Base case (n = 1):
• On the left side: 1/(1×2) = 1/2
• On the right side: 1/(1 + 1) = 1/2
Induction hypothesis: Assume the statement is true for n = k ; that is,
1/(1×2) + 1/(2×3) + … + 1/(k × (k + 1))) = k/(k + 1)
Inductive step (n = k + 1):
1/(1×2) + 1/(2×3) + … + 1/(k × (k + 1))) + 1/((k + 1) × (k + 2)))
= k/(k + 1) + 1/((k + 1) × (k + 2))
= (k × (k + 2) + 1) / ((k + 1) × (k + 2))
= (k ² + 2k + 1) / ((k + 1) × (k + 2))
= (k + 1)² / ((k + 1) × (k + 2))
= (k + 1) / (k + 2)
and this is what we wanted to show.
n(AnB)=3 and n(AuB)=10, then find (p(A∆B))?
I assume A ∆ B denotes the symmetric difference of A and B, i.e.
A ∆ B = (B - A) U (A - B)
where - denotes the set difference or relative complement, e.g.
B - A = {b ∈ B : b ∉ A}
It can be established that
A ∆ B = (A U B) - (A ∩ B)
so that
n(A ∆ B) = n(A U B) - n(A ∩ B) = 10 - 3 = 7
Not sure what you mean by p(A ∆ B), though... Probability?
Damaris will be working at the local pool over his ten-week summer break. His net pay will be $167.30 each week. He hopes to have enough money to purchase a new pair of shoes that cost $175 by the end of his break. What percent of his net pay does Damaris need to save each week to reach his goal? Round to the nearest hundredth. (2 points)
1.05%
10.46%
11.37%
Damaris needs to save 10.46% of his net pay each week to purchase the new pair of shoes by the end of his break.
Given:
Net pay per week is $167.30Cost of new pair of shoes is $175Summer break is for 10 weeksTo find: The percentage of his net pay that Damaris needs to save each week to purchase the shoes by the end of his break
Let us assume that Damaris needs to save x% of his net pay each week to buy the shoes by the end of his break.
Then, savings per week is x% of $167.30, that is,
[tex]\frac{x}{100}\times 167.30[/tex]
Then, his savings for 10 weeks is,
[tex]10 \times \frac{x}{100}\times 167.30[/tex]
Since the summer break is for 10 weeks, Damaris' savings for the entire summer break is,
[tex]10 \times \frac{x}{100}\times 167.30[/tex]
Damaris wants to buy the new pair of shoes by then end of the break. Then, his savings for the entire summer break should equal the cost of the new pair of shoes.
It is given that the cost of the new pair of shoes is $175.
Then, according to the problem,
[tex]10 \times \frac{x}{100}\times 167.30 =175[/tex]
[tex]x=\frac{175\times 100}{10\times167.30}[/tex]
[tex]x=10.460251[/tex]
Rounding to the nearest hundredth, we have,
[tex]x=10.46[/tex]
Thus, Damaris needs to save [tex]10.46[/tex]% of his net pay each week to buy the shoes by the end of his break.
Learn more about percentage here:
https://brainly.com/question/22400644
15 points work out ratio for x
Answer:
x = 25
Step-by-step explanation:
x : (x+10) = 5:7
Fractional form
x / x+10 = 5/7
Cross multiply:
x * 7 = (x+10) * 5
7x = 5x + 50
7x - 5x = 5x + 50 - 5x
2x = 50
x = 25
Check:
25 : 25 + 10
25 : 35
25/35 = 5/7
If my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
Which power does this expression simplify to?
[(7)(7)
1
- -
ооо
74
O
Step-by-step explanation:
Answer is in attached image...
hope it helps
Answer:
its a
Step-by-step explanation:
just did it
1) Sử dụng phương pháp diện tích chứng minh định lí Pitago: “Trong một tam giác vuông, bình phương cạnh huyền bằng tổng bình phương hai cạnh góc vuông”.
2) Chứng minh rằng tứ giác có một và chỉ một đường nối trung điểm hai cạnh đối chia tứ giác thành hai phần có cùng diện tích là hình thang.
Answer:
hmm i thought abt it and i think the answer is no
Step-by-step explanation:
write your answer in simplest radical form
Answer:
z = √3
Step-by-step explanation:
sin (30°) = z / 2√3
z = sin (30°) 2√3
z = √3
Suppose h(x)=3x-2 and j(x) = ax +b. Find a relationship between a and b such that h(j(x)) = j(h(x))
Probably a simple answer, but I'm completely lost at what I'm being asked here.
