Answers
Answer:
[tex]\displaystyle \lim_{x\rightarrow 0^{+}} \frac{x\ln x}{\tan x}=-\infty\implies \text{B. Nonexistent (best answer)}[/tex]
Step-by-step explanation:
Recall L'Hopital's rule:
[tex]\displaystyle \lim_{x\rightarrow c}\frac{f(x)}{g(x)}=\lim_{x\rightarrow c}\frac{f'(x)}{g'(x)}[/tex]
First derivative of [tex]x\ln x[/tex]:
Recall the product rule:
[tex](f\cdot g)'=f'\cdot g+g'\cdot f[/tex]
[tex]\displaystyle \frac{d}{dx} (x\ln x)=\frac{d}{dx}(x)\cdot \ln (x)+\frac{d}{dx}(\ln x)\cdot x[/tex]
Note that:
[tex]\displaystyle \frac{d}{dx}(x)=1,\\\frac{d}{dx}(\ln (x))=\frac{1}{x}[/tex]
Simplifying, we get:
[tex]\displaystyle \frac{d}{dx} (x\ln x)=1\cdot \ln x+\frac{1}{x}\cdot x,\\\frac{d}{dx}(x\ln x)=\ln x+1[/tex]
First derivative of [tex]\tan x[/tex]:
[tex]\displaystyle \frac{d}{dx}(\tan x)=\sec^2 x[/tex]
Therefore, we have:
[tex]\displaystyle \lim_{x\rightarrow 0^{+}}\frac{x\ln x}{\tan x}=\lim_{x\rightarrow 0^{+}}\frac{\ln x+1}{\sec^2{x}}[/tex]
By definition, [tex]\cos x=\frac{1}{\sec x}[/tex]. Therefore,
[tex]\displaystyle \lim_{x\rightarrow 0^{+}}\frac{x\ln x}{\tan x}=\lim_{x\rightarrow 0^{+}}\frac{\ln x+1}{\sec^2{x}}=\lim_{x\rightarrow 0^{+}}\cos^2x(\ln x+1)[/tex]
Note:
[tex]\displaystyle \lim_{x\rightarrow 0^{+}}\cos^2x=1,\\\lim_{x\rightarrow 0^{+}}\ln x+1=-\infty[/tex]
Substitute:
[tex]\displaystyle \lim_{x \rightarrow0^{+}} \cos^2x(\ln x+1)=1\cdot (-\infty)=-\infty[/tex]
Therefore, we have:
[tex]\displaystyle \lim _{x\rightarrow 0^{+}}\frac{x\ln x}{\tan x}=-\infty \text{}[/tex], which best corresponds with [tex]\boxed{\text{B. Nonexistent}}[/tex]
*Commentary:
Technically speaking, a limit exists only if it is equal to a real number. By proper definition, infinity is not a number. With that being said, you will see limits expressed as infinity or negative infinity.
Here's what I will say about this specific problem.
The problem is stipulating that we approach [tex]x[/tex] from the right side. Because of this condition, it may be unorthodox to say this limit doesn't exist. However, if the problem just asked for [tex]\displaystyle \lim_{x\rightarrow 0}\frac{x\ln x}{\tan x}[/tex], it is common and preferred to say this limit does not exist, since [tex]\displaystyle \lim_{x\rightarrow 0^{-}}\frac{x\ln x}{\tan x}\neq \displaystyle \lim_{x\rightarrow 0^{+}}\frac{x\ln x}{\tan x}[/tex].
For example, [tex]\displaystyle \lim_{x\rightarrow 0}\frac{1}{x}=\text{DNE}[/tex], because [tex]\displaystyle \lim_{x\rightarrow 0}\frac{1}{x}[/tex] diverges. In other words, [tex]\displaystyle \lim_{x\rightarrow 0^{-}}\frac{1}{x}=-\infty \neq \displaystyle \lim_{x\rightarrow 0^{+}}\frac{1}{x}=\infty[/tex].
But again, the problem is asking for the limit as [tex]x[/tex] approaches from the right, in which case [tex]\displaystyle \lim _{x\rightarrow 0^{+}}\frac{x\ln x}{\tan x}=-\infty }[/tex]. It's really a pedagogical choice whether to say a limit equal to infinity or negative infinity exists or not since infinity implies there is no limit, so saying the limit of something is infinity becomes an oxymoron. In this case, the person who wrote the answer choices chose to express a limit of infinity as nonexistent, but it is worth mentioning that someone else solving this problem might express [tex]-\infty[/tex] as the answer, and they would be just as, if not more, correct.
