What is the domain and range?

[tex]\displaystyle \lim_{x\to 0}\left(\frac{2x\ln \left(1+3x\right)+\sin \left(x\right)\tan \left(3x\right)-2x^3}{1-\cos \left(3x\right)}\right)[/tex]

Both the numerator and denominator approach 0, so this is a candidate for applying L'Hopital's rule. Doing so gives

[tex]\displaystyle \lim_{x\to 0}\left(2\ln(1+3x)+\dfrac{6x}{1+3x}+\cos(x)\tan(3x)+3\sin(x)\sec^2(x)-6x^2}{3\sin(3x)}\right)[/tex]

This again gives an indeterminate form 0/0, but no need to use L'Hopital's rule again just yet. Split up the limit as

[tex]\displaystyle \lim_{x\to0}\frac{2\ln(1+3x)}{3\sin(3x)} + \lim_{x\to0}\frac{6x}{3(1+3x)\sin(3x)} \\\\ + \lim_{x\to0}\frac{\cos(x)\tan(3x)}{3\sin(3x)} + \lim_{x\to0}\frac{3\sin(x)\sec^2(x)}{3\sin(3x)} \\\\ - \lim_{x\to0}\frac{6x^2}{3\sin(3x)}[/tex]

Now recall two well-known limits:

[tex]\displaystyle \lim_{x\to0}\frac{\sin(ax)}{ax}=1\text{ if }a\neq0 \\\\ \lim_{x\to0}\frac{\ln(1+ax)}{ax}=1\text{ if }a\neq0[/tex]

Compute each remaining limit:

[tex]\displaystyle \lim_{x\to0}\frac{2\ln(1+3x)}{3\sin(3x)} = \frac23 \times \lim_{x\to0}\frac{\ln(1+3x)}{3x} \times \lim_{x\to0}\frac{3x}{\sin(3x)} = \frac23[/tex]

[tex]\displaystyle \lim_{x\to0}\frac{6x}{3(1+3x)\sin(3x)} = \frac23 \times \lim_{x\to0}\frac{3x}{\sin(3x)} \times \lim_{x\to0}\frac{1}{1+3x} = \frac23[/tex]

[tex]\displaystyle \lim_{x\to0}\frac{\cos(x)\tan(3x)}{3\sin(3x)} = \frac13 \times \lim_{x\to0}\frac{\cos(x)}{\cos(3x)} = \frac13[/tex]

[tex]\displaystyle \lim_{x\to0}\frac{3\sin(x)\sec^2(x)}{3\sin(3x)} = \frac13 \times \lim_{x\to0}\frac{\sin(x)}x \times \lim_{x\to0}\frac{3x}{\sin(3x)} \times \lim_{x\to0}\sec^2(x) = \frac13[/tex]

[tex]\displaystyle \lim_{x\to0}\frac{6x^2}{3\sin(3x)} = \frac23 \times \lim_{x\to0}x \times \lim_{x\to0}\frac{3x}{\sin(3x)} \times \lim_{x\to0}x = 0[/tex]

So, the original limit has a value of

2/3 + 2/3 + 1/3 + 1/3 - 0 = 2

If U = {1, 2, 3, …, 12} is the universal set, and

A = {1, 2, 6, 10}

B = {3, 6, 9, 10, 11}

C = {1, 2, 4, 7, 11}

then

(1) A U B is the set containing all elements from A and B,

A U B = {1, 2, 3, 6, 9, 10, 11}

(2) A ∩ B is the set of elements that are contained in both A and B,

A ∩ B = {6, 10}

(3) Unfortunately, A ∩ B U C is somewhat ambiguous. It could mean (A ∩ B) U C or A ∩ (B U C ). Then either

(A ∩ B) U C = {6, 10} U {1, 2, 4, 7, 11} = {1, 2, 4, 6, 7, 10, 11}

or

A ∩ (B U C ) = {1, 2, 6, 10} ∩ {1, 2, 3, 4, 6, 7, 9, 10, 11} = {1, 2, 6, 10}

The first interpretation is probably the intended one, since that essentially reads the set operations from left to right.

