I believe the given limit is
[tex]\displaystyle \lim_{x\to\infty} \bigg(\sqrt[3]{3x^3+3x^2+x-1} - \sqrt[3]{3x^3-x^2+1}\bigg)[/tex]
Let
[tex]a = 3x^3+3x^2+x-1 \text{ and }b = 3x^3-x^2+1[/tex]
Now rewrite the expression as a difference of cubes:
[tex]a^{1/3}-b^{1/3} = \dfrac{\left(a^{1/3}-b^{1/3}\right)\left(a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}\right)}{\left(a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}\right)} \\\\ = \dfrac{a-b}{a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}}[/tex]
Then
[tex]a-b = (3x^3+3x^2+x-1) - (3x^3-x^2+1) \\\\ = 4x^2+x-2[/tex]
The limit is then equivalent to
[tex]\displaystyle \lim_{x\to\infty} \frac{4x^2+x-2}{a^{2/3}+(ab)^{1/3}+b^{2/3}}[/tex]
From each remaining cube root expression, remove the cubic terms:
[tex]a^{2/3} = \left(3x^3+3x^2+x-1\right)^{2/3} \\\\ = \left(x^3\right)^{2/3} \left(3+\dfrac3x+\dfrac1{x^2}-\dfrac1{x^3}\right)^{2/3} \\\\ = x^2 \left(3+\dfrac3x+\dfrac1{x^2}-\dfrac1{x^3}\right)^{2/3}[/tex]
[tex](ab)^{1/3} = \left((3x^3+3x^2+x-1)(3x^3-x^2+1)\right)^{1/3} \\\\ = \left(\left(x^3\right)^{1/3}\right)^2 \left(\left(3+\dfrac3x+\dfrac1{x^2}-\dfrac1x\right)\left(3-\dfrac1x+\dfrac1{x^3}\right)\right)^{1/3} \\\\ = x^2 \left(9+\dfrac6x-\dfrac1{x^3}+\dfrac4{x^4}+\dfrac1{x^5}-\dfrac1{x^6}\right)^{1/3}[/tex]
[tex]b^{2/3} = \left(3x^3-x^2+1\right)^{2/3} \\\\ = \left(x^3\right)^{2/3} \left(3-\dfrac1x+\dfrac1{x^3}\right)^{2/3} \\\\ = x^2 \left(3-\dfrac1x+\dfrac1{x^3}\right)^{2/3}[/tex]
Now that we see each term in the denominator has a factor of x ², we can eliminate it :
[tex]\displaystyle \lim_{x\to\infty} \frac{4x^2+x-2}{a^{2/3}+(ab)^{1/3}+b^{2/3}} \\\\ = \lim_{x\to\infty} \frac{4x^2+x-2}{x^2 \left(\left(3+\dfrac3x+\dfrac1{x^2}-\dfrac1{x^3}\right)^{2/3} + \left(9+\dfrac6x-\dfrac1{x^3}+\dfrac4{x^4}+\dfrac1{x^5}-\dfrac1{x^6}\right)^{1/3} + \left(3-\dfrac1x+\dfrac1{x^3}\right)^{2/3}\right)}[/tex]
[tex]=\displaystyle \lim_{x\to\infty} \frac{4+\dfrac1x-\dfrac2{x^2}}{\left(3+\dfrac3x+\dfrac1{x^2}-\dfrac1{x^3}\right)^{2/3} + \left(9+\dfrac6x-\dfrac1{x^3}+\dfrac4{x^4}+\dfrac1{x^5}-\dfrac1{x^6}\right)^{1/3} + \left(3-\dfrac1x+\dfrac1{x^3}\right)^{2/3}}[/tex]
As x goes to infinity, each of the 1/x ⁿ terms converge to 0, leaving us with the overall limit,
[tex]\displaystyle \frac{4+0-0}{(3+0+0-0)^{2/3} + (9+0-0+0+0-0)^{1/3} + (3-0+0)^{2/3}} \\\\ = \frac{4}{3^{2/3}+(3^2)^{1/3}+3^{2/3}} \\\\ = \frac{4}{3\cdot 3^{2/3}} = \boxed{\frac{4}{3^{5/3}}}[/tex]
Find an equation of the tangent line to the curve y = 10/(1 + e^-x) at the point (0,5)
Compute the derivative, dy/dx (use the chain rule):
y = 10/(1 + e ⁻ˣ) = 10 (1 + e ⁻ˣ)⁻¹
===> dy/dx = -10 (1 + e ⁻ˣ)⁻² (-e ⁻ˣ) = 10e ⁻ˣ/(1 + e ⁻ˣ)²
Evaluate the derivative for x = 0 to get the slope of the tangent to the curve at (0, 5) :
dy/dx (0) = 10e ⁰/(1 + e ⁰)² = 10/(1 + 1)² = 10/4 = 5/2
Use the point-slope formula to get the equation of the tangent line:
y - 5 = 5/2 (x - 0)
===> y = 5/2 x + 5
Your parents gave you P1,000.00 to cover food
provisions for 3 days for your family. Make a meal
plan and search for the specific amount of the
food you will buy.