Answer:
[tex]\displaystyle a = \frac{1}{3} \text{ and } b = \frac{2}{3}[/tex]
Step-by-step explanation:
We can use the definition of inverse functions. Recall that if two functions, f and g are inverses, then:
[tex]\displaystyle f(g(x)) = g(f(x)) = x[/tex]
So, we can let j be the inverse function of h.
Function h is given by:
[tex]\displaystyle h(x) = y = 3x-2[/tex]
Find its inverse. Flip variables:
[tex]x = 3y - 2[/tex]
Solve for y. Add:
[tex]\displaystyle x + 2 = 3y[/tex]
Hence:
[tex]\displaystyle h^{-1}(x) = j(x) = \frac{x+2}{3} = \frac{1}{3} x + \frac{2}{3}[/tex]
Therefore, a = 1/3 and b = 2/3.
We can verify our solution:
[tex]\displaystyle \begin{aligned} h(j(x)) &= h\left( \frac{1}{3} x + \frac{2}{3}\right) \\ \\ &= 3\left(\frac{1}{3}x + \frac{2}{3}\right) -2 \\ \\ &= (x + 2) -2 \\ \\ &= x \end{aligned}[/tex]
And:
[tex]\displaystyle \begin{aligned} j(h(x)) &= j\left(3x-2\right) \\ \\ &= \frac{1}{3}\left( 3x-2\right)+\frac{2}{3} \\ \\ &=\left( x- \frac{2}{3}\right) + \frac{2}{3} \\ \\ &= x \stackrel{\checkmark}{=} x\end{aligned}[/tex]
Ell takes the 17 apples home, and the bakes as many apple pies
as he can. He uses 7 apples in each ple. How many apple pies does
El bake? How many apples are left?
Counters
17:7
10
10
c
Boles
pies
apples are en
Answer:
Tedyxhcj eydyfhxrstetdhsawe
Need help answer plz help
Answer:
BONANA MY NANA
Step-by-step explanation:
What is the answer to it
No question?
Why not add one!
Nadia is ordering cheesecake at a restaurant, and the server tells her that she can have up to five toppings: caramel, whipped cream, butterscotch sauce, strawberries, and hot fudge. Since she cannot decide how many of the toppings she wants, she tells the server to surprise her. If the server randomly chooses which toppings to add, what is the probability that Nadia gets just caramel, butterscotch sauce, strawberries, and hot fudge
Answer:
The probability that Nadia gets just caramel, butterscotch sauce, strawberries, and hot fudge is P = 1/32 = 0.03125
Step-by-step explanation:
There are up to 5 toppings, such that the toppings are:
caramel
whipped cream
butterscotch sauce
strawberries
hot fudge
We want to find the probability that, If the server randomly chooses which toppings to add, she gets just caramel, butterscotch sauce, strawberries, and hot fudge.
First, we need to find the total number of possible combinations.
let's separate them in number of toppings.
0 toppins:
Here is one combination.
1 topping:
here we have one topping and 5 options, so there are 5 different combinations of 1 topping.
2 toppings.
Assuming that each topping can be used only once, for the first topping we have 5 options.
And for the second topping we have 4 options (because one is already used)
The total number of combinations is equal to the product between the number of options for each topping, so here we have:
c = 4*5 = 20 combinations.
But we are counting the permutations, which is equal to n! (where n is the number of toppings, in this case is n = 2), this means that we are differentiating in the case where the first topping is caramel and the second is whipped cream, and the case where the first topping is whipped cream and the second is caramel, to avoid this, we should divide by the number of permutations.
Then the number of different combinations is:
c' = 20/2! = 10
3 toppings.
similarly to the previous case.
for the first topping there are 5 options
for the second there are 4 options
for the third there are 3 options
the total number of different combinations is:
c' = (5*4*3)/(3!) = (5*4*3)/(3*2) = 10
4 toppings:
We can think of this as "the topping that we do not use", so there are only 5 possible toppings to not use, then there are 5 different combinations with 4 toppings.
5 toppings:
Similar to the first case, here is only one combination with 5 toppings.
So the total number of different combinations is:
C = 1 + 5 + 10 + 10 + 5 + 1 = 32
There are 32 different combinations.
