(b). Find the values of a and b that make f continuous everywhere.
X^2+6x + 5/x^2-3x-4, X<-1
ax^2- bx +3, -1<=x<=3
2x-a+b, x>=3

First, rewrite

[tex]\dfrac{x^2+5x+4}{x^2-3x+7} = \dfrac{(x^2-3x+7)+(8x-3)}{x^2-3x+7} = 1 + \dfrac{8x-3}{x^2-3x+7}[/tex]

Next, we have

[tex]\dfrac{x^2-3x+7}{8x-3} = \dfrac x8 - \dfrac{21}{64} + \dfrac{385}{512x-192}[/tex]

so we can further rewrite

[tex]\dfrac{x^2+5x+4}{x^2-3x+7} = 1 + \dfrac1{\frac x8 - \frac{21}{64} + \frac{385}{512x-192}}[/tex]

The idea behind all this is to get the limand into a form resembling the limit definition of the constant e, which is

[tex]\displaystyle e = \lim_{x\to\infty} \left(1+\frac1x\right)^x[/tex]

The next step is to make the substitution,

[tex]y = \dfrac x8 - \dfrac{21}{64} + \dfrac{385}{512x-192}[/tex]

Solving for x gives two solutions,

[tex]x = \dfrac{8y+3\pm\sqrt{64y^2+36y-19}}2[/tex]

but we want to have y approaching infinity, just like in the definition for e, which is achieved only by the solution with the positive square root. (You can check for yourself that the other solution instead approaches 3/4.)

So, our limit is equivalent to

[tex]\displaystyle \lim_{y\to\infty} \left(1+\frac1y\right)^{\frac{8y+3+\sqrt{64y^2+36y-19}}2}[/tex]

[tex]\displaystyle \lim_{y\to\infty} \left(1+\frac1y\right)^{\frac{8y+3+8y\sqrt{1+\frac{36}{64y}-\frac{19}{64y^2}}}2}[/tex]

[tex]\displaystyle \lim_{y\to\infty} \left(1+\frac1y\right)^{4y\left(1+\sqrt{1+\frac{36}{64y}-\frac{19}{64y^2}}\right)+\frac32}[/tex]

[tex]\displaystyle \lim_{y\to\infty} \left(\left(1+\frac1y\right)^y\right)^{4\left(1+\sqrt{1+\frac{36}{64y}-\frac{19}{64y^2}}\right)+\frac32}[/tex]

[tex]\displaystyle \lim_{y\to\infty} \left(\left(1+\frac1y\right)^y\right)^{4\left(1+\sqrt{1+\frac{36}{64y}-\frac{19}{64y^2}}\right)} \cdot \lim_{y\to\infty} \left(1+\frac1y\right)^{\frac32}[/tex]

Next, we use the fact that

[tex]\displaystyle \lim_{x\to\infty} f(x)^{g(x)} = \left( \lim_{x\to\infty}f(x)\right)^{\lim\limits_{x\to\infty}g(x)}[/tex]

Now, the "base" limit is e, while the "exponent" limit is

[tex]\displaystyle \lim_{y\to\infty} 4\left(1+\sqrt{1+\frac{36}{64y}-\frac{19}{64y^2}}\right) = 4(1+\sqrt{1+0+0}) = 8[/tex]

and the remaining limit is simply

[tex]\displaystyle \lim_{y\to\infty} \left(1+\frac1y\right)^{\frac32} = (1+0)^{\frac32} = 1[/tex]

So, we end up with

[tex]\displaystyle \lim_{x\to\infty} \left(\frac{x^2+5x+4}{x^2-3x+7}\right)^x = e^8 \cdot 1 = \boxed{e^8}[/tex]

Step-by-step explanation:

In pictures answered .

Answer:

1,200

Step-by-step explanation:

3x2=6 and 6x200=1,200

Answer:

Step-by-step explanation:

Geometric sequence.

Each term is obtained by multiplying 2. So, it is geometric sequence

2 = 1*2

4 = 2*2

8 = 4 *2

16 = 8 *2

Answer:

For 9., the answer is x = -6

For 10., the answer is x = 6

For 11., the answer is x = -7

For 12., the answer is x = -6

Answer:

2 hours 6 minutes

Step-by-step explanation:

40% of 315 is 126 [miles]

126 miles = 126 minutes, since he went at 60 miles per hour [one mile per minute]

126 minutes = 2 hours 6 minutes

So, 2 hours 6 minutes, or 126 minutes.

Answer:

Step-by-step explanation:

The simplest line you can draw with a slope of 4 is y = 4x. Use The equation to make a table

y = 4x

x    1     3     5

y    4    12    20

So I just multiplied 4 by  3 and 5 red graph below shows the three points on the line. (1,4),(3,12),(5,20)

What about a more complex line.

y = 4x + 5

x 1  3  5

y 9 17 25

The green line show (1,9),(3,17),(5,25)

Answer:

I was learning that to but it is way to blurred out and I can not see the text.

You need to make a clear picture if you need answers or no one can help you because we don't know what it say's.

Step-by-step explanation:

Answer:

Step-by-step explanation:

Let x be the number of hours worked

Let C = the charge made by the first company

Let c  = the charge made by the second company

You want to know when the costs will be =.

Equations

C = 376 + 12x

c = 280 + 15x

Solution

Make the two right sides equal

376 + 12x = 280 + 15x            Subtract 12x from both sides

376 = 280 + 15x - 12x             Combine like terms on the right

376 = 280 + 3x                       Subtract 280 from both sides

376 - 280 = 3x                        Combine

96 = 3x                                    Divide both sides by 3

32 = x

Each company would have to work 32 hours to get the same amount to be charged.

C = 376 + 12x

C = 376 + 12*32

C = 376 + 384

C = 760

The amount each would charge is 760

Answer:

mn

Step-by-step explanation:

Answer:

[tex]y = \frac{4}{3} x - 4[/tex]

Hope it helped:)

Hello! The missing number is -1.

Notice that we add 1 every time:

-3+1=-2

-2+1=-1

-1+1=0

Hope this helps!

~Just a felicitous girlie

#HaveAnAwesomeDay

[tex]SilentNature :)[/tex]

Answer:

it's a negative answer Because there's one negative and one positive and if there are your two negatives that makes a positive and it's also the same with two positives

==========================================================

Explanation:

For the original two digit number, we have:

The original number is 10x+y. For instance, if x = 5 and y = 7, then we have 10x+y = 10*5+7 = 57.

This original number 10x+y swaps its digits to 10y+x. Then we add up the two expressions:

(original)+(swapped) = (10x+y)+(10y+x) = 11x+11y = 11(x+y)

This sum is stated to be 154

11(x+y) = 154

x+y = 154/11

x+y = 14

Then we're told that "the ones digit in it (the original number) is 2 less than the tens digit". Meaning that y = x-2 is another equation we can use. We'll plug this into the previous equation we found and solve for x

x+y = 14

x+x-2 = 14 ... replace y with x-2

2x-2 = 14

2x = 14+2

2x = 16

x = 16/2

x = 8 is the tens digit of the original number

y = x-2

y = 8-2

y = 6 is the units digit of the original number

The original number is 10x+y = 10*8+6 = 86

As a check, adding 86+68 = 154 which confirms our answer.

Answer:

Well I'm not sure what your answer choices are but based on the question I can tell you this:

If you have 4 wood and 5 other materials then you have 9 total materials.

So if you have 4 wood then you have a 4/9 chance of choosing wood.

And you have a 5/9 chance of choosing another material

So based on that, you would be more likely to choose another material, that is not wood.

Answer:

  $13,259.30

Step-by-step explanation:

The account balance with compound interest is ...

  A = P(1 +r/n)^(nt)

  A = $12,000(1 +0.05/12)^(12·2) ≈ $13,259.30 . . . . will be owed in 2 years

__

It's a straight compound interest problem. "No payments for 2 years" means the account continues to accrue compounded interest. It's basically a future-value problem, as opposed to an amortization or annuity problem where payments are involved.

Answer:

-0.9³

Step-by-step explanation:

that's the answer hope it helps

Answer:

(-0.9)×(-0.9)×(-0.9)

(-0.9)^3

Answer:

[tex]\frac{x^2+19x-56}{4x^2-32x}[/tex]

Step-by-step explanation:

To add two fractions we need a common denominator. Just by looking at the two, there are no common factors. This means the denominator would be the product of the two: 4x(x-8). In order to obtain this, we multiply the first fraction by (x-8)/(x-8) - 1, and the second by 4x/4x - 1:

[tex]\frac{x+7}{4x} *\frac{x-8}{x-8} +\frac{5}{x-8}*\frac{4x}{4x}[/tex]

We can use the distributive property to simplify:

[tex]\frac{x^2-x-56}{4x(x-8)} +\frac{20x}{4x(x-8)}[/tex]

The sum of the two is:

[tex]\frac{x^2+19x-56}{4x^2-32x}[/tex]

Answer:

last option is correct

............................

9514 1404 393

Answer:

Step-by-step explanation:

The translation of a parent function f(x) by (h, k) gives the function ...

  g(x) = f(x -h) +k . . . . . h units right, k units up

Reflection across the x-axis multiplies the function value by -1.

__

a) The square root function is translated 4 left and 2 down. The equation of the graph is ...

  y = √(x +4) -2

__

b) The equation of the parabola is reflected across the x-axis and translated right 2 and down 3.

  y = -(x -2)^2 -3

Answer:

Recipes 1 and 3 will make the same amount

Answer:

It is 15 but I don't know what the steps are since I took a test on this question and failed to get the right answer.

Answer:

Step-by-step explanation:

you first have to find two points that pass directly through the graph in this case we have (1,4) and (4,-4)

Now we use y2-y1 divided by x2-x1

in this case -4 - 4 (y2-y1) and 4 - 1 (x2-x1)

giving you -8 and 3

so try putting -8/3 as your answer this is what i would put sorry if it is incorrect

Answer:

1/35

Step-by-step explanation:

plz mark brainllest

Answer:

1/7 divided by 5 =

1/35

in decimal form, its:

0.0285714...