Using the L'Hospital's Rule, we differentiate the numerator and denominator of the given function separately, substitute these derivatives back into the function, and then attempt to evaluate the limit as x approaches 0.
Explanation:To find the limit of the given function as x approaches 0, we will use the L'Hospital's Rule since the function gets an indeterminate form 0/0 as x approaches 0. L'Hospital's Rule states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.
First, we differentiate the numerator and the denominator separately. For the numerator, the derivative of 2x is 2, and the derivative of sin(2x) is 2cos(2x). So the derivative of the numerator is 2 - 2cos(2x).
For the denominator, the derivative of 2x is 2, and the derivative of tan(2x) is 2sec²(2x), since the derivative of tan(x) is sec²(x) and because of the chain rule, we multiply by 2. So, the derivative of the denominator is 2 - 2sec²(2x).
Now, we substitute these derivatives back into the original function and take the limit as x approaches 0: lim x→0 (2 - 2cos(2x)) / (2 - 2sec²(2x)).
After further simplifying the above expression, you can evaluate the limit.
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The limit is 0.
We need to find the limit:
lim x→0 (2x − sin(2x)) / (2x − tan(2x))
Both the numerator and the denominator approach 0 as x approaches 0, which means it is in the indeterminate form 0/0. This suggests that we can use L'Hôpital's Rule.
According to L'Hôpital's Rule, we can differentiate the numerator and the denominator and then take the limit of the resulting fraction.
First, let's compute the derivative of the numerator:
Numerator: 2x - sin(2x)Derivative: 2 - 2cos(2x)Next, let's compute the derivative of the denominator:
Denominator: 2x - tan(2x)Derivative: 2 - 2sec²(2x)Now, we apply L'Hôpital's Rule:
lim x→0 (2 - 2cos(2x)) / (2 - 2sec²(2x))
As x approaches 0, cos(2x) approaches 1 and sec²(2x) also approaches 1, so:
lim x→0 (2 - 2(1)) / (2 - 2(1)) = 0 / 0
This fraction again gives an indeterminate form. Applying L'Hôpital's Rule a second time will be helpful. We need to differentiate the numerator and the denominator again:
Second derivative of the numerator: d/dx [2 - 2cos(2x)] = 4sin(2x)
Second derivative of the denominator: d/dx [2 - 2sec²(2x)] = -8sec²(2x)tan(2x)
Applying L'Hôpital's Rule once more, we get:
lim x→0 (4sin(2x)) / (-8sec²(2x)tan(2x))
Substitute x = 0:
lim x→0 (4sin(2x)) / (-8sec²(2x)tan(2x)) = 0
Therefore, the limit is 0.
what is 50 divided by 2.5 divided by 0.04
The value of the 50 divided by 2.5 divided by 0.04 is equal to 500.
To calculate 50 divided by 2.5 divided by 0.04, should follow the order of operations,
which is parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right).
However, there are no parentheses or exponents in this expression, so we can proceed directly to multiplication and division:
50 ÷ 2.5 ÷ 0.04
First, divide 50 by 2.5:
50 ÷ 2.5 = 20
Next, divide the result (20) by 0.04:
20 ÷ 0.04 = 500
Therefore, 50 divided by 2.5 divided by 0.04 equals 500.
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A shirt company has 3 designs that can be made with short or long sleeves. there are 6 color patterns available. how many different types of shirts are available from this company?
Answer: 36 Types
Step-by-step explanation:
3 designs and 6 color options so that would be
3 x 6 = 18
they also have the option of either long or short sleeve so that would be
18 x 2 = 36
Solve this expression 5(4x-13).
what is 5/6 - (2/3) = ?
Which unit would you use to measure the amount of water in a swimming pool mL, kg, L or cm
in six years, clem kadiddlehopper will be 5 times as old as he was 26 years ago how old is he now?
The question involves setting up an equation to determine Clem Kadiddlehopper's current age based on his age in six years and 26 years ago. By solving the equation x + 6 = 5(x - 26), we find that Clem is currently 34 years old.
Explanation:Explanation:
The question asks us to determine Clem Kadiddlehopper's current age based on information about his age in future and past years. Let's define Clem's current age as x years old.
In six years, he will be x + 6 years old. It is stated that in six years, Clem will be five times as old as he was 26 years ago. Since 26 years ago he was x - 26 years old, we can write the equation: x + 6 = 5(x - 26).