And we want to find the probability of getting one particular combination (all of them have the same probability)
Then the probability is the quotient between one and the total number of different combinations.
p = 1/32
The probability that Nadia gets just caramel, butterscotch sauce, strawberries, and hot fudge is P = 1/32 = 0.03125
SCALCET8 4.7.011. Consider the following problem: A farmer with 950 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens
Answer:
For any rectangle, the one with the largest area will be the one whose dimensions are as close to a square as possible.
However, the dividers change the process to find this maximum somewhat.
Letting x represent two sides of the rectangle and the 3 parallel dividers, we have 2x+3x = 5x.
Letting y represent the other two sides of the rectangle, we have 2y.
We know that 2y + 5x = 750.
Solving for y, we first subtract 5x from each side:
2y + 5x - 5x = 750 - 5x
2y = - 5x + 750
Next we divide both sides by 2:
2y/2 = - 5x/2 + 750/2
y = - 2.5x + 375
We know that the area of a rectangle is given by
A = lw, where l is the length and w is the width. In this rectangle, one dimension is x and the other is y, making the area
A = xy
Substituting the expression for y we just found above, we have
A = x (-2.5x+375)
A = - 2.5x² + 375x
This is a quadratic equation, with values a = - 2.5, b = 375 and c = 0.
To find the maximum, we will find the vertex. First we find the axis of symmetry, using the equation
x = - b/2a
x = - 375/2 (-2.5) = - 375/-5 = 75
Substituting this back in place of every x in our area equation, we have
A = - 2.5x² + 375x
A = - 2.5 (75) ² + 375 (75) = - 2.5 (5625) + 28125 = - 14062.5 + 28125 = 14062.5
Step-by-step explanation:
Solve each system by graphing.
Answer:
it is 2 te he
Step-by-step explanation:
ONCE THE 5 6 = 7 10 .. ?% =1 x 7 =2 te he
The average weight of a professional football player in 2009 was pounds. Assume the population standard deviation is pounds. A random sample of professional football players was selected.
Required:
a. Calculate the standard error of the mean.
b. What is the probability that the sample mean will be less than 230 pounds?
c. What is the probability that the sample mean will be more than 231 pounds?
d. What is the probability that the sample mean will be between 248 pounds and 255 pounds?
Answer:
6.286;
0.0165
0.976
0.1995
Step-by-step explanation:
Given that :
Mean, μ = 243. 4
Standard deviation, σ = 35
Sample size, n = 31
1.)
Standard Error
S. E = σ / √n = 35/√31 = 6.286
2.)
P(x < 230) ;
Z = (x - μ) / S.E
P(Z < (230 - 243.4) / 6.286))
P(Z < - 2.132) = 0.0165
3.)
P(x > 231)
P(Z > (231 - 243.4) / 6.286))
P(Z > - 1.973) = 0.976 (area to the right)
4)
P(x < 248)
P(Z < (248 - 243.4) / 6.286))
P(Z < 0.732) = 0.7679
P(x < 255)
P(Z < (255 - 243.4) / 6.286))
P(Z < 1.845) = 0.9674
0.9674 - 0.7679 = 0.1995
Suppose that a customer is purchasing a car. He conducts an experiment in which he puts 10 gallons of gas in the car and drives it until it runs out of gas. He conducts this experiment 15 times on each car and records the number of miles driven.
Car 1 Car 2
214 220
245 221
239 244
224 225
220 258
295 259
Describe each data set, that is determine the shape, center, and spread
i. Sample mean for Car 1
ii. Sample mean for Car 2
Answer:
Kindly check explanation
Step-by-step explanation:
Given the data :
Car 1 Car 2
214 220
245 221
239 244
224 225
220 258
295 259
Ordered data:
Car 1 : 214, 220, 224, 239, 245, 295
Sample mean = ΣX/ n ; n = sample size = 6
Sample mean = 1437 / 6 = 239.5
Median = 1/2(n+1)th term = 1/2(7) = 3.5th term
Median = (3rd + 4th) /2 = (224 + 239) /2 = 231.5
Sample standard deviation; √(Σ(x - xbar)²/n-1 ) = 29.60 (using calculator)
Car 2 : 220, 221, 225, 244, 258, 259
Sample mean = ΣX/ n ; n = sample size = 6
Sample mean = 1427 / 6 = 237.833
Median = 1/2(n+1)th term = 1/2(7) = 3.5th term
Median = (3rd + 4th) /2 = (225 + 244) /2 = 234.5
Sample standard deviation; √(Σ(x - xbar)²/n-1 ) = 18.21 (using calculator)
here's a graph of a linear function write the equation that describes the function express it in slope-intercept form
Answer:
y = 3/4 x - 3
Step-by-step explanation:
the slope of a line is the factor of x in the equation and is expressed as ratio of y/x : defining how many units y changes, when x changes a certain number of units.
