Lim x-> vô cùng ((căn bậc ba 3 (3x^3+3x^2+x-1)) -(căn bậc 3 (3x^3-x^2+1)))

Answers

Answer 1

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]


Related Questions

Find an equation of the tangent line to the curve y = 10/(1 + e^-x) at the point (0,5)

Answers

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.

Answers

How much money is that?? In America

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?

Answers

Answer:

1-23.5

2-2

3-5.59

4-0.278

Please help! Suppose f(x)=-8x^7 +8x-6

then f(-x)=________

Answers

Answer:

f(-x) = 8x⁷ - 8x - 6

Step-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 - 6

A 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? *

Answers

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 +, -, + ...

Answers

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

Answers

Answer:

y(3r)

Step-by-step explanation:

a ____________ is sometimes a rectangle

a. parallelogram
b. rhombus
c. square
d. trapezoid ​

Answers

Answer:

parallelogram

Step-by-step explanation:

Parallelogram is the answer,
rhombus, square, and trapezoid are all their own shapes, parallelogram is a four-sided plane rectilinear figure with opposite sides parallel. So basically a parallelogram is a shape with four sides.
Hope this helps.

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

Answers

[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

Answers

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?​

Answers

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?

Answers

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 scores

Compute 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 = 8.6

Compute 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 better

d: -6x + 2 = -15 , help pls fasttt

Answers

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

Answers

Step-by-step explanation:

212 sit-ups = 8 minutes

1 minutes = 212÷8

1 minutes = 26.5

212/8=26.5
Answer=26.5

i need help from anyone i would be happy ​

Answers

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?

Answers

Estimate is KEY word! 975 rounds up to 1,000. 1,000/20=50 so yes his answer is reasonable.

evaluate -(2-^6)(2^3)

Answers

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?​

Answers

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?

Answers

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

Answers

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)

L=q(B-z)
q=L over B-7 put L as the numerator and B-7 as the dominator on the fraction

Question THREE and FOUR

Answers

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​

Answers

Answer:

y≤|x+1|

Step-by-step explanation:

Convert the percent to a decimal
24 1/2%


Answers

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

Answers

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

Answers

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

Answers

4 and 9 is the answer to this question

Find the exact circumference of the circle. please help?

Answers

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?

Answers

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

Answers

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

Answers

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. ❤️❤️

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