To find Clem's age, we will solve the equation:
x + 6 = 5(x - 26)x + 6 = 5x - 1304x = 136x = 34Therefore, Clem Kadiddlehopper is currently 34 years old.
solve the system by graphing or using a table 3x+y=5 x-y=7
A biker traveled 20 miles in 4 hours. A unit rate to describe this would be 5 miles per hour. What is another unit rate in this situation?
Lorenzo bought 15 / 16 pounds of ground beef.He wants to make hamburgers that weigh 3 / 16 pounds each.How many hamburgers can he make?
FIrst of all, i recommends you to do a model. Of 15/16 and 3/16 then compare them and take away the same denominators. What would you get? 15 and 3 so you would divide both of them And you will get 5 as your answer, remember to do an explanation on how you got it. :)
List all the factor pairs in the table factors of 25
What is the solution of the proportion 5/x+4=3/x-2
Jay went to an amusement park. The park charges an entrance fee of $10.50 and $4.50 for every ride. Jay spent $46.50 on entrance fees and rides. Which fuction can be used to find the number of rides he went on?
The function to find the number of rides Jay went on is [tex]\( 10.50 + 4.50x = 46.50 \)[/tex]
Let x be the number of rides Jay went on.
The cost of the entrance fee is $10.50, and the cost per ride is $4.50. The total amount Jay spent is $46.50. The equation representing this scenario is:
[tex]\[10.50 + 4.50x = 46.50\][/tex]
To find the number of rides, you can solve for x:
1. Subtract the entrance fee from the total amount:
[tex]\[ 46.50 - 10.50 = 36.00 \][/tex]
2. Divide by the cost per ride:
[tex]\[ \frac{36.00}{4.50} = 8 \][/tex]
So, Jay went on 8 rides.
The function to find the number of rides x is:
[tex]\[f(x) = 10.50 + 4.50x\][/tex]
Setting this equal to the total expenditure ($46.50), the function to find x is:
[tex]\[10.50 + 4.50x = 46.50\][/tex]
Given the following triangle, if c = 17 and a = 15, find the measure of A. 28° 41° 62°
Which represents the first three terms of the sequence: a1 = 2 and an = 4(an-1)2? 2, 16, 1024
4, 16, 1024
2, 16, 32
4, 16, 32
92 divided by 4 use partial and quotients to divide
The formula for glue says to add 50ML
of hardener to each container of resin. How much hardener should be added to 15 containers of resin?
Final answer:
To find out the total hardener needed for 15 containers of resin, multiply 15 by the amount per container, 50ML, which equals 750ML of hardener in total.
Explanation:
To calculate the total amount of hardener needed for 15 containers of resin, we use a simple multiplication. We have been given that each container requires 50ML of hardener. Therefore, we multiply the number of containers by the amount of hardener needed per container.
Calculation:
Number of containers × Amount of hardener per container = Total amount of hardener
15 containers × 50ML/container = 750ML
Thus, for 15 containers of resin, you would need to add a total of 750ML of hardener.
In the last two years, Mari grew 2 1/4 inches, Kim grew 2.4 inches, and Kate grew 2 1/8 inches. Write the amounts they grew in order from least to greatest
Answer:
[tex]\text{Growth in Kate}<\text{Growth in Mari}<\text{Growth in Kim}[/tex]
Step-by-step explanation:
We are given the following information in the question:
Growth in Mari =
[tex]2\displaystyle\frac{1}{4}\text{ inches} = \frac{2\times 4 + 1}{4} = \frac{9}{4} = 2.25\text{ inches}[/tex]
Growth in Kim =
2.4 inches
Growth in Kate =
[tex]2\displaystyle\frac{1}{8}\text{ inches}= \frac{2\times 8 + 1}{8} = \frac{17}{8} = 2.125\text{ inches}[/tex]
Growth in order from least to greatest:
[tex]2.125<2.25<2.4\\\text{Growth in Kate}<\text{Growth in Mari}<\text{Growth in Kim}[/tex]
Find the solution of the given initial value problem y'+2/ty=cost/t^2
A bag contains 20 red tickets and 12 blue tickets. What is the probability of randomly selecting a blue ticket from the bag? A. 4/5 B. 3/11 C. 3/8 D. 1/2
The ray GJ is the angle bisector of angle FGH and Measurement FGH =50 degrease what is the measurement of angle FGJ
Where e is the region that lies inside the cylinder x2 + y2 = 16 and between the planes z = −4 and z = 1?
