Use the old exp-log trick and properties of the logarithm to rewrite the limit as
[tex]\displaystyle \lim_{n\to\infty} \left(\frac{n^n \left(x+\frac n1\right) \left(x+\frac n2\right) \cdots \left(x+\frac nn\right)}{n! \left(x^2+\left(\frac n1\right)^2\right) \left(x^2+\left(\frac n2\right)^2\right) \cdots \left(x^2+\left(\frac nn\right)^2\right)}\right)^{\frac xn}[/tex]
[tex]\displaystyle = \exp\left[\lim_{n\to\infty} \frac xn \ln \left(\frac{n^n \left(x+\frac n1\right) \left(x+\frac n2\right) \cdots \left(x+\frac nn\right)}{n! \left(x^2+\left(\frac n1\right)^2\right) \left(x^2+\left(\frac n2\right)^2\right) \cdots \left(x^2+\left(\frac nn\right)^2\right)}\right)\right][/tex]
[tex]\displaystyle = \exp\left[x \lim_{n\to\infty}\frac1n\ln\left(\frac{n^n}{n!}\right) + \frac1n \sum_{k=1}^n \ln\left(x+\frac nk\right) - \frac1n \sum_{k=1}^n \ln\left(x^2+\left(\frac nk\right)^2\right)\right][/tex]
[tex]\displaystyle = \exp\left[x \lim_{n\to\infty}\frac1n\ln\left(\frac{n^n}{n!}\right) + \frac1n \sum_{k=1}^n \ln\left(x+\frac1{k/n}\right) - \frac1n \sum_{k=1}^n \ln\left(x^2+\frac1{\left(k/n\right)^2}\right)\right][/tex]
The first limit converges to 0, since n! asymptotically behaves like nⁿ, so ln(nⁿ/n!) → ln(1) = 0.
The two remaining sums converge to definite integrals:
[tex]\displaystyle \lim_{n\to\infty} \frac1n \sum_{k=1}^n \ln\left(x + \frac1{\frac kn}\right) = \int_0^1 \ln\left(x + \frac1y\right) \, dy = \frac{(x+1) \ln(x+1)}x[/tex]
[tex]\displaystyle \lim_{n\to\infty} \frac1n \sum_{k=1}^n \ln\left(x^2 + \left(\frac1{\frac kn}\right)^2\right) = \int_0^1 \ln\left(x^2 + \frac1{y^2}\right) \, dy = \frac{2\tan^{-1}(\sqrt x)}{\sqrt x} + \ln(1+x)[/tex]
It follows that
[tex]f(x) = \exp\left[x\left(0 + \dfrac{(x+1)\ln(x+1)}x - \dfrac{2\tan^{-1}(\sqrt x)}{\sqrt x} - \ln(x+1)\right)\right][/tex]
[tex]f(x) = \exp\left[\ln(x+1) - 2\sqrt x \tan^{-1}(\sqrt x)\right][/tex]
[tex]f(x) = (x+1) e^{-2\sqrt x \tan^{-1}(\sqrt x)}[/tex]
By computing f'(x) and f''(x), it's easy to show that f'(x) ≤ 0 and f''(x) ≥ 0 for all x > 0. So f(x) is decreasing and f'(x) is increasing, and
• (A) f(1/2) ≥ f(1) is true
• (B) f(1/3) ≤ f(2/3) is false
• (C) f'(2) ≤ 0 is true
• (D) f'(3)/f(3) ≥ f'(2)/f(2) ⟺ f'(3)/f'(2) ≥ f(3)/f(2) ⟺ (something larger than 1) ≥ (something smaller than 1) is true
Sketch a graph of x−y=−4
Answer:
x-intercept(s):
(
4
,
0
)
y-intercept(s):
(
0
,
−
4
)
Step-by-step explanation:
find the equation of a line passes through a point (0,4) and has a gradient of 2. leave your answer in the form y=mx+c
Answer:
y = 2x + 4
Step-by-step explanation:
the equation of a line in point- slope form is
y = mx + c ( m is the slope and c the y- intercept )
here m = 2 and passes through point (0, 4 ) ⇒ c = 4
y = 2x + 4 ← equation of line
Answer:
y = 2x + 4
Step-by-step explanation:
General form of linear equation: y = mx + c
where m is the gradient and c is the y-intercept
Given gradient = 2
⇒ m = 2
⇒ y = 2x + c
y-intercept is when x = 0
Given (0, 4), so y = 4 when x = 0
⇒ c = 4
⇒ y = 2x + 4
3 Consider two vectors Az 31 - 2; and Bz-i-4;. alculated B C) A+B and A-B B G) / Atb 1 and (A-Blu | - B (i fra) Direction of (A+B) and|A-b] B) B + ดู B BJ
Step-by-step explanation:
1. A+B=Just add the J hat with J hat and add the I hat with the I hat.
[tex](3i + ( - 1)i + ( - 2) + ( - 4)j = 2i - 6j[/tex]
A-B=
[tex]3 - ( - 1)i + ( - 2) - ( - 4)j = 4i + 4j[/tex]
2. Take the magnitude of A+B
[tex]2 {}^{2} + ( - 6) {}^{2} = {r}^{2} [/tex]
[tex]40 = {r}^{2} [/tex]
[tex]r = \sqrt{40} [/tex]
So the magnitude is root of 40.
So the magnitude of A+B is
The magnitude of A-B
[tex] {4}^{2} + {4}^{2} = 4 \sqrt{2} [/tex]
So the magnitude is
[tex]4 \sqrt{2} [/tex]
c. To find direction, we need to find the angle.
Use this rule,
[tex] \tan(x) = \frac{bj}{bi} [/tex]
For A+B, we get
[tex] \tan(x )= \frac{ - 6}{2} [/tex]
[tex] \tan(x) = - 3[/tex]
Since this is a southeast, the degree is 288.43.
The direction is southeast, 288.43 degrees
For A-B, we get
[tex] \tan(x) = \frac{4}{4} [/tex]
[tex] \tan(x) = 1[/tex]
Since this is northeast, the degree is 45 degrees.
The direction is northeast, 45 degrees.
In a sample of 150 batteries, 18 did not work properly. Choose the percent of the sample that was defective
Divide the number of defective by the total number of batteries then multiply by 100 to get the percent.
18/150 = 0.12
0.12 x 100 = 12%
Answer: 12%
A container built in the shape of a cylinder has a circular base with the radius of 1.5m. the altitude of the container is 3.5m. how any cudic meters of fluid can this container hold?
Answer:
the container will hold 3m cudic meters
I need help with B find the roots
Answer:
-1 and 3
Step-by-step explanation:
The roots of the quadratic function can be determined by [tex]f(x')=0[/tex] where x' is the roots of the quadratic function. In that graph, x = -1 and x = 3 makes the value of f(-1) = 0 and f(3) = 0. So, x = -1 and x = 3 are the roots of the f(x).
-9j -7m-8j?????? I need help plz
Answer:
-17j - 7m
Step-by-step explanation:
combine like terms -9j - 8j = -17j, there is no like term for -7m
Angle 1 and Angle 2 are complimentary angles. Find x and the measure of Angle 1.
Angle 1 = (3x+2)°
Angle 2 = (4x +4)º
x = ?
The measure of Angle 1 = ?
degrees
Answer:
x=12 pls give brainliest
Step-by-step explanation:
Answer: x = 12, Angle 1 = 38°
Step-by-step explanation: Complimentary angles add up to 90°.
Add the two angles and set them equal to 90°:
(3x + 2) + (4x + 4) = 90
Add like terms:
7x + 6 = 90
Subtract 6 from both sides:
7x = 84
Divide both sides by 7:
x = 12
Now, to get Angle 1, plug x in:
Angle 1 = 3(12) + 2
Angle 1 = 38°
Hope this helps!
Write the polynomial in factored form. Show all the steps that you use to solve this problem.
x^3 - 3x^2 - 10x
Answer:
x( x - 5)( x + 2)
Step-by-step explanation:
The polynomial is x^3 - 3x^2 -10x. Since each term has at least one x, we can factor out an x.
This gives us x( x^2 - 3x - 10).
Now, we may factor the second degree polynomial.
_ × _ = -10? (or a × c, where ax^2 + bx + c?)
_ + _ = -3? (or b?)
Well, -5 times 2 equals 10, and -5 plus 2 equals -3.
We use these values to factor further.
So, our final answer becomes x( x - 5 )( x + 2 ).
You can check this by simply multiplying and distributing the terms!
I hope this helped! :)
Solve for p: 5p+1/8 =3p-5/4 *Make sure you show all your work.
HELPP PLS!!!
Step-by-step explanation:
I think this is the answers. hope it helps you
Consider Nibbles, a very specialized brand of parakeet chow.
When its price is $16 per package, it sells 40,000 units. When its price rises to $20, it sells 30,000
units. Calculate its price elasticity of demand, carefully following all numeric directions. You will
interpret this result in the next question.
The price elasticity of demand for the parakeet chow is 1.33.
What is price elasticity of demand?Price elasticity of demand is the change in product quantity demanded to change in price.
