limit using L'Hopital's rule . I just want to make sure if my answer is correct or not?
Answers
In order to use L'Hopital's rule, it is necessary to rewrite the limit as the quotient of two functions. Notice that:
[tex]\begin{gathered} 6x^{\sin (4x)}=e^{\ln (6x^{\sin (ex)})^{}} \\ =e^{\sin (4x)\cdot\ln (6x)} \end{gathered}[/tex]Since the exponential function is a continuous function, then:
[tex]\lim _{\text{x}\rightarrow0}e^{\sin (4x)\cdot\ln (6x)}=e^{\lim _{x\rightarrow0}\sin (4x)\cdot\ln (6x)}[/tex]Find the following limit using L'Hopital's rule:
[tex]\lim _{x\rightarrow0}\sin (4x)\cdot\ln (6x)[/tex]Write the function as a fraction:
[tex]\lim _{x\rightarrow0}\frac{\ln (6x)}{(\frac{1}{\sin (4x)})}[/tex]Use L'Hopital's rule to rewrite the limit as the limit of the quotient of the derivatives:
[tex]\begin{gathered} \lim _{x\rightarrow0}\frac{(\frac{1}{x})}{(-\frac{4\cos(4x)}{\sin^2(4x)})}=\lim _{x\rightarrow0}-\frac{\sin ^2(4x)}{4x\cdot\cos (4x)} \\ =\lim _{x\rightarrow0}\sin (4x)\cdot\frac{\sin(4x)}{4x}\cdot\frac{-1}{\cos (4x)} \\ =\lim _{x\rightarrow0}\sin (4x)\cdot\lim _{x\rightarrow0}\frac{\sin(4x)}{4x}\cdot\lim _{x\rightarrow0}\frac{-1}{\cos (4x)} \\ =0\cdot1\cdot-1 \\ =0 \end{gathered}[/tex]Therefore:
[tex]\lim _{x\rightarrow0}6x^{\sin (4x)}=e^0=1[/tex]Related Questions
897 mL to liters ? L
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We have to convert 897 mL to L.
We have to take into account the following equivalency between the two units:
[tex]1L=1000mL[/tex]If we write this as a unit factor, we have:
[tex]\frac{1L}{1000mL}=1[/tex]So we can use this factor to multiply the quantity and change the units without changing the actual measure:
[tex]897mL\cdot(\frac{1L}{1000mL})=\frac{897}{1000}L=0.897L[/tex]NOTE: this unit factors can be used to change multiple units in the same operation.
Answer: 0.897 liters
Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form.Passing through (5,3) with x-intercept 6Write an equation for the line in point-slope form.
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In general, the equations of a line in point-slope form and slope-intercept form are:
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ y=mx+b \end{gathered}[/tex]respectively. Where m is the slope of the line, b is a constant, and (x_1, y_1) is a point on the line.
Thus, the point-slope form of the line described by the problem is:
[tex]y-3=m(x-5)[/tex]We simply need to calculate the slope of the line. For that, we simply require 2 points, we already have (5, 3) and, since the x-intercept is 6, we can deduce that the line goes through (0,6).
Therefore, the slope is:
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-3}{0-5}=-\frac{3}{5}[/tex]Then, the solution is:
[tex]y-3=-\frac{3}{5}(x-5)[/tex]On your own paper, solve the system of equations using substitution and identify the solution.4=16 +=2
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Answer: (4,-2)
Explanation:
Given:
4x=16
x+y=2
We solve for x in 4x=16.
4x=16
Then, divide both sides by 4.
4x/4=16/4
Calculate
x=4
Substitute x=4 into x+y=2.
x+y=2
4+y=2
Simplify and rearrange
y=2-4
Calculate
y=-2
Therefore, x=4 and y=-2.
x^2+3x-6 Find the Discriminat and state how many solutions and type of zero
Answers
Solution
Discriminant
- The formula for the discriminant of a quadratic equation is:
[tex]\begin{gathered} \text{ Given,} \\ ax^2+bx+c \\ \\ \text{ The Discriminant is:} \\ D=b^2-4ac \end{gathered}[/tex]- Applying the formula, we have:
[tex]\begin{gathered} a=1,b=3,c=-6 \\ \\ \therefore D=3^2-4(1)(-6) \\ D=9+24 \\ D=33 \end{gathered}[/tex]- Discriminant is 33
How many solutions
- If the discriminant is > 0, then, the Quadratic equation has 2 solution.
