It looks like the limit you want to compute is
[tex]\displaystyle L = \lim_{x\to0}\left(\frac x{\tan(x)}\right)^{\cot^2(x)}[/tex]
Rewrite the limand with an exponential and logarithm:
[tex]\left(\dfrac{x}{\tan(x)}\right)^{\cot^2(x)} = \exp\left(\cot^2(x) \ln\left(\dfrac{x}{\tan(x)}\right)\right) = \exp\left(\dfrac{\ln\left(\dfrac{x}{\tan(x)}\right)}{\tan^2(x)}\right)[/tex]
Now, since the exponential function is continuous at 0, we can write
[tex]\displaystyle L = \lim_{x\to0} \exp\left(\dfrac{\ln\left(\dfrac{x}{\tan(x)}\right)}{\tan^2(x)}\right) = \exp\left(\lim_{x\to0}\dfrac{\ln\left(\dfrac{x}{\tan(x)}\right)}{\tan^2(x)}\right)[/tex]
Let M denote the remaining limit.
We have [tex]\dfrac x{\tan(x)}\to1[/tex] as [tex]x\to0[/tex], so [tex]\ln\left(\dfrac x{\tan(x)}\right)\to0[/tex] and [tex]\tan^2(x)\to0[/tex]. Apply L'Hopital's rule:
[tex]\displaystyle M = \lim_{x\to0}\dfrac{\ln\left(\dfrac{x}{\tan(x)}\right)}{\tan^2(x)} \\\\ M = \lim_{x\to0}\dfrac{\dfrac{\tan(x)-x\sec^2(x)}{\tan^2(x)}\times\dfrac{\tan(x)}{x}}{2\tan(x)\sec^2(x)}[/tex]
Simplify and rewrite this in terms of sin and cos :
[tex]\displaystyle M = \lim_{x\to0} \dfrac{\dfrac{\tan(x)-x\sec^2(x)}{\tan^2(x)}\times\dfrac{\tan(x)}{x}}{2\tan(x)\sec^2(x)} \\\\ M= \lim_{x\to0}\dfrac{\sin(x)\cos^3(x) - x\cos^2(x)}{2x\sin^2(x)}[/tex]
As [tex]x\to0[/tex], we get another 0/0 indeterminate form. Apply L'Hopital's rule again:
[tex]\displaystyle M = \lim_{x\to0} \frac{\sin(x)\cos^3(x) - x\cos^2(x)}{2x\sin^2(x)} \\\\ M = \lim_{x\to0} \frac{\cos^4(x) - 3\sin^2(x)\cos^2(x) - \cos^2(x) + 2x\cos(x)\sin(x)}{2\sin^2(x)+4x\sin(x)\cos(x)}[/tex]
Recall the double angle identity for sin:
sin(2x) = 2 sin(x) cos(x)
Also, in the numerator we have
cos⁴(x) - cos²(x) = cos²(x) (cos²(x) - 1) = - cos²(x) sin²(x) = -1/4 sin²(2x)
So we can simplify M as
[tex]\displaystyle M = \lim_{x\to0} \frac{x\sin(2x) - \sin^2(2x)}{2\sin^2(x)+2x\sin(2x)}[/tex]
This again yields 0/0. Apply L'Hopital's rule again:
[tex]\displaystyle M = \lim_{x\to0} \frac{\sin(2x)+2x\cos(2x)-4\sin(2x)\cos(2x)}{2\sin(2x)+4x\cos(2x)+4\sin(x)\cos(x)} \\\\ M = \lim_{x\to0} \frac{\sin(2x) + 2x\cos(2x) - 2\sin(4x)}{4\sin(2x)+4x\cos(2x)}[/tex]
Once again, this gives 0/0. Apply L'Hopital's rule one last time:
[tex]\displaystyle M = \lim_{x\to0}\frac{2\cos(2x)+2\cos(2x)-4x\sin(2x)-8\cos(4x)}{8\cos(2x)+4\cos(2x)-8x\sin(2x)} \\\\ M = \lim_{x\to0} \frac{4\cos(2x)-4x\sin(2x)-8\cos(4x)}{12\cos(2x)-8x\sin(2x)}[/tex]
Now as [tex]x\to0[/tex], the terms containing x and sin(nx) all go to 0, and we're left with
[tex]M = \dfrac{4-8}{12} = -\dfrac13[/tex]
Then the original limit is
[tex]L = \exp(M) = e^{-1/3} = \boxed{\dfrac1{\sqrt[3]{e}}}[/tex]
least multiple of 2983548
Answer:
it must be 2983548×2............
