Presumably, the first limit is some finite number
[tex]\displaystyle \lim_{x\to\pi} \frac{f(x)}{x-\pi} = c[/tex]
Since x - π clearly approaches 0 as x approaches π, we must also have f(x) approaching 0,
[tex]\displaystyle \lim_{x\to\pi} f(x) = 0[/tex]
Recall the double angle identity,
sin(2t) = 2 sin(t) cos(t)
and rewrite the limit as
[tex]\displaystyle \lim_{x\to\pi} \frac{\sin(2 f(x))}{x - \pi} = \lim_{x\to\pi} \frac{2 \sin(f(x)) \cos(f(x))}{x - \pi} = 2 \lim_{x\to\pi} \frac{\sin(f(x))}{f(x)} \times \frac{f(x) \cos(f(x))}{x - \pi}[/tex]
Recall that
[tex]\displaystyle \lim_{x\to0}\frac{\sin(x)}x=1[/tex]
which means
[tex]\displaystyle \lim_{x\to\pi}\frac{\sin(f(x))}{f(x)} = \lim_{f(x)\to0}\frac{\sin(f(x))}{f(x)} = 1[/tex]
and by continuity,
[tex]\displaystyle \lim_{x\to\pi} \cos(f(x)) = \cos\left(\lim_{x\to\pi} f(x)\right) = \cos(0) = 1[/tex]
Then
[tex]\displaystyle 2 \lim_{x\to\pi} \frac{\sin(f(x))}{f(x)} \times \frac{f(x) \cos(f(x))}{x - \pi} \\\\ = 2 \left(\lim_{x\to\pi}\frac{\sin(f(x))}{f(x)}\right) \left(\lim_{x\to\pi} \frac{f(x)}{x-\pi}\right) \left(\lim_{x\to\pi}\cos(f(x))\right) \\\\ = 2 \times 1 \times c \times 1 = \boxed{2c}[/tex]
The perimeter of a rectangle is 60 cm. The ratio of length to width is 3:2. Find the length and width of the rectangle.
Given :
The Perimeter of the rectangle is 60 cmThe ratio of the length to the width is 3:2.To Find :
The Length and width of the rectangle .⠀
Solution :
We know that,
[tex]\qquad{ \bold{ \pmb{2(Length + Breadth ) = Perimeter_{(rectangle)}}}}[/tex]
Let's assume the length of the rectangle as 3x inches. and the width is 2x inches.
⠀
Now, Substituting the given values in the formula :
[tex]\qquad \dashrightarrow{ \sf{2(3x + 2x )= 60}}[/tex]
[tex]\qquad \dashrightarrow{ \sf{2(6x)= 60}}[/tex]
[tex]\qquad \dashrightarrow{ \sf{12x= 60}}[/tex]
[tex]\qquad \dashrightarrow{ \sf{x= \dfrac{60}{12} }}[/tex]
[tex]\qquad \dashrightarrow{ \bf{x= 5}}[/tex]
Therefore,
[tex]\qquad { \pmb{ \bf{ Length _{(rectangle)} = 3x \: = 3(5) = 15 \: inches}}}\:[/tex]
[tex]\qquad { \pmb{ \bf{ Width _{(rectangle)} = 2x = 2(5) = 10 \: inches}}}\:[/tex]
if you are wearing a black sweatshirt, what happens to most of the light that strikes it? (this is science but i cant find the science subject)
Answer:
The black colour will absorb the light.
Write the ratio as a ratio of whole numbers in lowest terms $1.50 to $1.20
Answer:
[tex]\frac{5}{4}[/tex]
Step-by-step explanation:
Ratio of $1.50 to $1.20
=[tex]\frac{1.50}{1.20}[/tex]=[tex]\frac{1.5}{1.2}[/tex]=[tex]\frac{15}{12}[/tex]=[tex]\frac{5}{4}[/tex]
Ratio will be [tex]\frac{5}{4}[/tex]
In rectangle ABCD, the diagonals intersect each other at point 0 and mABD=30°.
Find BC if AC = 16 in.
Answer:
Step-by-step explanation:
The length of the diagonal BC is equal to 18.47in
Data;
m∠ABD = 30°
BC = ?