in our graph here we can see that when increasing x from e.g. 0 to 4 (the x-axis intercept point, a change of +4), y changes from -3 to 0 (a change of +3).
so, the slope and factor of x is y/x = 3/4
and for x=0 we get y=-3 as y-axis intercept point.
so, the line equation is
y = 3/4 x - 3
cách tính tổng
12+25+45+65+34
12+25+45+65+34
= 181
Must click thanks and mark brainliest
Question 1 of 10
What is the value of n?
144
O A. 36
O B. 23
O C. 95°
D. 590
Answer:
Option C, 95°
Step-by-step explanation:
180-121 = 59
180-144 = 36
third angle of the triangle is, 180-59-36 = 85,
missing angle n = 180-85 = 95°
Answered by GAUTHMATH
Find the surface area of the cylinder and round to the nearest tenth and its recommended that you use pie or 3.14 also the radius is half the diameter
Diameter=d=2ft
Radius=d/2=2/2=1ftHeight=h=2ftWe know
[tex]\boxed{\sf Lateral\:Surface\:Area=2πrh}[/tex]
[tex]\\ \sf\longmapsto Lateral\: Surface\:Area=2\times 3.14\times 2\times 1[/tex]
[tex]\\ \sf\longmapsto Lateral\;Surface\:Area=4(3.14)[/tex]
[tex]\\ \sf\longmapsto Lateral\:Surface\:Area=12.56ft^3[/tex]
[tex]\begin{gathered} {\underline{\boxed{ \rm { \purple{Surface \: \: area \: = \: 2 \: \pi \: r \: h \: + \: 2 \: \pi \: {r}^{2} }}}}}\end{gathered}[/tex]
r represents radius of cylinder.h denotes height of cylinder.Solution[tex]\large{\bf{{{\color{navy}{h \: = \: 2 \: ft. }}}}}[/tex]
[tex]\bf \large \longrightarrow \: \: r \: = \: \frac{Diameter}{2} [/tex]
[tex]\bf \large \longrightarrow \: \: r \: = \: \frac{2}{2} \\ [/tex]
[tex]\bf \large \longrightarrow \: \: r \: = \: \cancel\frac{2}{2} \: ^{1} \\ [/tex]
[tex]\large{\bf{{{\color{navy}{r \: = \: 1 \: ft. \: }}}}}[/tex]
☛ Now , Substuting the values[tex]\bf \hookrightarrow \: \: \: 2 \: \times \: 3.14 \times \: 1 \: ft \: \times \: 2 \: ft \: + \: 2 \: \times \: 3.14 \: \times \: {(1 \: ft)}^{2}[/tex]
[tex]\bf \hookrightarrow \: \: \:6.28 \: ft \: \times \: 2 \: ft\: \: + \: 6.28 \: ft[/tex]
[tex]\bf \hookrightarrow \: \: \:12.56 \: {ft}^{2} \: + \: 6.28 \: ft[/tex]
[tex]\bf \hookrightarrow \: \: \:18.84 \: {ft} \: ^{2} [/tex]
Hence , the surface area of cylinder is 18.84 ft²
Round to the nearest 10 of 18.84 is 18.8
Full-time Ph.D. students receive an average of $12,837 per year. If the average salaries are normally distributed with a standard deviation of $1500, find these probabilities. a. The student makes more than $15,000. b. The student makes between $13,000 and $14,000.
Answer:
a) 0.0749 = 7.49% probability that the student makes more than $15,000.
b) 0.227 = 22.7% probability that the student makes between $13,000 and $14,000.
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Full-time Ph.D. students receive an average of $12,837 per year.
This means that [tex]\mu = 12837[/tex]
Standard deviation of $1500
This means that [tex]\sigma = 1500[/tex]
a. The student makes more than $15,000.