Solve 2/5 x + 1/2 = 5/6
Answer:
5/6
Step-by-step explanation:
I'm assuming the x in your equation is used as a variable.
5/6-1/2= 2/6=1/3
2/5x=1/3
x=5/6
Use the Distributive Property to multiply. 2(b – 7)
Answer:
2b - 14
Step-by-step explanation:
2(b – 7)
2b - 14
Answer:
[tex]2b-14[/tex]
Step-by-step explanation:
Using the distributive property, [tex]2(b-7)[/tex] can be written as [tex](2\cdot b)-(2\cdot7)[/tex] which solved/simplified is [tex]2b-14[/tex]
Man, it's late! Well, hope this helped you out and answered your question!
Complete the equation. (x-2)(x^2 + 2x + 4) = ( , )
Answer:
Step-by-step explanation:
x(x² + 2x + 4)-2((x² + 2x + 4)
x³+2x²+4x -2x²-4x-8
x³-8
A number is multiplied by 16, and that product is added to 4. The sum is equal to the product of 4 and 9. Find the number.
Answer:
The number is 2
Step-by-step explanation:
x * 16 + 4 = 9*4
x * 16 + 4 = 36
x * 16 + 4 - 4 = 36 - 4
x * 16 = 32
x * 16 / 16 = 32 / 16
x = 2
Answer:
The number is 2
Step-by-step explanation:
let the number be "a"
16a+4=4×9
16a+4=36
16a=36-4
16a=32
a=2
Find the area of each shaded region. Show your reasoning.
maybe we could answer if there waas a pic=]
Barry is checking his bank account for a few transactions that he completed. He had $230
in his account to start and then contributed a check for $360. He paid his water, electric and
cable bill, which were $100, $46 and $107. How much money is in his bank account?
A. $253
B. $383
C. $590
D. $337
Answer:
D.
Step-by-step explanation:
$230 + $360 = $590
$590 - $253* = $337
$337
*( total sum of all the bills )
Add or Subtract: show all written work 30 POINTS!!
Answer:
I hope it helped U
stay safe stay happy
PLEASE HELP NOW
write and solve an equation to find the value of X. please explain in details and show your work!
Answer:
Solution=96°+x+35°
=x+96°+35°
=x=96°-35°
=x=61°
I need help ASAP whit this question
Answer:
B. 7
Step-by-step explanation:
The original box must be 1 by 1 by 1.
So then the new box would be 2 by 2 by 2.
2×2×2=8
8-1=7
I hope this helps!
pls ❤ and mark brainliest pls!
Let A = {a, b, c, d}, B = {a, c, d, e}. Let f : A → R and g : B → R such that f = {(a, 1),(b, 2),(c, 2),(d, 4)} and g = {(a, 0),(c, 4),(d, 8),(e, 0)}. State the domain and find: f/g
Answers:
Domain = {c, d}f/g = { (c, 1/2), (d, 1/2) }============================================================
Explanation:
The domain of f: {a,b,c,d} which is drawn from set A.
The domain of g: {a,c,d,e} which is drawn from set B
So far, we can see the possible inputs are: a,b,c,d,e
Let's go through each lowercase letter of the given sets
If 'a' is plugged into f/g, then we'll have 0 in the denominator due to the ordered pair (a,0) in function g. This causes a division by zero error so the input 'a' is not allowed in the domain of f/g.If we tried b, then it would work for function f, but it doesn't work for function g. If we tried to compute g(b), then the result is undefined. This is because the lowercase b input is not found in set uppercase B which is where function g selects its inputs from. So b is eliminated.The input c does work however. f(c) = 2 because of the ordered pair (c,2) in the function f. Similarly, g(c) = 4. This means f/g = 2/4 = 1/2 when the input is c. This shows that c is a valid input in the domain of f/g.The input d is pretty much the same story as input c. Moreover, the input d leads to the same output as c does. This time we have f/g = 4/8 = 1/2 when the input is d. Therefore, d is another valid input in the domain of f/g.The input e is not valid because (e,0) will lead to a division by zero error, as it's part of the g(x) function. So we must eliminate 'e' as a valid option in the domain. Even if we didn't have a division by zero error, we still have the fact that 'e' is not in the domain of the first function f. This is another reason why f/g is not defined when the input is 'e'.To summarize, only inputs c and d are valid in the domain.