AC = 16 in
Length BC
Since the length of AC = 16in and angle m∠ABD = 30°, let us consider that the edges form a right angle triangle.
We can use trigonometric ratio to find the length BC since it is the length of the diagonal.
Since we have the value of opposite and angle, we can find the hypothenuse using sine angle.
The length of the diagonal BC is equal to 18.47in
What equation does the model below
represent?
Answer:
...
Step-by-step explanation:
Answer:
There is no model therefore the equation does not represent the model below
Step-by-step explanation:
Hope this helps mark brainliest please ;)
Please put the letter and the answer choice that goes with it so i know which answer is for that question. (example 1. 3+3=6 or 2.T)
Answers:
Choice DChoice DChoice C=======================================================
Explanations:
Problem 1
A = (4,4)
B = (-12,8)
Vector u starts at A and ends at B. The arrow points to B.
Subtract x coordinates: xB - xA = -12-4 = -16
Subtract y coordinates: yB - yA = 8-4 = 4
The vector is u = <-16, 4> which is in quadrant Q2.
Use the pythagorean theorem to find the length of the vector.
a^2 + b^2 = c^2
c = sqrt( a^2 + b^2 )
||u|| = sqrt( (-16)^2 + 4^2 )
||u|| = 16.492
Now compute the angle
theta = arctan(b/a)
theta = arctan(4/(-16))
theta = -14.036
Add on 180 degrees so the angle lands in Q2.
-14.036+180 = 165.964
The angle is roughly theta = 165.964 degrees
------------------------------
Problem 2
A = start point = (4, 3)
B = end point = (-4, -1)
Subtract x coordinates: xB - xA = -4 - 4 = -8
Subtract y coordinates: yB - yA = -1-3 = -4
v = <a, b> = <-8, -4> which is in Q3
Use the pythagorean theorem to find the length of the vector.
a^2 + b^2 = c^2
c = sqrt( a^2 + b^2 )
||v|| = sqrt( (-8)^2 + (-4)^2 )
||v|| = 8.944
And,
theta = arctan(b/a)
theta = arctan(-4/(-8))
theta = 26.565
We need to add on 180 so we move from Q1 to Q3
26.565+180 = 206.565
------------------------------
Problem 3
The notation <-3, 4> means "move 3 units left, 4 units up".
Only vector r fits the description as we move from the initial point (5,3) to the terminal point (2,7)
The concentration of platinum in a necklace is 7%. The necklace weighs 18 g. Find the amount of platinum in the necklace.
Answer:
1.26 grams
Step-by-step explanation:
18 g * 0.07 = 1.26 g
The monthy net gains sales for a new sporting goods store were: $7195, $1484, $3224, $8807, $7185, $1730, $2573, $7071, $5360, S4331. Find the median
Answer:
$7,185 see below ↓
explanation:
With median, it is about the number that appears the most, and since the closest it gets is between $7,071 and $7,195, we have to pick the number that averages in-between which in this case is 7,185.
Below, this ↓ array proves the logic.
[tex]\left[\begin{array}{ccc}7195&7185&7071\end{array}\right][/tex]
Jeff can ran 3 miles in 22 minutes. Assuming he can run at a constant rate, how long would it take Jeff to run 8 miles?
Find the lengths of the missing sides if side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse.
cos(B) = 4/5 a=90
Answer:
Step-by-step explanation:
cosB = a/c = 4/5
so c = 5a/4 = 5*90/4 = 112.5
b² = c² - a² = 112.5² - 90² = 4556.25
b = 67.5
My mom's friend bought cookbooks and travel books at a book sale. She spent $107 and bought 31 books. The cookbooks cost $2 apiece and the travel books cost $5 apiece. How many of each type of book did the friend buy?
MUST BE CORRECT OR ANSWER DELETED AND YOU WILL BE REPORTED.