(4) A' U B is the union of A' and B, where A' is the complement of A, or all elements in U that are not in A. We have

A' = U - A = {3, 4, 5, 7, 8, 9, 11, 12}

and so

A' U B = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

(5) We have

A U C = {1, 2, 4, 6, 7, 10, 11}

so that

(A U C )' = U - (A U C ) = {3, 5, 8, 9, 12}

(6) We have

B' = U - B = {1, 2, 4, 5, 7, 8, 12}

and so

A ∩ B' = {1, 2}

(7) Using the complements found in (4) and (6), we have

A' U B' = {1, 2, 3, 4, 5, 7, 8, 9, 11, 12}

Alternatively, we can use the fact that

A' U B' = (A ∩ B)'

and since we know from (2) that A ∩ B = {6, 10}, we end up with the same result,

(A ∩ B)' = U - (A ∩ B) = {1, 2, 3, 4, 5, 7, 8, 9, 11, 12}

(8) We have

A U B U C = {1, 2, 3, 4, 6, 7, 9, 10, 11}

so that

(A U B U C )' = {5, 8, 12}

Answer:

1) 6n+9

2) -309

Step-by-step explanation:

Let say 2n+1,2n+3,2n+5 the three consecutive odd numbers (n in Z)

1) sum= (2n+1)+(2n+3)+(2n+5=6n+9

2) 2n+5=-101

2n=-106

n=-53

S=6n+9=6*(-53)+9=-309

8 - (3 + b) = b - 9

Opening the brackets changes the symbol to it's opposite symbol, which gives us

8 - 3 - b = b - 9

By using the transposition method, we get

8 - 3 + 9 = b + b

or, 14 = 2b

or, 14/2 = b

or, 7 = b

The value of b is 7.

Step-by-step explanation:

0.0003÷0.0000000009=

333,333.33333333

Answer:

C=78

D=102

Step-by-step explanation:

co-interior angles are sepplementary

Answer:

Step-by-step explanation:

8(2)^2 + 8(2) + 9 = 57

your answer is 57

have a nice semester :)

Answer:

57

Step-by-step explanation:

8(2)^2 + 8(2) + 9

8×4+16+9

32+16+9=57

Answer:

Yes.

Step-by-step explanation:

While there are no labels to be COMPLETLY sure, they both APPEAR to be 90 deg, which checks out as they APPEAR to combine to make a straight angle.

Hope this helps!

Answer:

If you have a calculator with a "Y^X" function, you can raise 27 to the (1/3) power to get the cube root. Or, take the logarithm of 27, divide it by 3, and then find the anti log. That is, find the base 10 log of 27, divide it by three, and then compute 10 ^ that number. (Or of course you can use the base e log and raise e to the number.)

If you are willing to get into complex numbers, you can find that there are actually three cube roots of 27. One of them is 3. To find the other two, use de Moivre's formula (from Wikipedia):

If z is a complex number, written in polar form as

then the n nth roots of z are given by

where k varies over the integer values from 0 to n − 1.

This formula is also sometimes known as de Moivre's formula.

This leads to the two complex roots,

3 x ( (cos 120 degrees) + i x sin(120 degrees) ) and

3 x ( (cos 240 degrees) + i x sin(240 degrees) ).

These work out to be approximately:

3 x (-0.5 + i x 0.866) and

3 x (-0.5 - i x 0.866).

Step-by-step explanation:

Using function concepts, it is found that:

-------------------------------

The BMI B, for a person of weight w(in pounds) and height h(in inches) is given by:

[tex]B = \frac{703w}{h^2}[/tex]

-------------------------------

Item a:

To solve the formula for w, we isolate w in the equation above, thus:

[tex]B = \frac{703w}{h^2}[/tex]

[tex]703w = Bh^2[/tex]

[tex]w = \frac{Bh^2}{703}[/tex]

-------------------------------

Item b:

The weight is:

[tex]w = \frac{21.45(64)^2}{703} = 125[/tex]

The weight is of 125 pounds.