Without resorting to L'Hopital's rule, recall that
[tex]\displaystyle \lim_{x\to0}\frac{\sin(ax)}{ax} = 1[/tex]
for a ≠ 0. Then
[tex]\displaystyle \lim_{x\to0^+} \frac{x \ln(x)}{\tan(x)} = \lim_{x\to0^+}\frac x{\sin(x)} \times \lim_{x\to0^+}\cos(x) \times \lim_{x\to0^+}\ln(x)[/tex]
The first two limits exist and are equal to 1, but the last limit is -∞.
Related Questions
If UW = 9x - 9, what is UW in units?
5
6
30
36
Answers
Explanation:
UW is composed of UV and VW, which we have values for. So, let’s make an equation.
(2x - 4) + (4x + 10) = 9x - 9
Solve the left side
6x + 6 = 9x - 9
Subtract 6 from both sides
6x = 9x - 15
Subtract 9x from both sides
-3x = -15
Divide both sides by -3
x = 5
Plug 5 into the equation for UW.
9(5) - 9
45 - 9
36
Your college professor tells you he will give a 30%-30%-40% split on grades between homework, quizzes, and test. However whichever category you have the highest average in will be the 40% category.
What will your final grade be if your homework average is 53, you quiz average is 66, and your test average is 96?
Answers
Explanation:
53 x 30/100 = 15.9
66 x 30/100 = 19.8
96 x 40/100 = 38.4
38.4 + 19.8 + 15.9 = 74.1
Therefore, the final grade is 74.1
37. Verify Green's theorem in the plane for f (3x2- 8y2) dx + (4y - 6xy) dy, where C is the boundary of the
C
region defined by: (a) y = fx , y = x2 ; (b) x = 0, y = 0, x +y = 1 .
Ans. (a) common value = 3/2 (b) common value = 5/3
38. Evaluate f (3x +4y)dx + (2x --3y)dy where C, a circle of radius two with center at the origin of the xy
C plane, is traversed in the positive sense. Ans. - 87T
39. Work the previous problem for the line integral f (x2+y2)dx + 3xy2 dy. Ans. 127T
C 40. Evaluate f (x2-2xy)dx +(x2y+3)dy around the boundary of the region defined by y2 = 8x and x = 2
(a) directly, (b) by using Green's theorem. Ans. 128/5
(TT.2) 41. Evaluate f (6xy - y2) dx + (3x2 --- 2xy) dy along the cycloid x = 6 - sin 6, y = 1 - cos 6.
Answers
I'll only look at (37) here, since
• (38) was addressed in 24438105
• (39) was addressed in 24434477
• (40) and (41) were both addressed in 24434541
In both parts, we're considering the line integral
[tex]\displaystyle \int_C (3x^2-8y^2)\,\mathrm dx + (4y-6xy)\,\mathrm dy[/tex]
and I assume C has a positive orientation in both cases
(a) It looks like the region has the curves y = x and y = x ² as its boundary***, so that the interior of C is the set D given by
[tex]D = \left\{(x,y) \mid 0\le x\le1 \text{ and }x^2\le y\le x\right\}[/tex]
• Compute the line integral directly by splitting up C into two component curves,
C₁ : x = t and y = t ² with 0 ≤ t ≤ 1
C₂ : x = 1 - t and y = 1 - t with 0 ≤ t ≤ 1
Then
[tex]\displaystyle \int_C = \int_{C_1} + \int_{C_2} \\\\ = \int_0^1 \left((3t^2-8t^4)+(4t^2-6t^3)(2t))\right)\,\mathrm dt \\+ \int_0^1 \left((-5(1-t)^2)(-1)+(4(1-t)-6(1-t)^2)(-1)\right)\,\mathrm dt \\\\ = \int_0^1 (7-18t+14t^2+8t^3-20t^4)\,\mathrm dt = \boxed{\frac23}[/tex]
*** Obviously this interpretation is incorrect if the solution is supposed to be 3/2, so make the appropriate adjustment when you work this out for yourself.