Answer:
2
Step-by-step explanation:
$52 7books $6 CB which means she brought 2 which is $12$8 TB and then 7-2 because he already brought 2 CB so that means that he brought 5TB which is 40 Then add 40 +12 = $52 for 7 books
cookbooks be x
travel books be y
x+y=31--(1)2x+5y=107--(2)(1)×2
2x+2y=62--(3)Subtract (3) from (1)
3y=45y=15Now
x=31-15x=16write an equation that represents each circle
the answers circle with circle with hat with circle
I need help with this question please
Answer : It Is B Reflection Across The Y Axis
Explanation : The Triangles Are At The Exact Same Spot On The Negative And Positive Sides
Hope This Helps
◊ YusuCr ◊
QRST WXYZ. The area of QRST is given. Find the area of WXYZ.
Answer:
b. 1296 in^2
Step-by-step explanation:
The area of a rectangle is calculated by multiplying width and length:
The length of the QRST is 144 since its width is given as 8 to find the length:
144/8 = 18 is the length of the QRST
it is stated that the two rectangles are similar so their widths and lengths are proportional.
Let x represent the width of WXYZ:
8/24 = 18/x cross multiply expressions
8x = 432 divide both sides by 8
x = 54 now we can calculate area of WXYZ
24*54 = 1296
Peter deposited 5000 ssp in the bank with an interest rate of 5% for 3 years what was the interest
Answer:
$250Step-by-step explanation:
First calculate what 1% of 5,000 is (50). Second multiply 50 by 5 (250) Now you got your answer! :)PS:Can you please mark me brainly? It would mean the world to me.
rational expression,, please explain!
I understand mostly, but the subtraction sign throws me off...
Answer:
x = 2
Step-by-step explanation:
What is needed to solve this equation is a common denominator. With a common denominator, you can get rid of subtracted fractions.
First, factor the denominator on the left side of the equation:
[tex]\frac{3}{(x + 3)(x - 2)} = \frac{2}{x+3} - \frac{1}{x-2}[/tex]
Then, create a common denominator on the right:
[tex]\frac{3}{(x + 3)(x - 2)} = \frac{2}{x+3}(\frac{x-2}{x-2}) - \frac{1}{x-2}(\frac{x+3}{x+3})[/tex]
[tex]\frac{3}{(x + 3)(x - 2)} = \frac{2(x - 2) - 1(x + 3)}{(x+3)(x-2)}[/tex]
Put everything on one side:
[tex]0 = \frac{2(x - 2) - 1(x + 3)}{(x+3)(x-2)} - \frac{3}{(x + 3)(x - 2)}[/tex]
[tex]0 = \frac{2(x - 2) - 1(x + 3) - 3}{(x+3)(x-2)}[/tex]
[tex]0 = \frac{2x - 4 - x - 3 - 3}{(x+3)(x-2)}[/tex]
[tex]0 = \frac{x - 10}{(x+3)(x-2)}[/tex]
set each term equal to zero:
x - 10 = 0
x = 10
x + 3 = 0
x = -3
x - 2 = 0
x = 2 is the answer, since it is the only answer which is listed in the multiple choice question.
Afia bought shoes that were on sale for `30\%` off the regular price. Afia saved `\$12`. What was the regular price of the shoes
Answer:
$40
Step-by-step explanation:
30% = .30
$12/0.30 = 40
[RevyBreeze]
[tex][Hello,BrainlyUser][/tex]
Answer:
$40
Step-by-step explanation:
Original Price = P
30% = .30 or .3
30% of P = 12
.3P = 12
P =12/.3 = $40
~[CloudBreeze]~
Can somebody help me figure out the tax bracket please!
Answer:
Hmmmm, very tricky question but I'll figure it out and type it in the comments
Answer my question and get points
How many solutions do two parallel lines have?
Parallel lines don't have any solutions since they never intersect
Answer:
When the lines are parallel, there are no solutions, and sometimes the two equations will graph as the same line, in which case we have an infinite number of solutions.
In a certain normal distribution, find the mean μ when σ = 5 and 5.48% of the area lies to the left of 78.
The mean of the normal distribution, using the formula for the z-score, considering the given data, is of 86.