A similar problem is given at https://brainly.com/question/5123313

Explanation:

There's a 9 in the thousandths place. The next digit to the right is 1, which is not 5 or more. So we'll just drop that 1 and that's why the answer is 3.569

Answer:

3.569

Step-by-step explanation:

Rounding is the process of shortening a number to a certain number of digits by finding the closest number with the desired number of digits. When rounding, look one digit to the right of the desired decimal place. In this case, you want the thousandths place. The thousandths place is 3 digits after the decimal, so look at the number next to the 9, the 1. If the number is 5 or greater round up, if it is 4 or less round down. Therefore, 3.5691 should be rounded down to 3.569.

Other examples include,

4.8736 being rounded up to 4.874

6.3168 being rounded up to 6.317

9.7340 being rounded down to 9.734

Answer: the answer is in the screenshot

Step-by-step explanation:

The hypotenuse of each set of legs of a right triangle are:

1. 5 and 6 => √61

2. 4 and 1 => √17

3. 2 and 5 => √29

4. 9 and 12 => 15

In a right triangle, the hypotenuse is the longest side that is opposite to the right angle. The hypotenuse can be determined by finding the square root of the sum of the other two legs.

1. √(5² + 6²) = √61

2. √(4² + 1²) = √17

3. √(2² + 5²) = √29

4. √(9² + 12²) = 15

Learn more about the hypotenuse on:

https://brainly.com/question/2217700

The text of the Pythagorean Theorem The Pythagorean Theorem states: “The sum of the squares of the lengths of the two sides of the right triangle, which are the shortest sides in a right-angled triangle, is equal to the square of the length of the hypotenuse, which is the longest side of the triangle.”

And in symbols: Pythagorean theorem = ( A² + B² = C²)

where ;

● A, B: the two sides of a right triangle ABC.

● C: The hypotenuse of a right triangle is ABC, which is the longest side.

It should be noted that the inverse of the theory is also true; Since the triangle to which the Pythagorean theorem applies, namely: A² + B² = C², is necessarily a right-angled triangle.

Nội dung của Định lý Pitago Định lý Pitago phát biểu: “Tổng bình phương độ dài hai cạnh của tam giác vuông, là cạnh ngắn nhất của tam giác vuông, bằng bình phương độ dài của cạnh huyền, là cạnh dài nhất của tam giác. "

Và trong các ký hiệu: Định lý Pitago = (A² + B² = C²)

ở đâu ;

● A, B: hai cạnh của tam giác vuông ABC.

● C: Cạnh huyền của tam giác vuông ABC là cạnh dài nhất.

Cần lưu ý rằng điều ngược lại của lý thuyết cũng đúng; Vì tam giác áp dụng định lý Pitago, cụ thể là: A² + B² = C², nhất thiết phải là tam giác vuông.

I hope I helped you^_^

Answer:

-5

Step-by-step explanation:

put the 10 on the other side of the = wich will change it to -10 then do -10/2 and x2/2 to get x=-5

Answer:

39.9 grees

Step-by-step explanation:

Since 4.2 kribs = 18 grees

Then,

1 krib = 18/4.2 grees

9.3 kribs = (18/4.2) x 9 3

9.3 kribs = 39.9 grees.

Step-by-step explanation:

Q1 . (f+g)(x) = f(x) + g(x)

=4x-4 +2x^2 -3x

= 2x^2 + x -4

Q2. (f-g)(x) = f(x) - g(x)

= 2x^2−2 - (4x+1)

= 2x^2 -2 -4x -1

= 2x^2 - 4x -3

Q3. h(x)=3x−3 and g(x)=x^2+3

(h.g)(x) = h(x) × g(x)

= (3x-3) × (x^2 + 3)

=3x^3 -3x^2 + 9x -9

Q4.f(x)=x+4 and g(x)=x+6

(f/g)(x) = f(x) ÷ g(x)