• Compute the same integral using Green's theorem:
[tex]\displaystyle \int_C (3x^2-8y^2)\,\mathrm dx + (4y-6xy)\,\mathrm dy = \iint_D \frac{\partial(4y-6xy)}{\partial x} - \frac{\partial(3x^2-8y^2)}{\partial y}\,\mathrm dx\,\mathrm dy \\\\ = \int_0^1\int_{x^2}^x 10y\,\mathrm dy\,\mathrm dx = \boxed{\frac23}[/tex]
(b) C is the boundary of the region
[tex]D = \left\{(x,y) \mid 0\le x\le 1\text{ and }0\le y\le1-x\right\}[/tex]
• Compute the line integral directly, splitting up C into 3 components,
C₁ : x = t and y = 0 with 0 ≤ t ≤ 1
C₂ : x = 1 - t and y = t with 0 ≤ t ≤ 1
C₃ : x = 0 and y = 1 - t with 0 ≤ t ≤ 1
Then
[tex]\displaystyle \int_C = \int_{C_1} + \int_{C_2} + \int_{C_3} \\\\ = \int_0^1 3t^2\,\mathrm dt + \int_0^1 (11t^2+4t-3)\,\mathrm dt + \int_0^1(4t-4)\,\mathrm dt \\\\ = \int_0^1 (14t^2+8t-7)\,\mathrm dt = \boxed{\frac53}[/tex]
• Using Green's theorem:
[tex]\displaystyle \int_C (3x^2-8y^2)\,\mathrm dx + (4y-6xy)\,\mathrm dx = \int_0^1\int_0^{1-x}10y\,\mathrm dy\,\mathrm dx = \boxed{\frac53}[/tex]
Find the solution set
2(x + 4 = 8
Answers
Answer:
x=0
Step-by-step explanation:
0+4=4
4x2=8
it's pretty simple once you get the hang of it
Well the answer is 6
The sum of three times a number, x, and 12
Answers
Answer:
The sum of three times a number and twelve is fewer than twenty translates into an inequality
3x+12 < 20
Step-by-step explanation:
it's not the perfect answer of your question but it's kind of similar to it
(2.5*10^3)*3.4*10^2 (in scientific notation)
Answers
Answer:
8.5 × 10^5
Step-by-step explanation:
Move the decimal so there is one non-zero digit to the left of the decimal point. The number of decimal places you move will be the exponent on the 10. If the decimal is being moved to the right, the exponent will be negative. If the decimal is being moved to the left, the exponent will be positive.
[tex]\\ \rm\Rrightarrow (2.5\times 10^3)^3\times 4\times 10^2[/tex]
[tex]\\ \rm\Rrightarrow 2.5\times 3.4 10^3 \times 10^2[/tex]
[tex]\\ \rm\Rrightarrow 8.5\times 10^{3+2}[/tex]
[tex]\\ \rm\Rrightarrow 8.5\times 10^5[/tex]
y = 10 when x = - 4 , find y when x = 12
Answers
Answer:
y = -30
Step-by-step explanation:
Set the ratio: [tex]\frac{x_1}{y_1} = \frac{x_2}{y_2}[/tex]
Let:
[tex](x_1 , y_1) = (-4 , 10)\\(x_2 , y_2) = (12 , y)\\[/tex]
Plug in the corresponding terms to the corresponding variables:
[tex]\frac{-4}{10} = \frac{12}{y}[/tex]
First, cross multiply.
[tex](-4 * y) = (10 * 12)\\-4y = 10 * 12\\-4y = 120[/tex]
Next, isolate the variable, y. Divide -4 from both sides of the equation:
[tex]\frac{-4y}{-4} = \frac{120}{-4}\\y = \frac{120}{-4}\\y = -30[/tex]
-30 is your answer for y.
what is the perimeter of an equilateral triangular lamina of side length 12 cm
Answers
Answer:
It is 36 cm
Step-by-step explanation:
In equilateral triangle, all sides are equal.
[tex]{ \sf{perimeter = side + side + side}}[/tex]
but side = 12 cm
[tex] { \sf{p = (12 + 12 + 12)}} \\ { \sf{p = 36 \: cm}}[/tex]
Answer: Perimeter = 36 cm
Concept:
Here, we need to know how to find the perimeter of an equilateral triangle.
An equilateral triangle is a triangle in which all three sides have the same length.
Perimeter (equilateral triangle) = 3s
s = side length
Solve:
Given information
Side length = 12 cm
Given expression
Perimeter = 3s
Substitute values into the expression
Perimeter = 3 (12)
Simplify by multiplication
Perimeter= [tex]\boxed{36cm}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Evaluate the expression (-243)^2/5
Answers
Answer:
Step-by-step explanation:
The fifth root of -243 = - 3
If you now square - 3, you get (-3)(-3) = 9
Find the value of x.
Answers
Answer:
x=60
Step-by-step explanation:
x=180-120
20 points if these are correct pls.
Answers
Answer:
16. 42
21. 18
Step-by-step explanation:
find the slope of the line
Answers
Answer:
m = (6-5)/(4-1)
m = 1/3
.............
g(x) = sqr root of x+3
What is the domain of g?
Answers
Answer:
[tex]all \: real \: numbers \: except \: \\ x + 3 < 0 \\ \: \: \: \: x < - 3[/tex]
Answer:
X is the domain and sqr root of X+3 is the range.