This is 1 subtracted by the p-value of Z when X = 15000.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{15000 - 12837}{1500}[/tex]
[tex]Z = 1.44[/tex]
[tex]Z = 1.44[/tex] has a p-value of 0.9251.
1 - 0.9251 = 0.0749
0.0749 = 7.49% probability that the student makes more than $15,000.
b. The student makes between $13,000 and $14,000.
This is the p-value of Z when X = 14000 subtracted by the p-value of Z when X = 13000.
X = 14000
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{14000 - 12837}{1500}[/tex]
[tex]Z = 0.775[/tex]
[tex]Z = 0.775[/tex] has a p-value of 0.7708.
X = 13000
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{13000 - 12837}{1500}[/tex]
[tex]Z = 0.11[/tex]
[tex]Z = 0.11[/tex] has a p-value of 0.5438.
0.7708 - 0.5438 = 0.227
0.227 = 22.7% probability that the student makes between $13,000 and $14,000.
7.49% of the student makes more than $15,000, while 23.85% of the student makes between $13,000 and $14,000
What is z score?
Z score is used to determine by how many standard deviations the raw score is above or below the mean. It is given by:
z = (raw score - mean) / standard deviation
Given that:
Mean = $12837, standard deviation = $1500
a) For >15000:
z = (15000 - 12837)/1500 = 1.44
P(z > 1.44) = 1 - P(z < 1.44) = 1 - 0.9251 = 0.0749
b) For >13000:
z = (13000 - 12837)/1500 = 0.11
For <14000:
z = (14000 - 12837)/1500 = 0.78
P(0.11 < z < 0.78) = P(z < 0.78) - P(z < 0.11) = 0.7823 - 0.5438 = 0.2385
7.49% of the student makes more than $15,000, while 23.85% of the student makes between $13,000 and $14,000
Find out more on z score at: https://brainly.com/question/25638875
7. 20x + 10 = 110
a. X= 1
b. X= 5
c. x= 12
Answer:
b x=5
Step-by-step explanation:
20x+10=110
20x+10-10=110-10
20x/20=100/20
x=5
Answer:
x=5
Step-by-step explanation:
20x + 10 = 110
Subtract 10 from each side
20x +10-10 = 110-10
20x = 100
divide by 20
20x/20 =100/20
x= 5
Max has 3 fiction books and 6 nonfiction books to donate to the community center. He wants to package them so that there is an equal number of fiction and nonfiction books in each group. He also wants to have as many packages as possible. How many books are in each group?
Answer:
Each group has 1 fiction book and 2 nonfiction book(s).
Simplify your answer as much as possible.
Find the midpoint of the segment with the given endpoints.
(7,10) and (-1,- 8)
Answer:
(3,1) is the midpoint
Step-by-step explanation:
To find the x coordinate of the midpoint, average the x coordinates of the endpoints
(7+-1)/2 = 6/2 =3
To find the y coordinate of the midpoint, average the y coordinates of the endpoints
(10+-8)/2 = 2/2 = 1
(3,1) is the midpoint
Answer:
(3, 1)
Step-by-step explanation:
We can use the formula [ (x1+x2)/2, (y1+y2/2) ] to solve for the midpoint.
7+(-1)/2, 10+(-8)/2
6/2, 2/2
3, 1
Best of Luck!
Is this a function help
Ahmed bought a TV for his room in 2016 for AED 1,500. he decided to sell it in 2020 for AED 900. what is the rate of depreciation when he bought the TV and when he sold it
Answer:
40% depreciation over the 4 years
10% depreciation per year
Step-by-step explanation:
The number of years between buying and selling is:
2020 - 2016 = 4
4 years
The amount of depreciation in the 4 years is:
AED 1,500 - AED 900 = AED 600
The percent depreciation for the 4 years is:
(1500 - 900)/1500 * 100% = 40%
The percent depreciation per year is:
40%/4 = 10%
I want to know the distance
here's the answer to your question
Instructions: Find the missing length indicated.