= x+4 / x+6

the domain restriction is x>-6

x<-6

x doesn't equal (-6)

If the two function f(x) and g(x) are to be combined, use basic arithmetic rules,

(f + g)(x) = f(x) + g(x)

(f - g)(x) = f(x) - g(x)

[tex](\frac{f}{g})(x)=\frac{f(x)}{g(x)}[/tex]

(f . g)(x) = f(x) × g(x)

Question 1

f(x) = 4x - 4 and g(x) = 2x² - 3x

[tex](f+g)(x)=f(x)+g(x)[/tex]

Therefore, [tex](f+g)(x)=(4x-4)+(2x^2-3x)[/tex]

                                   [tex]=2x^2+x-4[/tex]

Question 2

f(x) = 2x² - 2 and g(x) = 4x + 1

Since, (f - g)(x) = f(x) - g(x)

Therefore, (f - g)(x) = (2x² - 2) - (4x + 1)

                               = 2x² - 2 - 4x - 1

                               = 2x² - 4x - 3

Question 3

h(x) = 3x - 3 and g(x) = x² + 3

Since, (h - g)(x) = h(x) - g(x)

Therefore, (h - g)(x) = (3x - 3) - (x² + 3)

                                = 3x - 3 - x² - 3

                                = -x² + 3x - 6

Question 4

F(x) = x + 4 and g(x) = x + 6

Since, [tex](\frac{f}{g})(x)=\frac{f(x)}{g(x)}[/tex]

Therefore, [tex](\frac{f}{g})(x)=\frac{x+4}{x+6}[/tex]

Learn more,

https://brainly.com/question/7447002

Answer:

10 remainder 0

Step-by-step explanation:

60/8 = 7.5

6/6 = 10 remainder 0

Answer:

x = 8

Step-by-step explanation:

AN = NC,

4x - 7 = 5x -15,

-7 = x -15,

8 = x

is 4.23 less than 4.135

its not its larger

Hey there!

Just do PEMDAS to solve

Parentheses

Exponents

Multiplication

Division

Addition

Subtraction

8 + (7 - 1)^2 / 20 - 9 * 2

= 8 + 6^2/ 20 - 9 * 2

= 8 + 36/10 - 9 * 2

= 8 + 9/5 - 18

= 49/5 - 18

= - 41/5

Therefore, your answer is:

-41/5 ≈ 8 1/5 or 8.2

Good luck on your assignment and enjoy your day!

~Amphitrite1040:)

Answer:

9 sec

Step-by-step explanation:

because it decends 12 feet evry second so the answer is nine sec

Answer:

4-10.5

5-no because the nearest thousandth is not 5 or more

6- Tu is correct because the tenth is 6 which is more than 5 so it can be rounded up to $79

Answer:

Step-by-step explanation:

Answer:

280cm^2

Step-by-step explanation:

[tex]area \: of \: rhombus = \frac{1}{2} \times b \times h \\ = \frac{1}{2} \times 35cm \times 16m \\ = 280 {cm}^{2} [/tex]

Answer:

280

Step-by-step explanation:

→ State the rhombus formula

( p × q ) ÷ 2

→ Substitute in the values

( 35 × 16 ) ÷ 2

→ Simplify

280

Answer:

below

Step-by-step explanation:

this is very simple one I don't study mathematics but also know this

-18<-15<-5<0<9<18

Answer:

LN = 12

Step-by-step explanation:

L______M_____N

LM = x+3

MN = 2x

LN = 4x

M is the midpoint of LN

LN= ?

LM + MN = LN

x + 3 + 2x = 4x

x + 2x + 3 = 4x

3x + 3 = 4x

-3x         -3x

----------------------

3 = 1x

/1    /1

----------------------

3 = x

LN = 4x

= 4 (3)

= 12

LN = 12

Answer:

LN = 12

Step-by-step explanation:

If LM = X+3, MN = 2x, LN = 4x, and M is between L and N, then LN = 12.