these options?
Answers
Answer:
96
Step-by-step explanation:
12 colors times 2 possibilities of with or without air conditioning is 24.
24 times two possibilities of with or without automatic transmission is 48.
48 times two possibilities of with or power windows is 96.
96 times two possibilities of with or CD player is 192.
reflection across y = -1
Please I need help ASP I will mark you brainless please
Answers
Answer:
See picture below.
Step-by-step explanation:
9. A submarine is 140 ft
below sea level. It rose 30 ft.
What integer describes the
new depth of the submarine?
Answers
Answer:
it's easy if you set your mind to it, think about it angain