- If the discriminant is = 0, then, the Quadratic equation has 1 solution
- If the discriminant is < 0, then, the Quadratic equation has no real solutions.
- The discriminant is 33 > 0, thus, the Quadratic equation has 2 solutions
Type of zero
- Since there are 2 solutions, then, it has real solutions
Final Answer
OPTION B
The figure below shows a circular lawn. It’s diameter is 72 ft.a.Use 3.14 for n in your calculations,and do not round your answer.Make sure to include the correct units.B.Which measure would be used in finding the amount of fertilizer needed? C.Which measure would be used in finding the amount of tape needed?
Answers
Answers:
a) Area = 4069.44 ft²
Circumference = 226.08 ft
b) Area
c) Circumference
Explanation:
The area of the circular lawn can be calculated as:
[tex]\text{Area}=\pi\cdot r^2[/tex]Where π is 3.14 and r is the radius of the circular lawn.
The radius is half the diameter, so the radius is equal to:
[tex]r=\frac{\text{Diameter}}{2}=\frac{72\text{ ft}}{2}=36\text{ ft}[/tex]Then, the area of the lawn is equal to:
[tex]\begin{gathered} \text{Area = 3.14}\cdot(36ft)^2 \\ \text{Area}=3.14(1296ft^2) \\ \text{Area}=4069.44ft^2 \end{gathered}[/tex]On the other hand, the circumference of the lawn can be calculated as:
[tex]\text{Circumference = 2}\cdot\pi\cdot r[/tex]So, the circumference is equal to:
[tex]\begin{gathered} \text{Circumference = 2}\cdot\text{(3.14)}\cdot(36\text{ ft)} \\ \text{Circumference = }226.08\text{ ft} \end{gathered}[/tex]Finally, the fertilizer is applied to the region, so the measure that you would use to find the amount of fertilizer is the area.
In the same way, to surround the lawn, the measure that would be used to find the amount of tape is the circumference.
So, the answers are:
a) Area = 4069.44 ft²
Circumference = 226.08 ft
b) Area
c) Circumference
What is the distance between points (-8,5) and (7,-3)? (Hint. Use the distance formula)
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SOLUTION:
We are to find the distance between points (-8,5) and (7,-3).
[tex]\begin{gathered} \sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ \\ \text{where x}_1=-8,x_2=7,y_1=5andy_2=-3_{} \end{gathered}[/tex][tex]\begin{gathered} \sqrt[]{(7-(-8))^2+(-3-5)^2} \\ \\ \sqrt[]{(7+8)^2+(-8)^2} \\ \\ \sqrt[]{15^2_{}+(-8)^2} \\ \\ \sqrt[]{225+64} \\ \\ \sqrt[]{289} \\ 17\text{ units} \end{gathered}[/tex]CONCLUSION:
The distance between points (-8,5) and (7,-3) is 17 units.
Can you help me solve the one with mrs Jones I want to know if I’m right
Answers
We know that
• 11 students own a cat.
,• 12 students own a dog.
,• 6 students own both a cat and a dog.
,• 3 students own neither.
First, let's draw a Venn diagram to visualize the problem.
First, we have to fund the total number of students inside the sets Cat and Dog. We need to subtract the number 6 once, otherwise, we'll count it twice.
[tex]11+12-6=17[/tex]Then, we include the students that own neither.
[tex]17+3=20_{}[/tex]Therefore, the total number of students is 20.pls help for brainliest
Answers
Answer:
9 nickels, 8 dimes
Step-by-step explanation:
Let n = number of nickels, and let
d = number of dimes.
n + d = 17--------->.05n + .05d = .85
.05n + .10d = 1.25---->.05n + .10d = 1.25
------------------------
.05d = .40
d = 8, n = 9
Solve for a.69a = = √ [?]Pythagorean Theorem: a2 + b² = c²a
Answers
1) In this problem, we need to find the leg "a". Note that in the Pythagorean Theorem the hypotenuse, the largest leg is opposed to the right angle.
2) So, we can write out the following:
[tex]\begin{gathered} 9^2=6^2+a^2 \\ 81=36+a^2 \\ 81-36=a^2 \\ a^2=45 \\ a=\sqrt{45} \end{gathered}[/tex]Note that we could simplify that, but since the question wants it all under the radical.