The domain is the set {c, d}
Both inputs in the domain lead to the output 1/2. This means the range of f/g is {1/2}. Despite having just one element inside, we must use curly braces to indicate we have a set.
The function f/g is { (c, 1/2), (d, 1/2) }
We simply list all of the (x,y) ordered pairs to describe the function. In this case, there are only two such pairs.
PLEASEEE HELP!!!!!!!
Temperatures in Verkhoyansk, Russia have been recorded as low as -90°F and
as high as 98°F. What is the difference between the highest and lowest recorded
temperatures in Verkhoyansk?
How many cubes of side 2cm can be fitted into a box 12 cm long, 8cm wide and 4cm high?
Answer:
there are 48cubes fetted into a box
this isn’t a question but to everyone who’s actually good at math thank you so much for taking the time to answer these questions.
Answerurdsrd d e
Step-by-step explanation:
Which of the following fractions converts to a repeating decimals
Step-by-step explanation:
The answer is a option.
hope it helps
The following dotplot shows the centuries during which the 11 castles whose ruins remain in Somerset, England were constructed. Each dot represents a different castle.
According to the 1.5.IQR rule for outliers, how many high outliers are there in the data set?
Choose 1 answer:
Taking into account the rule 1.5 IQR, there aren't high outliers in the displayed data set.
Calculation of the Interquartile Range and the rule 1.5 IQR.
To calculate the Interquartile Range IQR, you must subtract the value of the first quartile Q1, to the third quartile Q3, as you can see in the formula below:
IQR = Q3 - Q1Using the five-number summary, you can replace the formula:
IQR = 17 - 13IQR = 4Before to apply the rule 1.5 IQR, you must multiply the IQR by 1.5:
4 * 1.5 = 6Now, to apply the rule you must add the value obtained (6), to the third quartile (17), if any data is over this, theoretically it's an outlier:
17 + 6 = 23 (As you do not have any data over this value, it is identified that there are no outliers in the upper section).Finally, you must subtract the value obtained (6) to the first quartile (13), if any data is under this, theoretically it's an outlier:
13 - 6 = 7 (Since you don't have any castle data built before this century, there wouldn't be any theoretically outliers either).By this reason, using the rule of 1.5 IQR, you can see there aren't outliers in the displayed data set.
If you want to study more exercises using data sets, you can see this link https://brainly.com/question/16445367?referrer=searchResults
Answer:
The answer is 0
Step-by-step explanation:
It said it was right on khan!
mathematics geniuses where are you ? help me answer this question please.
Step-by-step explanation:
There is 7 blocks that measure is 12.0-12.2
11 blocks that measure 12.2 to 12.4
18 blocks that measure 12.4 to 12.6
10 blocks that measure 12.6-12.8
4 blocks that measure 12.8 to 13.0
We can't find a mean by with Intervals so let find the midpoint of each interval,
12.1, 12.3, 12.5, 12.7, 12.9.
12.1 occurs 7 times so that equal 12.1 x 7=84.712.3 occurs 11 times so that equal 12.3 x 11=135.312.5 occurs 18 times so that equal 12.5x18=22512.7 occurs 10 times so that equal 12.7x10=12712.9 occurs 4 times 12.9×4=51.6Add up the products and divide them by 50
[tex]\frac{623.6}{50} = 12.472[/tex]
Standard Deviation is
0.23( it is alot of steps so I use a calculator to save time).
