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If \($f$\)is bounded and \($\displaystyle\lim_{x \rightarrow c} g(x)$\) exists, then the limit of function\($f(x)g(x)$\)as \($x$\) approaches domain\($c$\)also exists and is equal to \($Lf(c)$\)

Let \($M$\)be an upper bound of \($f$\) on the domain . Since \($\displaystyle\lim_{x \rightarrow c} g(x)$\) exists, there exists a number \($L$\) such that for all \($\epsilon > 0$\) there exists a \($\delta > 0$\) such that for \($x \in A$\) with \($0 < |x - c| < \delta |g(x) - L| < \epsilon\).

Now let \(\epsilon > 0Then $|f(x)g(x) - Lf(x)| = |f(x)||g(x) - L| < M\epsilon\)

for all \(x \in A$ with $0 < |x - c| < \delta$\). So \(\displaystyle\lim_{x \rightarrow c} f(x)g(x)$ exists and is equal to $Lf(c)\)

If \($f$\) is bounded and \($\displaystyle\lim_{x \rightarrow c} g(x)$\) exists, then the limit of \(f(x)g(x)$\)as \($x$\)approaches\($c$\)also exists and is equal to \($Lf(c)$\).

The complete question is:

Let \($f$\) and \($g$\) be functions defined on a domain\($A$\). Prove that if \($f$\) is bounded, and \($\displaystyle\lim_{x \rightarrow c} g(x)$\)exists, then \($\displaystyle\lim_{x \rightarrow c} f(x)g(x)$\) exists.

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Answer:

15a⁸b⁴/5a⁴b=3a⁴b³

...........

Answer:

D:y=5x-8

steps- by- step explanation

Explanation:

Rolling a number less than 7 is guaranteed to happen since all of the outcomes {1,2,3,4,5,6} are all less than 7. So we have a 100% probability here.

The number 100% converts to the decimal 1.00 or just 1 for short.

Answer:

the answer is observation study

Step-by-step explanation:

Answer: 1

Step-by-step explanation: Fractions are the same as division, so what this is really saying is 8 ➗ 8 which is 1.

Hence, The selling price of the article is $48 .

Given that ,

The cost price of article is = $40

And the profit = 20% = 0.2

We have to find,

The selling price of the article.;

Then,

According to the question,

Selling price = Cost price of article + 20% profit

                     = $40 + 0.2 × $40

                     = $40 + $8

                     = $48

Hence, The selling price of the article is $48 .

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Answer: Put equations into slope intercept form: * Add or Subtract the x-term * Divide all ... In this case, the slope is represented by the ratio 2 1 2, or 2 2.

Step-by-step explanation:

Answer:

6,000 total items

Step-by-step explanation:

3*600=1800   (total juice)

2*600=1200   (total fruit)

2*600=1200   (total burritos)

2*600=1200   (total doritos)

1*600=600     (total milk)

1800+1200+1200+1200+600=6000

The given equation,

\(: \: \implies \: \sf{- 4 \times (x - 1) = 2 (x - 4) }\)

\(: \: \implies \: \sf{- 4x + 4 \: = \: 2 (x - 4) }\)

\(: \: \implies \: \sf{- 4x + 4 \: = \: 2 \times (x - 4) }\)

\(: \: \implies \: \sf{- 4x + 4 \: = \: 2x - 8 }\)

\(: \: \implies \: \sf{- 4x - 2x + 4\: = \: - 8 }\)

\(: \: \implies \: \sf{- 4x - 2x\: = \: - 8 - 4}\)

\(: \: \implies \: \sf{- 4x - 2x\: = \: -12}\)

\(: \: \implies \: \sf{ - 6x\: = \: -12}\)

\(: \: \implies \: \sf{ \cancel- \: 6x\: = \: \cancel- \: 12}\)

\(: \: \implies \: \sf{6x\: = \: \: 12}\)

\(: \: \implies \: \sf{x\: = \: \dfrac{12}{6} }\)

\(: \: \implies \: \sf{x\: = \: \cancel\dfrac{12}{6} }\)

\(: \: \implies \: \red{\bf{x\: = \:2 }}\)

Remove the greatest common factor.

