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]
Solve the function.
Plz Help!!
[tex]1\pm\sqrt{2}i[/tex]
Step-by-step explanation:
We already know that one of the roots is x = -4 so we can factor this out to get
[tex]x^3+2x^2-5x+12=(x+4)(x^2-2x+3)[/tex]
Using the quadratic equation on the 2nd factor, we find the roots are
[tex]x= \dfrac{2\pm\sqrt{(2)^2 - 4(1)(3)}}{2}=1\pm\sqrt{2}i[/tex]
Use the Distributive property to solve this equation
-2(x-4)+8=2
Answer:
x=7
Step-by-step explanation:
-2(x-4)+8=2(remove brackets by multiplying with -2)
-2x+8+8=2(group like terms and simply)
-2x=2-16(change side change sign)
-2x = -14(divide both sides by -2)
x=7
20 bottles make 1 fleece jacket how much do 200 make
A committee of 3 is to be selected randomly from a group of 3 men and 2 women.
Let X represent the number of women on the committee. Find the probability
distribution of X.
Total number of ways to select 3 people from the 5 total: 5!/(3! (5 - 3)!) = 10
• Number of ways of picking 0 women:
[tex]\dbinom20\times\dbinom33 = 1\times1 = 1[/tex]
• Number of ways of picking 1 woman:
[tex]\dbinom21\times\dbinom32 = 2\times3 = 6[/tex]
• Number of ways of picking 2 women:
[tex]\dbinom22\times\dbinom31 = 1\times3 = 3[/tex]
• Number of ways of picking 3 women: 0, since there are only 2 to choose from
Then X has the probability mass function
[tex]P(X=x) = \begin{cases}\frac1{10}&\text{if x=0}\\\frac6{10}=\frac35&\text{if }x=1\\\frac3{10}&\text{if }x=2\\0&\text{otherwise}\end{cases}[/tex]
A rectangular board is 1400 millimeters long and 900 millimeters wide. What is the area of the board in square meters? Do not round your answer.
=__M
Answer:
1260000 millimeters
Step-by-step explanation:
You multiply 1400mm x 900mm and this is the correct answer.
Hope it helps c:
The area of the rectangular board which is 1400 millimeters long and 900 millimeters wide is 1.26 square meters.
What is the area of a rectangular board?If x and y be the length and width of a rectangular board respectively, then the area of that rectangular board is xy square unit.
How to solve this problem?Since 1000 millimeters = 1 meter.
i.e. 1 millimeter = 1/1000 meter.
Now, length of the board = 1400 millimeters = 1400/1000 meters = 1.4 meters
Width of the board = 900 millimeters = 900/1000 meters = 0.9 meter
Area of the board = 1.4 * 0.9 = 1.26 square meters.
Therefore, the area of the rectangular board which is 1400 millimeters long and 900 millimeters wide is 1.26 square meters.
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Solve: 1/3a^2-1/a=1/6a^2
Step-by-step explanation:
there are two answers for a
Answer:
The ANSWER IS 1/6
Step-by-step explanation:
A class of kindergarteners is making get well cards for children in the houses how many different cards can be made from 2 shapes 4 card colors 3 colors of glitter and 8 colors of markers
Answer:
192
Step-by-step explanation:
Multiply all numbers together.
2 * 4 * 3 * 8 = 192
Answer:
it's 192 because you have to multiply all of the numbers to get 192
Question 8 of 53
How much would $700 be worth after 8 years, if it were invested at 5%
interest compounded continuously? (Use the formula below and round your
answer to the nearest cent.)
A(t) = P•e^rt
A. $5887.12
B. $1044.28
C. $6432.11
D. $38,218.71
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Answer:
B. $1044.28
Step-by-step explanation:
Putting the given numbers into the given formula, we have ...
A(8) = $700•e^(0.05•8) ≈ $1044.28
Given the equation y/x = -6/7 the constant of variation is:
Answer:
[tex]{ \tt{ \frac{y}{x} = - \frac{6}{7} }} \\ { \tt{y = - \frac{6}{7}x }} \\ { \boxed{ \bf{constant = - \frac{6}{7} }}}[/tex]
A candidate running for public office receives 2,000 of the 8, 000 total votes cast from an election. Name the numerator and denominator
of the fraction that represents the candidate's fraction of the total votes.
A test has 6 multiple choice questions , each with four possible answers . How man different answer keys are possible?
Answer:
24
Step-by-step explanation:
6 x 4 = 24
Answer:
Step-by-step explanation:
4⁶=4096
The area of a rectangle is 3,878 square centimeters. If the rectangle has a width of 14 centimeters, what is its length?
Answer:
277 cm
Step-by-step explanation:
[tex]A=l*w\\3,878=l*14\\l=\frac{3,878}{14} \\l=277[/tex]
generate a table of values for the equation y = -4.5x - 0.5. Use values for x from -2 to 2, increment by 1 in each row
Answer:
x = -2: y = 8.5
x = -1: y = 4
x = 0: y = -0.5
x = 1: y = -5
x = 2: y = -9.5
Step-by-step explanation:
We find the numeric values for the function from x = -2 to x = 2.
x = -2:
[tex]y = -4.5(-2) - 0.5 = 9 - 0.5 = 8.5[/tex]
x = -1:
[tex]y = -4.5(-1) - 0.5 = 4.5 - 0.5 = 4[/tex]
x = 0:
[tex]y = -4.5(0) - 0.5 = 0 - 0.5 = -0.5[/tex]
x = 1:
[tex]y = -4.5(1) - 0.5 = -4.5 - 0.5 = -5[/tex]
x = 2:
[tex]y = -4.5(2) - 0.5 = -9 - 0.5 = -9.5[/tex]
Find an odd natural number x such that LCM (x, 40) = 1400
Answer:
175.
Step-by-step explanation:
40 = 2*2*2*5
1400 = 2*2*2*5*5*5*7
So by inspection we have x = 5*5*7 = 175
calculate the area of the following figure.
Answer:
77 cm²
Step-by-step explanation:
The figure can be decomposed into two rectangles.
✔️Area of rectangle 1:
Area of rectangle 1 = length * width
length = 3.5 cm
width = 8 cm
Area of rectangle 1 = 3.5*8 = 28 cm²
✔️Area of rectangle 2:
Area of rectangle 2 = length * width
length = 14 cm
width = 3.5 cm
Area of rectangle 2 = 14*3.5= 49 cm²
✔️Area of the figure = 28 + 49 = 77 cm²
Seth and Ted can paint a room in 5 hours if they work together. If Ted were to work by himself, it would take him 1 hours longer than it would take Seth working by himself. How long would it take Seth to paint the room by himself if Ted calls in sick
Answer:
Seth would need 10 hours to paint the room.
Step-by-step explanation:
Let's define:
S = rate at which Seth works
T = rate at which Ted works
When they work together, the rate is S + T
And we know that when they work together they can pint one room in 5 hours, then we can write:
(S + T)*5 h = 1 room.
We also know that Ted alone would need one hour more than Seth alone.
Then if Seth can paint the room in a time t, we have:
S*t = 1room
and
T¨*(t + 1h) = 1room
Then we have 3 equations:
(S + T)*5 h = 1
S*t = 1
T¨*(t + 1h) = 1
(I removed the "room" part so it is easier to read)
We want to find the value of S.
First, let's isolate one variable (not S) in one of the equations.
We can isolate t in the second one, to get:
t = 1/S
Now we can replace it on the third equation:
T¨*(t + 1h) = 1
T¨*( 1/S + 1h) = 1
Now we need to isolate T in this equation, we will get:
T = 1/( 1/S + 1h)
Now we can replace this in the first equation:
(S + T)*5h = 1
(S + 1/( 1/S + 1h) )*5h = 1
Now we can solve this for S
(S + 1/( 1/S + 1h) )= 1/5h
S + 1/(1/S + 1h) = 1/5h
Now we can multiply both sides by (1/S + 1h)
(1/S + 1h)*S + 1 = (1/5h)*(1/S + 1h)
1 + S*1h + 1 = 1/(S*5h) + 1/5
S*1h + 2 = (1/5h*S) + (1/5)
Now we can multiply both sides by S, to get:
(1h)*S^2 + 2*S = (1/5h) + (1/5)*S
Now we have a quadratic equation:
(1h)*S^2 + 2*S - (1/5)*S - (1/5h) = 0
(1h)*S^2 + (9/5)*S - (1/5h) = 0
The solutions are given by the Bhaskara's formula:
[tex]S = \frac{-(9/5) \pm \sqrt{(9/5)^2 - 4*(1h)*(-1/5h)} }{2*1h} = \frac{-9/5 \pm 2}{2h}[/tex]
Then the solution (we only take te positive one) is:
S = (-9/5 + 2)/2h
S = (-9/5 + 10/5)/2h = (1/5)/2h = 1/10h
Then Seth needs a time t to paint one room:
(1/10h)*t = 1
t = 1/(1/10h) = 10h
So Seth would need 10 hours to paint the room.