Normal Probability DistributionIn a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.In this problem:
The standard deviation is of [tex]\sigma = 5[/tex].5.48% of the area lies to the left of 78, hence when X = 78, Z has a p-value of 0.0548, that is, Z = -1.6.Then:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.6 = \frac{78 - \mu}{5}[/tex]
[tex]78 - \mu = -1.6(5)[/tex]
[tex]\mu = 86[/tex]
More can be learned about the normal distribution at https://brainly.com/question/24663213
I need help my teacher needs it tomorrow
[tex]\tt V=\dfrac{1}{3}\times base~area\times height\\\\V=\dfrac{1}{3}\times 8^2\times 12\\\\V=256~cm^3[/tex]
In the figure, m ║ n and p is a transversal. Which of the following are alternate interior angles?
A. ∠1 and ∠7, ∠2 and ∠8
B. ∠2 and ∠8, ∠1 and ∠6
C. ∠3 and ∠5, ∠4 and ∠6
D. ∠3 and ∠7, ∠4 and ∠8
HELP!! I dont know which one is the correct answer. I'll give brainliest
Answer:
[tex]\sf g = 13.18 \ units[/tex]
explanation:
the "g" refers to the opposite side of angle G
First solve for angle E:
E + 60° + 45° = 180°
E = 180° - 105°
E = 75°
Then using sine rule:
[tex]\sf \dfrac{a}{sin(A)} = \dfrac{b}{sin(B)}[/tex]
[tex]\rightarrow \sf \dfrac{18}{sin(75)} = \dfrac{g}{sin(45)}[/tex]
[tex]\rightarrow \sf \dfrac{18*sin(45)}{sin(75)} = g[/tex]
[tex]\rightarrow \sf g = 13.18 \ units[/tex]
he volume of this cube is 216 cubic yards. What is the value of d?
Answer:
do the cube root of 216
the answer is 6
because 6 * 6 * 6= 216
If the width of a rectangle plot is “x” and the length of the plot is 100-x/2, what is the maximum area of the plot?
Answer:Let x = length of the plot (in meters).
Then the two lengths use up 2x meters of the fencing,
. . . leaving 100 - 2x meters for the two widths.
This means each width is (100 - 2x)/2 = 50 - x meters.
The area of a rectangle is: Length x Width
. . . so we have: . A .= .x(50 - x)
We have a <u>parabola</u>: . A .= .- x<sup>2</sup> + 50x</sup>
We know that this parabola opens down (don't we?)
. . . so its vertex must be the highest point.
We're expected to know that the vertex is at: . x = -b/2a
For this problem, a = -1, b = 50, so we have: . x = -50/(2(-1) = 25
We have maximum A when x = 25.
This leaves 50 - x = 25 for the width.
Therefore, for maximum area, make the plot a 25-by-25 square.
Step-by-step explanation:
Answer:
1250 units squared
Step-by-step explanation:
if you're allowed to use a calculator;
whats the solution to
3/4 x8 1/2=
Answer:
3
Step-by-step explanation:
3. Margaret has a container on her dresser in the shape of a cube. It holds her earrings. Each
edge of the container is 6 cm long. What is the volume of the container?
clo
Step-by-step explanation:
Volume of cube = 6 × 6 × 6 cm³
= 216 cm³
Solve for x. Please look at the picture and answer it. Thank you.
Hope it helps ~
Answer:
x=24
Step-by-step explanation:
i cant quite explain how i did it
On a coordinate plane, 2 parallelograms are shown. Parallelogram 1 has points (0, 2), (2, 6), (6, 4), and (4, 0). Parallelogram 2 has points (2, 0), (4, negative 6), (2, negative 8), and (0, negative 2). How do the areas of the parallelograms compare? The area of parallelogram 1 is 4 square units greater than the area of parallelogram 2. The area of parallelogram 1 is 2 square units greater than the area of parallelogram 2. The area of parallelogram 1 is equal to the area of parallelogram 2. The area of parallelogram 1 is 2 square units less than the area of parallelogram 2.
Answer:
A. The area of parallelogram 1 is 4 square units greater than the area of parallelogram 2.
Step-by-step explanation:
I just took the test
Given two parallel lines, same side interior angles are supplementary. Fill
in the blanks with the correct answers.
1. Vertical angles are
2. Same side exterior angles are
3. Alternate side interior angles are
4. Alternate side exterior angles are
5. Corresponding angles are