The interqualtie range is 0.4
if SP is equal to rupees 400 and loss percent is 20% find CP
100-20=80%=400
[tex]\\ \rm\longmapsto 80\%\:of\:CP=400[/tex]
[tex]\\ \rm\longmapsto CP=400\times \dfrac{100}{80}[/tex]
[tex]\\ \rm\longmapsto CP=5(100)[/tex]
[tex]\\ \rm\longmapsto CP=500[/tex]
Step-by-step explanation:
Given,
S.p.=Rs400
Loss percent(L%) =20%
C.p.=?
We know,
S.p. =(100-L) % of c.p.
or,400=(100-20)% *c.p.
or,400=80% *c.p.
or, 400=80/100*c.p.
or,c.p.=Rs.500
Here, c.p.=Rs.500
solve for x of the given angle
Answer:
m<ABC = 92
Step-by-step explanation:
(3x + 23) and 4x are equal angle wise. So we have the equation:
(3x + 23) = 4x
3x = 4x - 23
-x = -23
x = 23
Multiply x by 4 which is 92
185 for 4 tickets whats the price per ticket
Can someone pls help me pls
Answer:
your fractions will go in order of 1/4, 2/5, 5/8, 4/6
Step-by-step explanation:
a. 2/5 < 1/2 and 4/6 > 1/2 so 2/5 < 4/6
b. 5/8 > 1/2 and 1/4 < 1/2 so 5/8 > 1/4
c. 4/6 > 1/2 and 4/6 < 2/2 and 5/8 > 1/2 and 5/8 < 2/2 and 5/8 ?? cannot see this
A shop in the mall is 6 yards by 11 yards. How much would it cost to recarpet the shop with carpet that costs $5.00 per square yard?
Answer:
$330
Step-by-step explanation:
To find the total area of the shop do 6 times 11
6*11 = 66 sq. yards
Now, do 5 * 66 because one yard is $5
5 * 66 = 330
It costs $330
What is the coefficient of the x-term in the expression 4x+5xy-2y
Factor x^2+x+6
Plz help
Answer:
2
Step-by-step explanation:
find the value of x.
9514 1404 393
Answer:
x = 4
Step-by-step explanation:
The parallel lines cut the transversals proportionally, so we have ...
(10 -x)/x = 3/2
2(10 -x) = 3x . . . . . . . multiply by 2x
20 -2x = 3x . . . . . . . eliminate parentheses
20 = 5x . . . . . . . . . . add 2x
4 = x . . . . . . . . . . . . divide by 5
For the two numbers listed, find two factors of the first number such that their product is the first number and their sum is the second number.
- 10,9
9514 1404 393
Answer:
10, -1
Step-by-step explanation:
If you assume the numbers are integers, you can look at the factor pairs that make -10:
-10 = 10(-1) = 5(-2) = 2(-5) = 1(-10)
The sums of these pairs are 9, 3, -3, -9, so the pair of interest is the first one:
10, -1
Which expression is equivalent to?
Answer:
b
Step-by-step explanation:
Answer:
Option(b)
Step-by-step explanation:
(x)^-5/3
=> 1/(3√(x)^5)
11. The temperature was 81°F at 4PM and steadily decreased 3° each hour. What was the
temperature at 8PM?
Answer:
69°
Step-by-step explanation:
4x3 = 12
81 - 12 = 69
69°
if the cost price of 10 tables is equal to the selling price of 16 tables , find out the gain or loss percentage
Hi,
To solve this problem, Let us take the LCM of 10 and 16 which will come 80.
Now suppose the cost price of 10 tables =₹n CP of 80 tables will be ₹ 8n
According to the question, CP of 10 tables is equal to the SP of 16 tables, then
the SP of 16 tables will also be ₹ n.
So, SP of 80 tables will be ₹ 5n
So, Loss = CP-SP
→ 8n - 5n = ₹ 3n
Loss%= (3n×100)/8n
Loss%= 37.5%.
Hence the correct answer will be a loss of 37.5%.
Please help!! 7th grade math! will mark brainlist!!!
Answer:
A: rectangle
B: still a rectangle
Step-by-step explanation:
Basically, if you cut it in half starting from the top straight down it would make it a rectangle because it can't be a square because all sides are not equal. And if you go from the far right top edge to the bottom left edge it will still be a rectangle but long.
Hope this helps love <3