Both have a multiple of 2 in them

Both have a multiple of x in them

Therefore, both have a multiple of 2x in them.

Factor 2x out.

2x(y^2 -5)

Use difference of squares to factor the second term.

(2x) (y-5) (y+5) is factored form.

Hope this helps :)

Answer:

2x (y+5) (y-5)

Step-by-step explanation:

2x (\(y^{2}\)-25)

2x (y+5) (y-5)

Answer:

130 and 211

Step-by-step explanation:

15+17= 32

17+32=49

32+49=81

49+81=130

81+130=211

Answer:

130 and 211

Step-by-step explanation:

15,17,32,49,81 _ _

15+2=17+15=32+17=49+32=81

that mean 81+ 49=130

and 81+130= 211

Hence, the answer is 130 and 211.

Answer:

1. x = 5

Remember that all angles add up to 180, and because you know that one of the angles = 90, you can add the other two angles like so.

6x + 15 + 10x - 5 = 90

2.

Angle 1 = 43 degrees

Angle 3 = 32 degrees

The answer would be 18.8

Answer:

Mr. A initially paid approximately N8000 for the land.

Step-by-step explanation:

Step 1: Let's assume Mr. A initially purchased the land for a certain amount, which we'll call "x" in currency units.

Step 2: Mr. A sold the land to Mr. B at a profit of 10%. This means Mr. A sold the land for 110% of the amount he paid (1 + 10/100 = 1.10). Therefore, Mr. A received 1.10x currency units from Mr. B.

Step 3: Mr. B sold the land to Mr. C at a gain of 5%. This means Mr. B sold the land for 105% of the amount he paid (1 + 5/100 = 1.05). Therefore, Mr. B received 1.05 * (1.10x) currency units from Mr. C.

Step 4: According to the given information, Mr. C paid N1240 more for the land than Mr. A paid. This means the difference between what Mr. C paid and what Mr. A paid is N1240. So we have the equation: 1.05 * (1.10x) - x = N1240

Step 5: Simplifying the equation: 1.155x - x = N1240

Step 6: Solving for x: 0.155x = N1240

x = N1240 / 0.155

x ≈ N8000

Therefore, in conclusion, Mr. A initially paid approximately N8000 for the land.

The rate at which the angle between the ladder and the wall changes is 30√75 ft/sec and the rate at which the base of the ladder slides away from the wall at this instant is 13.2 ft/sec.

When a 10-foot ladder slides down a wall at a rate of 3 ft/sec, the rate at which the (acute) angle between the ladder and the wall changes when the foot of the ladder is 5 feet from the wall and the rate at which the base of the ladder slides away from the wall are required to be calculated.

Let's mark the length of the ladder as ‘l’ and the distance of the ladder from the wall as ‘x’.If the ladder has slid x feet down the wall at time t, the triangle formed by the ladder, the wall, and the ground is such that l² = x² + y². The angle is θ, then sin θ = y/l, cos θ = x/l, and tan θ = y/x.

Thus, we have the following equation:(l)² = (x)² + (y)²Differentiate with respect to t.2(l)(dl/dt) = 2(x)(dx/dt) + 2(y)(dy/dt)

Solve for dy/dt when x = 5ft and dx/dt = 3 ft/sec.dy/dt = (l/x)(dx/dt)√[l² - x²]dy/dt = (10/5)(3)√[10² - 5²]dy/dt = 30√75 ft/sec

Now, let's differentiate the equation l² = x² + y² with respect to time:2l(dl/dt) = 2x(dx/dt) + 2y(dy/dt)

We have already calculated the value of dy/dt from the previous step. Substituting, l = 10, x = 5 and dx/dt = 3, we get: 20(dl/dt) = 10(3) + 2(5)(30√75)dl/dt = 13.2 ft/sec

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Answer:

The equation of the line is;

Explanation:

Given that the line has a slope of;

And passes through the point;

Applying the point-slope equation of straight line;

substituting the given values and solving;

Therefore, the equation of the line is;

Answer:the population of brook trout

Step-by-step explanation: Because population is half of a whole

Answer:

okay i'll try to help

Answer:

I'll help too.