Find the mean, median, and mode of the following data. If necessary, round to one more decimal place than the largest number of decimal placesgiven in the dataRate of Fatal Alcohol ImpairedCar Crashes per 100 MillionVehicle Miles of Travel0.29 0.45 0.69 0.53 0.600.37 0.59 0.43 0.61 0.300.54 0.43 0.70 0.40 0.760.38 0.72 0.43 0.60 0.73Copy DataPrevAnswer 2 PointsKeypadKeyboard ShortcutsSeparate multiple answers with commas, if necessary.Selecting a button will replace the entered answer value(s) with the button value. If the button is not selected, the entered answer is used.Mean:Median:Mode:O No mode
Answers
The mean of the given data set is 0.528, the median of the given data set is 0.535 and the mode of the given data set is 0.43.
The given data set is:
0.29, 0.45, 0.69, 0.53, 0.60, 0.37, 0.59, 0.43, 0.61, 0.30, 0.54, 0.43, 0.70, 0.40, 0.76, 0.38, 0.72, 0.43, 0.60, 0.73.
Arranging the given data set in ascending order:
0.29, 0.30, 0.37, 0.38, 0.40, 0.43, 0.43, 0.43, 0.45, 0.53, 0.54, 0.59, 0.60, 0.60, 0.61, 0.69, 0.70, 0.72, 0.73, 0.76.
The mean of the data is given by;
Mean = (Sum of the numbers)/(total numbers)
=(0.29 + 0.30 + 0.37 + 0.38 + 0.40 + 0.43 + 0.43 + 0.43 + 0.45 + 0.53 + 0.54 + 0.59 + 0.60 + 0.60 + 0.61 + 0.69 + 0.70 + 0.72 + 0.73 + 0.76) / 20
= 10.55 / 20
= 0.5275 ≈ 0.528
Median = (0.53 + 0.54) / 2
= 0.535
Mode = 0.43 (0.43 repeated most time)
Thus, the mean of the given data set is 0.528, the median of the given data set is 0.535 and the mode of the given data set is 0.43.
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is √4 a perfect square root
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A perfect square is a value that has a whole number square root. So, if the square root of anumber gives a whole number then the square root is called a perfect square root. The square root of 4, √4 is 2 . 2 is a whole number. So,Its square root is a whole number. Thus, √4 is a perfect square root.
3,400 mL. 34 L I need help which one is bigger
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Answer: 34L is bigger than 3400mL
Given data
3, 400 mL and 34L
Firstly, we need to convert mL to L for unit consistency
1000mL = 1L
3, 400Ml = xL
Cross multiply
1000 * x = 1 x 3400
1000x = 3400
Divide both sides by 1000
1000x / 1000 = 3400/ 1000
x = 3.4 L
Therefore, 3400mL = 3.4L
34L is bigger than 3400mL
A circle has a diameter of 12 m. What is its circumference? Use 3.14 for π, and do not round your answer. Be sure to include the correct unit in your answer. Explanation Check 12 m 4
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To find the circumference of a circle , we use the formula
[tex]C=\pi d[/tex]C = Circumference
d= diameter
[tex]\begin{gathered} C=3.14\times12 \\ C=37.68m^2 \end{gathered}[/tex]im on a bit of a time crunch so please go fast
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Solution
[tex]undefined[/tex][tex]Slope=\frac{24-12}{4-2}=\frac{12}{2}=6[/tex]The final answer
[tex]6\text{ feets}[/tex]-7(x - 2) = 38 - 3x
Answers
We need to solve the following expression:
[tex]-7(x-2)=38-3x[/tex]The first step to solve this problem is to apply the distributive property on the left side of the equation. This is given by the sum of the products. We have:
[tex]\begin{gathered} -7x-2\cdot(-7)=38-3x \\ -7x+14=38-3x \end{gathered}[/tex]We need to change the terms that have "x" from the right to the left. To do that we need to add "3x" on both sides.
[tex]\begin{gathered} -7x+14+3x=38-3x+3x \\ -7x+3x+14=38 \\ -4x+14=38 \end{gathered}[/tex]Then we need to subtract "14" on both sides to isolate the term with x on the left. We have:
[tex]\begin{gathered} -4x+14-14=38-14 \\ -4x=24 \end{gathered}[/tex]Then we need to divide both sides by "-4".
[tex]\begin{gathered} \frac{-4x}{-4}=\frac{24}{-4} \\ x=-6 \end{gathered}[/tex]The value of "x" that solves this equation is -6.