Final answer:
The region e refers to the inside of a cylinder defined by x² + y² = 16 and between the planes z = -4 and z = 1. It represents a three-dimensional space bounded both radially, by the cylinder, and vertically, by the planes.
Explanation:
The region described in the question is a cylindrical volume in three-dimensional space. To visualize this region, we remember that the cylinder is defined by the equation x2 + y2 = 16, which is the set of points in the xy-plane that are at a distance of 4 from the origin (since the radius is √16 = 4). This cylinder extends indefinitely in the positive and negative z-directions.
The planes z = -4 and z = 1 act as boundaries in the z-direction, trimming the cylinder down so that we're only considering the portion of the cylinder between these two planes. Thus the region e of interest is the set of points inside the cylinder that lie at a height between z = -4 and z = 1.
This type of region is common in problems involving volume calculations, especially when using integral calculus to determine the volume of the shape or when analyzing physical phenomena like fluid flow or electric fields within prescribed bounds.
You buy a pair of jeans at a department store.
What is the percent of sales tax to the nearest tenth of a percent?
The price of the jeans includes a 60% markup. After the discount, what is the percent of markup to the nearest percent?
The price of the jeans before the discount is $24.99. The sales tax rate is 6.5%. The percent of markup after the discount is 27.7%.
The price of the jeans before the discount can be found by dividing the total cost by the markup percentage plus 100% and then multiplying by 100%.
So, the price of the jeans before the discount is (39.99 / 160) * 100 = 24.994375.
Next, to find the sales tax, multiply the subtotal by the sales tax rate.
The sales tax rate can be found by dividing the sales tax amount by the subtotal and then multiplying by 100%.
So, the sales tax rate is (1.95 / 29.99) * 100 = 6.502167.
To find the percent of markup after the discount, divide the difference between the total cost and the price before the discount by the price before the discount and then multiply by 100%.
So, the percent of markup after the discount is ((31.94 - 24.994375) / 24.994375) * 100 = 27.67322.
The probable question may be:
You buy a pair of jeans at a department store.
What is the percent of sales tax to the nearest tenth of a percent?
The price of the jeans includes a 60% markup. After the discount, what is the percent of markup to the nearest percent?
Department store
Jeans =39.99
Discount =-10.00
Subtotal =29.99
Sales tax=1.95
Total = 31.94
If the CPI (consumer price index) of 141 increased 8.3 % this year. What was the end of year CPI?
If the CPI increased by 8.3%, therefore this means that there was a fractional increase of 0.083. So amount increased is:
CPI increase = 141 * 0.083 = 11.703
Therefore the end of year CPI is the sum of the original and the increase:
end of year CPI = 141 + 11.703 = 152.703
Answer:
152.703
In a right triangle, he measure of one acute angle is 3 times the sum of the other acute angle and 8, Find the measure of angle 1 and 2
The measure of angles 1 and 2 are 16.5 and 73.5 respectively.
The sum of the two acute angles is complementary.
Let the complementary acute angles be x and y
Since they form a right angle triangle, hence;
x + y = 90 ............................. 1
If the measure of one acute angle is 3 times the sum of the other acute angle and 8, this is expressed as x = 3(y + 8)
Substitute the value of x into the equation 1 as shown:
3(y+8) + y = 90
3y + 24 + y = 90
4y + 24 = 90
4y = 90 - 24
4y = 66
y = 66/4
y = 16.5
Get the other acute angle:
x = 90 - y
x = 90 - 16.5
x = 73.5 degrees
Hence the measure of angles 1 and 2 are 16.5 and 73.5 respectively.
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2.1, -2.1, 2 and 2 1/11, -2 from least to greatest show work
Evaluate the expression. [6(22 + 4)2 − 2(19 + 3)2]0
Answer:
the anwser is 1 so B)Step-by-step explanation:i just gottt it right on usa test prep
Rick works off commission. He earns 10 percent of all manufacturing equipment he sells. If he made a sale of $9,000, how much was his commission?
commission would be 9000 * 0.10 = 900
he would make $900
a person leaves their house to go to the store. it takes 15 minutes at 12 mph to get to the store. They return home at 31 mph. What is the average rate of the speed for the entire trip?
Q: Write the slope intercept form of the equation of the line through the given point with the given slope.
Both problems; provide an explanation, please! Thank you!
A science fair poster is a rectangle
3
feet long and
2
feet wide. What is the area of the poster in square inches?