Answer:
x = 65
Step-by-step explanation:
x = √(25×(25+144))
x = √(25×169)
x = 5×13
x = 65
Answered by GAUTHMATH
lim ₓ→∞ (x+4/x-1)∧x+4
It looks like the limit you want to find is
[tex]\displaystyle \lim_{x\to\infty} \left(\frac{x+4}{x-1}\right)^{x+4}[/tex]
One way to compute this limit relies only on the definition of the constant e and some basic properties of limits. In particular,
[tex]e = \displaystyle\lim_{x\to\infty}\left(1+\frac1x\right)^x[/tex]
The idea is to recast the given limit to make it resemble this definition. The definition contains a fraction with x as its denominator. If we expand the fraction in the given limand, we have a denominator of x - 1. So we rewrite everything in terms of x - 1 :
[tex]\left(\dfrac{x+4}{x-1}\right)^{x+4} = \left(\dfrac{x-1+5}{x-1}\right)^{x-1+5} \\\\ = \left(1+\dfrac5{x-1}\right)^{x-1+5} \\\\ =\left(1+\dfrac5{x-1}\right)^{x-1} \times \left(1+\dfrac5{x-1}\right)^5[/tex]
Now in the first term of this product, we substitute y = (x - 1)/5 :
[tex]\left(\dfrac{x+4}{x-1}\right)^{x+4} = \left(1+\dfrac1y\right)^{5y} \times \left(1+\dfrac5{x-1}\right)^5[/tex]
Then use a property of exponentiation to write this as
[tex]\left(\dfrac{x+4}{x-1}\right)^{x+4} = \left(\left(1+\dfrac1y\right)^y\right)^5 \times \left(1+\dfrac5{x-1}\right)^5[/tex]
In terms of end behavior, (x - 1)/5 and x behave the same way because they both approach ∞ at a proportional rate, so we can essentially y with x. Then by applying some limit properties, we have
[tex]\displaystyle \lim_{x\to\infty} \left(\frac{x+4}{x-1}\right)^{x+4} = \lim_{x\to\infty} \left(\left(1+\dfrac1x\right)^x\right)^5 \times \left(1+\dfrac5{x-1}\right)^5 \\\\ = \lim_{x\to\infty}\left(\left(1+\dfrac1x\right)^x\right)^5 \times \lim_{x\to\infty}\left(1+\dfrac5{x-1}\right)^5 \\\\ =\left(\lim_{x\to\infty}\left(1+\dfrac1x\right)^x\right)^5 \times \left(\lim_{x\to\infty}\left(1+\dfrac5{x-1}\right)\right)^5[/tex]
By definition, the first limit is e and the second limit is 1, so that
[tex]\displaystyle \lim_{x\to\infty} \left(\frac{x+4}{x-1}\right)^{x+4} = e^5\times1^5 = \boxed{e^5}[/tex]
You can also use L'Hopital's rule to compute it. Evaluating the limit "directly" at infinity results in the indeterminate form [tex]1^\infty[/tex].
Rewrite
[tex]\left(\dfrac{x+4}{x-1}\right)^{x+4} = \exp\left((x+4)\ln\dfrac{x+4}{x-1}\right)[/tex]
so that
[tex]\displaystyle \lim_{x\to\infty} \left(\frac{x+4}{x-1}\right)^{x+4} = \lim_{x\to\infty}\exp\left((x+4)\ln\dfrac{x+4}{x-1}\right) \\\\ = \exp\left(\lim_{x\to\infty}(x+4)\ln\dfrac{x+4}{x-1}\right) \\\\ =\exp\left(\lim_{x\to\infty}\frac{\ln\dfrac{x+4}{x-1}}{\dfrac1{x+4}}\right)[/tex]
and now evaluating "directly" at infinity gives the indeterminate form 0/0, making the limit ready for L'Hopital's rule.
We have
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\ln\dfrac{x+4}{x-1}\right] = -\dfrac5{(x-1)^2}\times\dfrac{1}{\frac{x+4}{x-1}} = -\dfrac5{(x-1)(x+4)}[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1{x+4}\right]=-\dfrac1{(x+4)^2}[/tex]
and so
[tex]\displaystyle \exp\left(\lim_{x\to\infty}\frac{\ln\dfrac{x+4}{x-1}}{\dfrac1{x+4}}\right) = \exp\left(\lim_{x\to\infty}\frac{-\dfrac5{(x-1)(x+4)}}{-\dfrac1{(x+4)^2}}\right) \\\\ = \exp\left(5\lim_{x\to\infty}\frac{x+4}{x-1}\right) \\\\ = \exp(5) = \boxed{e^5}[/tex]
The equation of a line is (3)/(5)x+(1)/(3)y=(1)/(15) . The x-intercept of the line is , and its y-intercept is .
bxf-mgii-whr
Step-by-step explanation:
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