Step-by-step explanation:

Answer:

D.  Multiply the input number by 7

Step-by-step explanation:

Make a table in order to help guide our thought on what relates these numbers.

Input  Output

 3         21

 6         42

 8         56

 11         77

 12        84

A first glance suggests the Input is multiplied by 7 to reach the Output.  This can be confirmed with a calculation for each Input value.  None of the other options even come close.

Answer:

83.33%

Step-by-step explanation:

Amount she eats everyday = 150g

Amount she eats now = 25g

By what percent did Amanda decrease the amount of sugar she eats?

Percentage decrease in the amount of sugar she eats = (150 g - 25g) / 150g × 100

= 125g / 150g × 100

= 0.8333333333333 × 100

= 83.33%

Percentage decrease in the amount of sugar she eats = 83.33%

Henry could only see one line since both lines had the same slope, which means that the graphs of both equations will be identical and hence overlap.

Linear equations in a system 3x + 2y = 4, and 9x + 6y = 12

We must demonstrate why Henry could only make out one line when he plotted the equations 3x-2y=4 and 9x-6y=12 on a graph.

Take the provided linear equation system into consideration.

3x - 2y = 4 ................(1)

9x - 6y = 12 ..................(2)

Due to the fact that equation (2) is a multiple of equation (1), 3 (3x - 2y = 4) = 9x - 6y = 12

The slopes of the provided equations are also same.

Difference with regard to x for equation (1) yields,

additional to equation (2),

With regard to x, we can differentiate to get,

The graphs of both equations will overlap since both lines have the same slope and hence have the same appearance on the graph.

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Answer:

135 degree angle

Step-by-step explanation:

1. Triangle Sum Theorem: Calculate the angles of the triangle.

Remember that the sum of all the angles of ANY triangle is 180 degrees.  To find the missing angle of the triangle all we need to do is subtract 180 degrees with 69 and 66 to get the missing angle of the triangle (45 degrees)

2. Angle Sum Theorem?: Note: the degrees of a straight line is 180 degrees

We now know that the missing angle of the triangle is 45 degrees.  We also know that the degree of a straight line will always equal to 180 degrees.  We can find angle x by doing 180 - 45, similarly to when we found the missing angle.

Answer: 123 degree angle

The expected earnings for the festival is  $128,700.

Probability is used to determine the chance that a random event would happen. The chance that the random event occurs lie between 0 and 1 or between 0% and 100%. The more likely it is that an event would happen, the more closer to one or hundred percent, the probability value is.

The chance it does not rain = 100% - chance it rains

100% - 46% = 54%

The expected earnings is a function of the chance that it rains, chance it does not rain and the amount that it would earn if it rains or it does not rain.

Expected earnings = (chance it would rain x amount you would earn if it rains) + chance it does rain x amount you would earn if does not rain)

($200,000 x 0.54) + ($45,000 x 0.46)

108,000 + 20,700 = $128,700

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Answer:

X=6. NBS=110

Step-by-step explanation:

12x+38=17x+8

The part of the quadratic equation that says it's shifting to the right is -3.

What is a quadratic equation?

Any equation that can be written in the standard form where x is an unknown value, a, b, and c are known quantities, and a ≠ 0 is a quadratic equation.

f(x) = ax² bx + c = 0

The given equation g(x) = (x − 3)² is the equation of a parabola.

The general equation of a parabola is given by y = a(x – h)²+ k.

The graph of y=x² is shifted to h units to the right and k units up to produce the graph of y=a(x-h)²+k.

The above equation is similar to the equation given in the question.

Comparing,

a = 1

h = 3

So the given quadratic equation states that a parabola of the form g(x) = x² is shifted to the right by 3 units to give the resulting equation

g(x) = (x − 3)²

Therefore the part of the quadratic equation that says it's shifting to the right is -3.

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Answer: is A

Step-by-step explanation: