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artcher [175]
4 years ago
10

Determine whether the given limit leads to a determinate or indeterminate form. HINT [See Example 2.]

Mathematics
1 answer:
Musya8 [376]4 years ago
8 0

Answer:

The limit leads to a determinate form.

\lim_{x \to \infty} \frac{2}{-x+3} = 0

Step-by-step explanation:

The following are indeterminate forms.

\frac{0}{0} \ and \ \frac{\infty}{\infty}

Given the limit of a function \lim_{x \to \infty} \frac{2}{-x+3}, to show if the given limit is determinate or indeterminate form, we will need to substitute the value of -\infty into the function as shown,

\lim_{x \to \infty} \frac{2}{-x+3}\\= \frac{2}{-(-\infty)+3}\\= \frac{2}{\infty+3}\\=  \frac{2}{\infty}\\\\Generally, \ \frac{a}{\infty} =0

where a is any constant, therefore \frac{2}{\infty} = 0

Since we are able to get a finite value i.e 0, this shows that the limit does exist and leads to a determinate form

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Find the area of the region that is inside r=3cos(theta) and outside r=2-cos(theta). Sketch the curves.​
raketka [301]

Answer:

3√3

Step-by-step explanation:

r = 3 cos θ

r = 2 - cos θ

First, find the intersections.

3 cos θ = 2 - cos θ

4 cos θ = 2

cos θ = 1/2

θ = -π/3, π/3

We want the area inside the first curve and outside the second curve.  So R = 3 cos θ and r = 2 - cos θ, such that R > r.

Now that we have the limits, we can integrate.

A = ∫ ½ (R² - r²) dθ

A = ∫ ½ ((3 cos θ)² - (2 - cos θ)²) dθ

A = ∫ ½ (9 cos² θ - (4 - 4 cos θ + cos² θ)) dθ

A = ∫ ½ (9 cos² θ - 4 + 4 cos θ - cos² θ) dθ

A = ∫ ½ (8 cos² θ + 4 cos θ - 4) dθ

A = ∫ (4 cos² θ + 2 cos θ - 2) dθ

Using power reduction formula:

A = ∫ (2 + 2 cos(2θ) + 2 cos θ - 2) dθ

A = ∫ (2 cos(2θ) + 2 cos θ) dθ

Integrating:

A = (sin (2θ) + 2 sin θ) |-π/3 to π/3

A = (sin (2π/3) + 2 sin(π/3)) - (sin (-2π/3) + 2 sin(-π/3))

A = (½√3 + √3) - (-½√3 - √3)

A = 1.5√3 - (-1.5√3)

A = 3√3

The area inside of r = 3 cos θ and outside of r = 2 - cos θ is 3√3.

The graph of the curves is:

desmos.com/calculator/541zniwefe

5 0
3 years ago
A triangle has angles measuring 45°, 55°, and 80°. It is dilated by a scale factor of 2. What are the angle measures, in order f
lbvjy [14]

Answer:

45°,55° and 80°.

Step-by-step explanation:

Dilations preserve angle measures, regardless of the scale factor. Therefore, the angle measures of the image will be the same of that of the pre-image, so our answer is still 45 °, 55 °, and 80 °.

to learn more about angles

visit;. brainly.com/question/28493506

#SPJ9

3 0
1 year ago
Read 2 more answers
Identify the area of the figure rounded to the nearest tenth
gtnhenbr [62]

Answer:

71.4 in ^2

Step-by-step explanation:

area of the circles = πr^2

3.14(4x4) = 50.24 but you only have half so divide by 2

= 25.12

second circle

3.14(2x2)= 13.56 again you only have half so divide by 2

= 6.28

square

5x8 = 40

add them all up

25.12+6.28+40 = 71.4

5 0
3 years ago
2x-6 = 3x+1-X-7<br> Solve the equation.
svet-max [94.6K]

Answer: x = all real numbers

Step-by-step explanation:

2x -6 = 3x + 1 - x -7

First let’s simplify

2x -6 = 2x -6

Then add 6 on each side

2x = 2x

X can equal any real number

3 0
3 years ago
Read 2 more answers
Which of the following is the surface area of the right cylinder below?
arlik [135]

Answer: OPTION C

Step-by-step explanation:

You need to use this formula for calculate the surface area of the right cylinder:

SA=2\pi r^2+2\pi rh

Where "r" is the radius and "h" is the height.

You can identify in the figure that:

r=6units\\h=14units

Knowing this, you can substitute these values into the formula SA=2\pi r^2+2\pi rh, therefore you get that the surface area of this right cylinder is:

SA=2\pi (6units)^2+2\pi (6units)(14units)

SA=240\pi\ units^2

8 0
4 years ago
Read 2 more answers
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