From the question:
[tex]\%\ Change\ in \ quantity=\frac{30000-40000}{30000}= -33.3\%\\\\\%\ Change\ in \ price=\frac{20-16}{16}= 25\%\\\\PED=|\frac{\%\ Change\ in \ quantity}{\%\ Change\ in \ price} |=|\frac{-33.3\%}{25\%}|=1.33[/tex]
The price elasticity of demand for the parakeet chow is 1.33.
Find out more on elasticity of demand at: https://brainly.com/question/24384825
A random sample from the 6th and 7th grade student population was taken to determine which clubs were the most popular.
6th Grade: Yearbook 10; Student Council 24; Jump Rope 6
7th Grade: Yearbook 18; Student Council 20; Jump Rope 2
What percent of 6th graders are involved in the yearbook club?
Using the percentage concept, it is found that 25% of 6th graders are involved in the yearbook club.
What is a percentage?The percentage of an amount a over a total amount b is given by a multiplied by 100% and divided by b, that is:
[tex]P = \frac{a}{b} \times 100\%[/tex]
In this problem, 10 out of 10 + 24 + 6 = 40 6th graders are involved in the yearbook club, hence the corresponding percentage is given by:
[tex]P = \frac{10}{40} \times 100\% = 25\%[/tex]
More can be learned about percentages at https://brainly.com/question/10491646
Can some help me please
Answer:
what's my 19 answering your question sorry I have no idea what the answer is bio
The question is in the picture, please help.
Answer:
how to understand the problem is picture the shape as a house and try to figure out what to do next
Step-by-step explanation:
I need help on it pls pls
Answer:
b,d
Step-by-step explanation:
The base of a triangle is 2 ft. The
height of the triangle is 15 in. What is the area of the triangle in square inches?
Answer:
180 in²
Step-by-step explanation:
base b = 2 ft = 2*12 in = 24 in
height h = 15 in
Area = bh/2 = 24 * 15/2 = 180 in²
1 and 3 fourths - 7/10
Answer:
1.05
There you go!
In an accelerated failure test, components are operated under extreme conditions so that a substantial number will fail in a rather short time. In such a test involving two types of microchips, 620 chips manufactured by an existing process were tested, and 125 of them failed. Then, 820 chips manufactured by a new process were tested, and 130 of them failed. Find a 90% confidence interval for the difference between the proportions of failures for chips manufactured by the two processes. (Round the final answers to four decimal places.) The 90% confidence interval is ( . , ).
Answer:
The 90% confidence interval is [tex] 0.0097 < p_1 -p_2 < 0.0773 [/tex]
Step-by-step explanation:
From the question we are told that
The first sample size is [tex]n_1 = 620[/tex]
The number of chips that failed is k = 125
The second sample size is [tex]n_2 = 820[/tex]
The number of chips that fail is [tex]u = 130[/tex]
Generally the confidence level is 90% , hence the level of significance is
[tex]\alpha = (100 - 90)\% [/tex]
=> [tex]\alpha = 0.10 [/tex]
Generally from the normal distribution table the critical value of [tex]\frac{\alpha }{2}[/tex] is
[tex]Z_{\frac{\alpha }{2} } = 1.645[/tex]
Generally the first sample proportion is
[tex]\r p _1 = \frac{125}{620}[/tex]
=> [tex]\r p _1 = 0.202[/tex]
Generally the second sample proportion is
[tex]\r p _1 = \frac{130}{820}[/tex]
=> [tex]\r p _1 = 0.1585[/tex]
Generally the standard error is mathematically represented as
[tex]SE = \sqrt{\frac{\r p_1 (1 - \r p_1)}{n_1} +\frac{\r p_2 (1 - \r p_2)}{n_2} }[/tex]
=> [tex]SE = \sqrt{\frac{0.202(1 - 0.202)}{620} +\frac{0.1585 (1 - 0.1585)}{820} }[/tex]
=> [tex]SE = 0.0206[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * SE[/tex]
=> [tex]E = 1.645 * 0.0206[/tex]
=> [tex]E =0.0338[/tex]
Generally 95% confidence interval is mathematically represented as
[tex](\r p_1 -\r p_2) -E < p_1 - p_2 < (\r p_1 -\r p_2) +E [/tex]
=> [tex](0.202 -0.1585) -0.0338 < p_1 - p_2 < (0.202 -0.1585) +0.0338 [/tex]
=> [tex] 0.0097 < p_1 - p_2 < 0.0773 [/tex]
Equation 1) 2x - 3y = 6 Equation 2) 4y - x = 12 *
Answer:
2x - 3y = 6
[tex]x = 3 + \frac{3y}{2}[/tex]
[tex]y = 2 + \frac{2x}{3}[/tex]
4y - x = 12
[tex]x = 12 + 4y\\y = 3 + \frac{x}{4}[/tex]
Step-by-step explanation:
Divide both sides of the equation to "solve for x and y."