A dog alte 5 feet away from its owner's 50-foot tall house. What equation could be used to find the angle of elevation? 60 8 tan = BO CON 60 bine BO 50 ban () = b
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Answer:
The equation that could be used to find the angle of elevation is;
[tex]\tan \theta=\frac{50}{5}[/tex]Explanation:
Given that the dog sits 5 ft away from its owner's 50 ft tall house.
it can be represented with the drawing below;
Using Trigonometry;
Recall that;
[tex]\tan \theta=\frac{opposite}{adjacent}[/tex]From the diagram;
opposite = 50 ft
adjacent = 5 ft
so, we have;
[tex]\tan \theta=\frac{50}{5}[/tex]Therefore, the equation that could be used to find the angle of elevation is;
[tex]\tan \theta=\frac{50}{5}[/tex]Where are you most likely to build up enough static charge to receive a shock? O A. In a rain forest B. On a concrete sidewalk on a dry day O c. On a nylon carpet in a dry area O d. On a nylon carpet in a hotel by the beach
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Where are you most likely to build up enough static charge to receive a shock?
The correct answer is option C on a nylon carpet in a dry area
Explanation:
As nylon is a good conductor of electricity, the static electricity accumulatedin the body does ntot flow to the ground. Also, when we touch a metal object directly connected to the ground, a difference in electric charges creates a static electric discharge.iAs a reult ,we receive a shock.
John has three parts that he mows each Park measures 2 and 1/2 miles by 2 3/4 miles how many square miles does he know in all
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We will determine the number of square miles he mows as follows:
[tex]A=(2\frac{1}{2})(2\frac{3}{4})\Rightarrow A=(\frac{4}{2}+\frac{1}{2})(\frac{8}{4}+\frac{3}{4})[/tex][tex]\Rightarrow A=(\frac{5}{2})(\frac{11}{4})\Rightarrow A=\frac{55}{8}\Rightarrow A=6\frac{7}{8}\Rightarrow A=6.875[/tex]So, he mows 55/8 square miles for each park.
what is the solution to the equation below? (round your answer to two decimal places.) 3•6^x= 15.57
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Simplify the given expression as shown below
[tex]\begin{gathered} 3*6^x=15.57 \\ \Rightarrow6^x=\frac{15.57}{3}=5.19 \\ \Rightarrow xln(6)=ln(5.19) \end{gathered}[/tex][tex]\Rightarrow x=\frac{ln(5.19)}{ln(6)}\approx0.92[/tex]Thus, the answer is x=0.92, option D.Recall the perimeter of a figure is the distance around figure
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To find the perimeter of the triangle you have to add the length of its sides:
[tex]\begin{gathered} P=a+b+c \\ P=\frac{4}{17}+\frac{5}{17}+\frac{8}{17} \\ P=\frac{4+5+8}{17} \\ P=\frac{17}{17} \\ P=1 \end{gathered}[/tex]The perimeter of the triangle is 1 inch
If angle A is a complement to angle B and the m
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If Angle A is a complement to Angle B, then mIf we know the value of m[tex]\begin{gathered} m\measuredangle a+m\measuredangle b=90 \\ 31+m\measuredangle b=90 \\ m\measuredangle b=90-31 \\ m\measuredangle b=59 \end{gathered}[/tex]The measure of Angle B is 59°,
Given that sino =V48and cotê is negative, determine 0 and coté. Enter the angle O in degrees from the interval [0°, 360). Write the exact answer. Do not round.
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In this problem
we have that
sin(theta) is positive and cos(theta) is negative
That means
the angle theta lies on the II quadrant
Remember that
[tex]\cot (\theta)=\frac{\cos(\theta)}{\sin(\theta)}[/tex]Find out the value of cos(theta)
[tex]\sin ^2(\theta)+\cos ^2(\theta)=1[/tex]substitute the given value
[tex](\frac{\sqrt[]{48}}{8})^2+\cos ^2(\theta)=1[/tex][tex]\cos ^2(\theta)=1-\frac{48}{64}[/tex][tex]\begin{gathered} \cos ^2(\theta)=\frac{16}{64} \\ \cos ^{}(\theta)=-\frac{4}{8} \end{gathered}[/tex]Find out the value of cot(theta)
substitute given values
[tex]\cot (\theta)=-\frac{4}{\sqrt[\square]{48}}[/tex]simplify
[tex]\cot (\theta)=-\frac{4}{\sqrt[\square]{48}}\cdot\frac{\sqrt[]{48}}{\sqrt[]{48}}=-\frac{4\sqrt[]{48}}{48}=-\frac{\sqrt[]{48}}{12}=-\frac{4\sqrt[]{3}}{12}=-\frac{\sqrt[]{3}}{3}[/tex]Find out the angle theta
using a calculator
angle in II quadrant
theta=120 degreesConvert to radians ---->Astrid works in the oil and gas industry near Fort McMurray, Alberta. She earns $28.30/h and makes double time for hours worked beyond 40 each week. Calculate Astrid’s gross income for 45.25 hours worked last week.