1.Round 23.542 to the nearest tenth.
2.Round 1.823 to the nearest whole number.
3.Round 5.592 to the nearest hundredth.
4. What is 0.27845 rounded to 3 decimal places?
Answer:
1-23.5
2-2
3-5.59
4-0.278
Please help! Suppose f(x)=-8x^7 +8x-6
then f(-x)=________
Answer:
f(-x) = 8x⁷ - 8x - 6Step-by-step explanation:
f(x) = -8x⁷ + 8x - 6
To find f(-x), we'll need to plug in -x instead of x.
f(x) = -8x⁷ + 8x - 6
f(-x) = -8(-x)⁷ + 8(-x) - 6
f(-x) = 8x⁷ - 8x - 6A department store pays an interior designer a 25% commission for
sales. How much will the department store have to pay the interior
designer if her weekly sales are $1,875? *
Answer:
25 percent of 1,875 is 468.75
If x-a is a factor of the difference of powers of equal odd exponent of the form x[tex]^{n}[/tex]-a[tex]^{n}[/tex] the signs of the second factor are:
A.
All negative
B.
All positive
C.
Alternating +, -, + ...
Answer:
Option 'D' is correct.
Step-by-step explanation:
Since we have given that
Both the first and second signs are negative,
Let α and β are the roots of the ,
then, we know that the relationship between the zeroes and coefficients of quadratic equations:
Since the product of roots is positive.
So, the signs of the factors will be one positive and one negative.
Hence, Option 'D' is correct.
Translate the following phrase into an algebraic expression. do not simplify.
y multiplied by the product of 3 and r
Answer:
y(3r)
Step-by-step explanation:
a ____________ is sometimes a rectangle
a. parallelogram
b. rhombus
c. square
d. trapezoid
Answer:
parallelogram
Step-by-step explanation:
The equation y = 18x describes the amount of money Spongebob earns as a fry cook, where x is the
number of hours he works and y is the amount of money he earns. Show work. Complete sentence.
The table shows the amount of money Squidward earns as a cashier for different numbers of hours worked.
Squidward’s Earnings
Time (h) 3 5 8 10
Money earned
($) 48 80 128 160
(a) How much money does Spongebob earn per hour?
(b) How much money does Squidward earn per hour?
(c) Who earns more per hour? Justify your answer
[tex]▪▪▪▪▪▪▪▪▪▪▪▪▪ {\huge\mathfrak{Answer}}▪▪▪▪▪▪▪▪▪▪▪▪▪▪[/tex]
a.) How much money does Spongebob earn per hour?
[tex] \$ \: 18[/tex]b.) How much money does Squidward earn per hour?
[tex] \$ \: 16[/tex]c) Who earns more per hour? Justify your answer
Spongebob earns more than that of squidward per hour.
Refer to attachment for Solution ~
Math 7 Review:Question 4
A car used 15 gallons of gasoline when driven 315 miles. Based on this
information, which expression should be used to determine the unit rate of
miles per gallon of gasoline?
Select one:
15 gallons
315 miles
O
315 miles x 15 gallons
O
315 miles
15 gallons
O
315 miles - 15 gallons
Answer: 315/15 gallons
Step-by-step explanation:
You have to divide 315 by 15
The ratio of girls to total students in math class is 13 to 20. Using this ratio, how many boys are in math class?
9514 1404 393
Answer:
girls : boys = 13 : 7
7 boys in the math class
Step-by-step explanation:
We assume that the number of boys is the total number of students less the number of girls. In ratio units, that is ...
girls : boys = girls : (total - girls) = 13 : (20 -13)
girls : boys = 13 : 7
There are 7 boys in the math class.
_____
Additional comment
We are only told the ratio, not any actual numbers. In order to answer the question, we must assume a total for the number of students in the class. It is convenient to assume that total is 20. There is nothing in the problem statement to indicate it is not some multiple of 20. The number of boys will be 7/20 of the total number of students in the math class.
Todd and Garrett began arguing about who did better on their tests, but they couldn't decide who did better given that they took different tests. Todd took a test in Math and earned a 74.6, and Garrett took a test in English and earned a 68.8. Use the fact that all the students' test grades in the Math class had a mean of 70.6 and a standard deviation of 11.9, and all the students' test grades in English had a mean of 63.7 and a standard deviation of 8.6 to answer the following questions.
Required:
a. Calculate the z-score for Todd's test grade.
b. Calculate the z-score for lan's test grade.
c. Which person did relatively better?
Part A
For Todd's class, we have this given info:
mu = 70.6 = population mean of math scoressigma = 11.9 = population standard deviation of math scoresCompute the z score for x = 74.6
z = (x-mu)/sigma
z = (74.6 - 70.6)/(11.9)
z = 0.34 approximately
Side note: Convention usually has z scores rounded to two decimal places. If your teacher instructs otherwise, then go with those instructions of course.
Answer: 0.34======================================================
Part B
For Garret's English class, we have:
mu = 63.7sigma = 8.6Compute the z score for x = 68.8
z = (x-mu)/sigma
z = (68.8 - 63.7)/(8.6)
z = 0.59 approximately
Answer: 0.59======================================================
Part C
Garret has the higher z score, which means that Garret did relatively better to his classmates compared to Todd's performance (in relation to his classmates). The z score is the distance, in units of standard deviation, the score is from the mean. Positive z values are above the mean, while negative z values are below the mean.
Answer: Garret did relatively betterd: -6x + 2 = -15 , help pls fasttt
Pigs love to eat bacon! I wonder why. But, you learn something new everyday! :)
Abby did 212 sit-ups in 8 minutes.Estimate the number of sit-ups she did in 1 minute
Step-by-step explanation:
212 sit-ups = 8 minutes
1 minutes = 212÷8
1 minutes = 26.5
i need help from anyone i would be happy
Answer:
The answers are given in the attachment.
Brenden collected 975 bottle caps for a school fundraiser over 20 days. He estimated that he collected an average of 50 bottle caps each day. Was Brendens answer reasonable?
evaluate -(2-^6)(2^3)
Answer:
The answer = -512
Step-by-step explanation:
To find -(2-^6)(2^3)
We have that -(2-^6) = {-(-2*-2*-2*-2*-2*-2)}=
-(64) which is -64
For (2^3) = (2*2*2) =8
Therefore -(2-^6)(2^) = (-64)(8)
which is = -512
So therefore -(2-^6)(2^3) = -512
If 6x-11y = 0, What will be the value of X:y?
Step-by-step explanation:
value i believe will be for
X= 11/6y
Y=6/11x
If there is 1/3 of quantity A for every 4/9 of quantity B, how much of quantity A is there for 2/5 of quantity B?
Answer:
The ratio equivalent to 5/3 is
For example, if we have a ratio 250 to 150, we can simplify it by dividing both numbers by 10 and then by 5 to get 5 to 3: 250 : 150 25 : 15 5:3 . The ratio 5 to 3 is the simplest form of the ratio 250 to 150, and all three ratios are equivalent.
L = q(B - z) I have to solve for q and I’m not sure how
Answer:
q=L/(B - z)
Step-by-step explanation:
It's actually very easy just make q the subject of the formula by dividing both sides by (B - z)
Question THREE and FOUR
Answer:
-10 out side the radical sign then 2 in the sign
2 out side the radical 3 inside the radical
Step-by-step explanation:
the graph shown is the solution set for which of the following inequalities
y≤x-1
y≤|x+1|
y≤|x|-1
Answer:
y≤|x+1|
Step-by-step explanation:
Convert the percent to a decimal
24 1/2%
Answer:
0.245
Step-by-step explanation:
24/1/2%=24.5%
=0.245
which is irrational?
-12/3 , 9.1293… , 10.55..?
-5/19 , checkmark81 , checkmark30
3.1415.. , -2pi , checkmark16/2
0.5 , - checkmark100 , 4.23
Answer:
3.1415.. , -2pi , checkmark16/2
Step-by-step explanation:
maybe this i dont really know im only here for the points since i need help with math too
Is the function even, odd, or neither?
A. even
B. odd
C. neither
D. both even and odd
it's neither
Step-by-step explanation:
It's not symmetrical
Answer:
C
Step-by-step explanation:
The function is neither
Fill in the blanks below
Find the exact circumference of the circle. please help?
Answer:
3rd option
Step-by-step explanation:
We first need to find the diameter of the circle.
We know this is a square because all sides are marked by a line so this means they are equal. Let find the diagonal of the square to find all parts of the square.
The diagonal of a square is
[tex]a \sqrt{2} [/tex]
where a is the side lengths of the square.
So this means the diagonal of the square is
[tex]18 \sqrt{2} [/tex]
The diagonal is also the diameter of the main circle. The formula for circumference is
[tex]d\pi[/tex]
where d is the diameter.
So the diameter is
[tex](18 \sqrt{2} )\pi[/tex]
or
[tex]18\pi \sqrt{2} [/tex]
The third option is correct
Andre's personal record for the long jump is 16 feet.
Then, he jumps 10% farther.
What is Andre's new personal record for the long jump?
Answer: 17.6
Step-by-step explanation:
16*10/100
160/100
1.6+16
17.6
Answer:
17.6 feet
Step-by-step explanation:
16*10%(or 0.10)= 1.6 feet which is 10% of his 16 foot jump
16+ the 10% increase of 1.6= 17.6 feet as his new record
You buy tomatoes at
$1.29 per pound and peppers at $3.99 per pound
to make salsa. Write an expression for the total
cost of the ingredients. Then find the total cost of
5 pounds of tomatoes and 2 pounds of peppers
Answer:
Expression:
1.20p + 3.99p
Now subsitutute p with the number of pounds given for both tomato and pepper.
1.20(5) + 3.99(2) = $13.98
Look for a pattem to find the value of n
Answer:
It's 0.
Step-by-step explanation:
When you started counting from the left to the right. You will notice that the numbers get increased. At the 3 as a x-value, there would be 0.
This might surely help you ☺️❤️.
Keep studying and rest doubts leave on us. ❤️❤️