Ibrahim wants to give each of his six friends 1 2/3 candy bars. How many candy bars does he need to buy?
Answer:
10 candy bars
Step-by-step explanation:
Since each of his friends needs a candy bar, you need to multiply 6 and 1 ⅔ together.
First, convert 1 ⅔ into an improper fraction: this gives us ⁵⁄₃.
Next, multiply ⁶⁄₁ and ⁵⁄₃ together. To do this, you can visualize 6 as ⁶⁄₁ (which is the same thing). Now you have ⁶⁄₁ x ⁵⁄₃.
Simplify:
The 6 in the numerator and the 3 in the denominator cancel out. This gives us ²⁄₁ x ⁵⁄₁ , which is 2 x 5.
2 x 5 = 10
If y is 2,851, 1% of Y is
Excellent question, but let's rephrase it.
Suppose you have a square of surface area of 2851.
What would be a hundredth of such square?
What would be surface area of that hundredth.
Why hundredth? Because percent denotes hundredths cent is a latin word for hundred. You would usually encounter similar word that describes 100 years: century.
Well it is actually very easy. Just divide 2851 into 100 pieces and look at what is the area of one piece.
[tex]2851/100=285.1=y[/tex]
So a single piece has an area of 258.1.
Hope this helps. :)
Supposed we saved 55$ and we saved 6$ each week what’s the total amount of t we will have after w weeks
Answer:
t=55+6w
Hope This Helps!!!
If an author wanted to inform readers about the different types of advertisements, how would this shape his essay’s content and delivery? He would take an explanatory tone with readers to reveal the different categories of advertisements. He would take a factual tone to show readers why most advertising is exaggerated or even false. He would take a formal, technical tone telling readers about the amount of advertising they see every day without even noticing it half the time. He would take an amusing tone showing readers the worst advertisments ever created.
Answer:
a He would take an explanatory tone with readers to reveal the different categories of advertisements
Step-by-step explanation:
a is formal and is the best tone for the paper
By examining past tournaments, it's possible to calculate the probability that a school wins their first game in the national college basketball tournament. Each school's rank going into the tournament is a strong indicator of their likelihood of winning their first game.
Find the linear regression equation that models this data.
The linear regression model which models the data is :
y = -6.41053X + 103.83509
Obtaining the regression equation could be performed using either the formula method or using technology (excel, calculator, online regression calculators )
Using technology :
• Enter the data into the columns provided ;
The regression equation obtained for the data is : y = -6.41053X + 103.83509
Where ;
Slope = -6.41053
Intercept = 103.83509
X = Rank
y = probability percentage
Hence, from the linear regression equation obtained, we could see that a negative linear relationship exists between rank and probability as implied from it's negative slope value.
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Please help, I really need this
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Answer:
(a) -- the correct choice is highlighted
Step-by-step explanation:
The units of specific heat tell you what quantities make up the ratio.
[tex]\dfrac{390\text{ J}}{1\text{ kg$\cdot^\circ$C}}=\dfrac{-12.0\text{ J}}{0.012\text{ kg}\cdot\Delta T}\\\\\Delta T=\dfrac{-12.0}{0.012\cdot390}\ ^\circ\text{C}\approx-2.56\text{ $^\circ$C}[/tex]
The temperature will decrease by 2.56 C.
Please help! Identify the recursive formula for the sequence 20, 28, 36, 44, . . . .
Answers below in picture:
Option A
Answered by GauthMath if you like please click thanks and comment thanks
30 POINTS PLEASE HELP
Answer:
Answer:
Solution given:
f(x)=5x-3
let
y=f(x)
y=5x-3
interchanging role of x and y
x=5y-3
x+3=5y
y=[tex]\frac{x+3}{5}[/tex]
$o,
f-¹(x)=[tex]\frac{x+3}{5}[/tex]
we conclude that
f-¹(x)≠g(x)
Each pair of function are not inverses.
g(x)=x/5+3
let g(x)=y
y=x/5+3
interchanging role of x and y
x=y/5+3
x-3=y/5
doing crisscrossed multiplication
5(x-3)=y
y=5x-15
g-¹(x)=5x-15
So
g-¹(x)≠f-¹(x)
Each pair of function are not inverses.
Rationalize the denominator of $\displaystyle \frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{7}}$, and write your answer in the form\[
\frac{A\sqrt{2} + B\sqrt{3} + C\sqrt{7} + D\sqrt{E}}{F},
\]where everything is in simplest radical form and the fraction is in lowest terms, and $F$ is positive. What is $A + B + C + D + E + F$?
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Answer:
57
Step-by-step explanation:
Apparently, you want to simplify ...
[tex]\displaystyle \frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{7}}[/tex]
so the denominator is rational. It looks like the form you want is ...
[tex]\dfrac{A\sqrt{2} + B\sqrt{3} + C\sqrt{7} + D\sqrt{E}}{F}[/tex]
And you want to know the sum A+B+C+D+E+F.
__
We can start by multiplying numerator and denominator by a conjugate of the denominator. Then we can multiply numerator and denominator by a conjugate of the resulting denominator.
[tex]\displaystyle =\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{7}}\cdot\frac{\sqrt{2} + \sqrt{3} - \sqrt{7}}{\sqrt{2} + \sqrt{3} - \sqrt{7}}=\frac{\sqrt{2} + \sqrt{3} - \sqrt{7}}{2\sqrt{6}-2}\\\\=\frac{\sqrt{2} + \sqrt{3} - \sqrt{7}}{2\sqrt{6}-2}\cdot\frac{\sqrt{6}+1}{\sqrt{6}+1}=\frac{(1+\sqrt{6})(\sqrt{2}+\sqrt{3}-\sqrt{7})}{10}\\\\=\frac{\sqrt{2}+\sqrt{3}-\sqrt{7}+2\sqrt{3}+3\sqrt{2}-\sqrt{42}}{10}=\frac{4\sqrt{2}+3\sqrt{3}-\sqrt{7}-\sqrt{42}}{10}[/tex]
Comparing this to the desired form we have ...
A = 4, B = 3, C = -1, D = -1, E = 42, F = 10
Then the sum is ...
A +B +C +D +E +F = 4 + 3 -1 -1 +42 +10 = 59 -2 = 57
The sum of interest is 57.
Find the critical numbers (x-values) of the function y equals 2 x to the power of 5 plus 5 x to the power of 4 minus 19. Enter your answers as a comma-separated list. Round answers to 2 decimal places, if necessary.
Answer:
[tex]x=0,x=-2[/tex]
Step-by-step explanation:
From the question we are told that:
[tex]y=2x^5+5x^4-19[/tex]
Generally the equation if differentiated is mathematically given by
[tex]y'=10x^4+20x^3-0[/tex]
Where
y'=0
[tex]10x^4+20x^3=0[/tex]
Factorizing,We have
[tex]x=0,x=-2[/tex]
Therefore
The critical points are
[tex]x=0,x=-2[/tex]
Use the unit circle to find tan 60°.
a. square root 3/3
c. 2 square root 3/3
b. square root 3/2
d. square root 3
Please select the best answer from the choices provided
A
B
C
D
the answer is d ( square root 3 )
tan = oposite / adjacent
tan 60° = √3 / 1
= √3
what is the minimum number of students who must be enrolledin a universitt to guarantee that there are at least two students from the same country
Answer:
Being that there are 195 countries, there would have to be 390 students
Step-by-step explanation:
I need to know what goes in the periods rate total amount and total interest boxes in why
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Answer:
periods: 4rate: 1%total amount: 598.35total interest: 23.35Step-by-step explanation:
"Periods" will be filled with the number of years (1) times the number of periods per year (quarterly = 4). periods = 4
"Rate" is likely filled with the rate of interest divided by the number of periods per year. rate = 4%/4 = 1%
Then the total amount is found by ...
total amount = principal × (1 + rate)^(periods)
= 575 × (1.01^4) = 598.35
The total interest is the difference between this amount and the principal.
total interest = total amount - principal
= 598.35 -575 = 23.35
_____
We believe we have made reasonable assumptions regarding the intent of the columns in the table. Your best bet is to compare this problem to the worked examples in your curriculum materials.
PLEASE HELPPP ASAPPPPP
Answer:
combination 91 ways
Step-by-step explanation:
This is a combination since order doesn't matter
Permutation are when order matter
14 choose 2 order doesnt matter
14*13
--------
2*1
91 different ways