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We have that the first 40h are paid at $28.30 each hour, so this first 40h give $1132
[tex]28.30\cdot40=1132[/tex]Now the other 5.25 hours that are beyond 40h the last week are paid at the double of a normal hour this means are paid at $56.60 ($28.30 x 2). Now the 5.25h give 297.15
[tex]56.60\cdot5.25=297.15[/tex]So the total is $1429,15. The answer is $1429,15
A flagpole 3m tall casts a shadow5m long at the same time a nearby hill casts a shadow62m long. How Tall is the hill.
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Explanation:
We are given information about the height of a flagpole and its shadow, and we also have information about the shadow of a hill.
This information is represented in the following diagram:
Where the symbol '?' represents what we are trying to find:
How tall is the hill
We will call this h for reference:
Since these shadows are happening at the same time, the ratio between these values must be the same, this is to say that
3/5 must be equal to h/62:
[tex]\frac{3}{5}=\frac{h}{62}[/tex]From this equation, we solve to find h:
[tex]\begin{gathered} \frac{3}{5}\times62=h \\ \end{gathered}[/tex]Solving the operations the result is:
[tex]\begin{gathered} 0.6\times62=h \\ \downarrow \\ 37.2=h \end{gathered}[/tex]The hill is 37.2 m tall.
Answer:
37.2 m
Noah finds an expression for V(x) that gives the volume of an open-top box in cubic inches in terms of the length x in inches of the cutout squares used to make it. This is the graph Noah gets if he allows x to take on any value between -1 and 5.What is the approximate maximum volume for his box?
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the real world domain would be 0 to 2.5, the maximum would be 15
Me. Gray is going to save money until she can afford to buy a new television that cost $4,189 including tax. If she saves $60 each month
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Given:
Ms. Gray wants to buy a new television that costs $4189.
She saves $60 each month.
To find the numbers of months will be required for her to save enough money to buy the television,
[tex]\frac{4189}{60}=69.8167\approx70[/tex]Verify,
[tex]\begin{gathered} \text{ \$ 60 for 70 months} \\ 60\times70=4200\text{ this is enough money to buy the television that costs \$4189} \end{gathered}[/tex]Answer: option J
Write an equation in slope-intercept form of a line passingthrough the given point and parallel to the given line.3. (-3, -1);2y- 3x= 8
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It's required to find the equation of a line that passes through (-3, -1) and is parallel to the line 2y - 3x = 8.
Solving for y:
[tex]y=\frac{3}{2}x+4[/tex]The slope of this line is 3/2 and the required line must have the same slope because they are parallel.
The point-slope form of a line passing through the point (h, k) and slope m is:
y = m(x - h) + k
Substituting:
[tex]\begin{gathered} y=\frac{3}{2}(x+3)-1 \\ Operate. \\ y=\frac{3}{2}x+\frac{9}{2}-1 \end{gathered}[/tex]Simplifying, the required line is:
[tex]y=\frac{3}{2}x+\frac{7}{2}[/tex]Can you help me with problem #1 I think I remember how to do it but just want to make sure
Answers
1)
[tex]3x-2y=-16[/tex]To convert this equation into slope-intercept form we have to isolate y. Subtracting 3x at both sides of the equation:
[tex]\begin{gathered} 3x-2y-3x=-16-3x \\ -2y=-3x-16 \end{gathered}[/tex]Dividing by -2 at both sides of the equation:
[tex]\begin{gathered} \frac{-2y}{-2}=\frac{-3x-16}{-2} \\ y=\frac{-3x}{-2}+\frac{-16}{-2} \\ y=\frac{3}{2}x+8 \end{gathered}[/tex]use the diagrams to answer the following questions Number 6
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Explanation:
86 is the far arc and 18 is near intercepted arc
x = 1/2(far arc - near arc)
[tex]\begin{gathered} x\text{ = }\frac{1}{2}(86-18) \\ \text{ =34} \end{gathered}[/tex]Answer: x = 34
To find the missing length below, would you use Law of Sines or Law of Cosines?Find the missing length. There are 2 answers, Law of _____ and missing side length.