Hope it helps!!!Brainliest pls!!!please help. i’m on a time limit.
Answer: 161 m^2
Step-by-step explanation:
The surface area composes of 4 triangles and 1 square.
The side length of the square is 7 m, so the area is 49 m^2 because the area of a square is the side length^2.
Each of the triangles has a base of length 7 m and a height of 8 m, so the area of each triangle is 7m*8m/2 = 28 m^2 because the area of a triangle is the base*height/2.
Since there are 4 triangles, the total surface area of the 4 triangles is 28 m^2 * 4 = 112 m^2.
Adding that to the area of the square, we get 49 m^2 + 112 m^2 = 161 m^2
What is the area of this trapezoid?
Enter your answer in the box.
__ft²
Answer:
468 ft²
Step-by-step explanation:
Area= (sum of the 2 parallel sides÷2)×perpendicular height
=(15+37)÷2×18
=52÷2×18
=468
which of the following is consistent with and could explain the changes in lowe's dollars of gross profit and gross profit percentages in fiscal 2019
The changes in lowe's dollars of gross profit and gross profit percentages as seen in fiscal 2019 is due to;
Sales volume increased in fiscal 2019, and unit costs increased. What does the gross profit margin means?The gross profit margin is known to be that which informs a person what their business has made after paying their direct cost of doing business.
This is made up of labor, materials and other kinds of direct production costs. It is known to be a key element in profitability ratios along with operating profit margin and net profit margin.
The Iowe's company was said to have posted that they made about 2.6 percent in terms of sales growth and the fiscal 2019 shows that they had the highest sales and adjusted diluted earnings per share since the start of the company.
See options below
a. Sales volume increased in fiscal 2019, but unit costs decreased.
b. Sales volume decreased in fiscal 2019, and per unit selling prices decreased.
c. Sales volume increased in fiscal 2019, and unit costs increased.
d. Sales volume and unit costs did not change, but per unit selling prices increased.
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The label on a can of paint states that one gallon of the paint will cover 560 feet. How many gallons of that paint are needed to cover 1400 square feet?
Answer:
its about 5 gallons
Step-by-step explanation:
because the feets how are for the times its 4 or 5 if its 4 it cant go until 1400 and if its 5 theres left paint
lab technician cuts a 12 in piece of glass tubing into two pieces. One piece is 5 in longer than the other. How long are the pieces?
Question content area bottom
Part 1
What is the length of the shorter piece?
enter your response here
Answer:
Part 1. 7 inches (longer piece)
Part 2: 5 inches (shorter piece)
Step-by-step explanation:
There is 12 inches in total, given that one piece is 5 inches longer, the other part of the 5 inches would be 7 inches which is the longer side.
5 inches would be the shorter side.
What is the value of x?
3
4
5
6
Answer:
x = 6
Step-by-step explanation:
Using the relationships between the entries in the table, you can find x two different ways.
1.The numbers down the Total column add to give the one at the bottom:
0.1 + 0.09x = 0.64
0.09x = 0.54 . . . . . . . subtract 0.1
x = 6 . . . . . . . . . . . . divide by 0.9
__
2.The product of the numbers in the first two columns is the number in the third. On the bottom row, this means ...
(2 +x)(0.08) = 0.64
2 +x = 8 . . . . . . . . . . . divide by 0.08
x = 6 . . . . . . . . . . . subtract 2
help me please it for math
Answer:
c is the answer
Step-by-step explanation:
i don't have one
how do you write 13. 085 in words
ANSWER
thirteen point zero eight five
HELLLLLP ILL GIVE BRAINLIEST!!!!
Given that each package contains 9 trading cards.
Packages Trading cards
2 18
4 36
7 63
which number line expression show how to find distance from 2 to -5?
the last one is the line expression who shows the distance from 2 to negative 5
Option D is the correct answer.
We need to identify which number line expression shows how to find the distance from 2 to -5.
What is a number line?A line on which numbers are marked at intervals is used to illustrate simple numerical operations.
If we see option D, it has been marked correctly at 2 and -5.
Therefore, option D is the correct answer.
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I WILL GIVE BRAINLIEST!! PLEASE HELP
Answer:
angle mnp
Step-by-step explanation:
i think it is helpful