Solve for x: 4 − (x + 2) < −3(x + 4)
Answers
Answer: x < 7
Step-by-step explanation:
Round each number to the nearest hundredth.
a. 8.7495980
b. 0.72091
C. 9.3113
d. 23.0368
e 6.9788
Answers
Answer:
a. 8.75
b. 0.72
c. 9.31
d. 23.04
e. 6.98
Step-by-step explanation:
a.
in this number 8.7495980
the digit in the hundredth place is 4
the digit in the thousandth place is 9
Since 9 ≥ 5 ,then 8.7495980 rounded to nearest hundredth is :
8,75
b.
in this number 0.72091
the digit in the hundredth place is 2
the digit in the thousandth place is 0
Since 0 < 5 ,then 0.72091 rounded to nearest hundredth is :
0.72
c.
in this number 9.3113
the digit in the hundredth place is 1
the digit in the thousandth place is 1
Since 1 < 5 ,then 9.3113 rounded to nearest hundredth is :
9.31
d.
in this number 23.0368
the digit in the hundredth place is 3
the digit in the thousandth place is 6
Since 6 ≥ 5 ,then 23.0368 rounded to nearest hundredth is :
23.04
e.
in this number 6.9788
the digit in the hundredth place is 7
the digit in the thousandth place is 8
Since 8 ≥ 5 ,then 6.9788 rounded to nearest hundredth is :
6.98
Find the sum.
1/xy^2 + x^2/y
Answers
Answer:
Step-by-step explanation:
HELP NOW PLEASE NOW ILL GIVE YOU BRAINLY IF YOU GET IT RIGHT NOW PLEASE!!!
Answers
Answer:
you can check the picture, i tried to make it as simple as possible. so I answered by the format.
Rose's rectangular backyard has an area of 672 square meters. Find the length of the backyard if the backyard is 7 meters wide.
Answers
[tex]\\ \rm\longmapsto Length(Breadth)=Area[/tex]
[tex]\\ \rm\longmapsto Length(7)=672[/tex]
[tex]\\ \rm\longmapsto Length=\dfrac{672}{7}[/tex]
[tex]\\ \rm\longmapsto Length=96m[/tex]
Convert 2 yards to centimeters. Round to the nearest whole
centimeter.
Answers
Answer:
183 centimeters
Step-by-step explanation:
2 multiply the length value by 91.44
11. Write in the expanded notation : (a) 85, 33, 48, 925 (b) 8, 25, 38, 201 (c) 2,00, 00, 63, 670 (d) 8,57,00, 07, 005 class 5
Answers
please mark this answer as brainlist
What is the largest amount which can be borrowed over three years at 4.5% APR if the largest
affordable monthly payment is $279? (Round to the nearest ten dollars.)
Answers
Answer:
$12.5 dollars
Step-by-step explanation:
cc
Plane P is a cross-section of the solid below. What shape is the cross section?
A. not enough information
B. hexagon
C. pentagon
D. rectangle
Answers
Answer:
B. hexagon
Step-by-step explanation:
ANSWeR RIGHT ANDIL ALSO GIVE 5 STAR AND BRAINLIEST
Answers
Answer:
3. Sanford Township, Minnesota
4. Inner Mongolia, China
5. Arunachal Pradesh, India
6.Almaty Region, Kazakhstan
7.Madhya Pradesh, India
Step-by-step explanation:
Help with number 7 picture above
Answers
9514 1404 393
Answer:
next: ←, ↑, →, ↓, ←
Step-by-step explanation:
It appears you want to identify a rule describing the sequence, then provide several additional elements in the sequence.
It looks to me like each successive element has been rotated 90° CW from the previous one.
Continuing that pattern, the arrows will point left, up, right, down, left.
Help!
Solve the equation: a/0.4 = –20
Answers
Answer:
i think this might be it? but most likely :) <3
Step-by-step explanation:
A box of 8 cellphones contains two yellow cellphones and six green cellphones. Complete parts (a) through (d) below.
a. If two cellphones are randomly selected from the box without replacement, what is the probability that both cellphones selected will be green?
b. If two cellphones are randomly selected from the box without replacement, what is the probability there will be one green cellphone and one yellow cellphone selected?
c. If three cellphones are selected with replacement (the first cellphone is returned to the box after it is selected), what is the probability that all three will be yellow?
d. If you were sampling with replacement (the first cellphone is returned to the box after it is selected), what would be the answers to (a) and (b)?
Answers
Probabilities are used to determine the chance of an event. The following are the summary of the solution.
The probability that the two selected cellphones are green (without replacement) is 15/28The probability that one green and one yellow is selected (without replacement) is 3/7The probability that all three cellphones are yellow (with replacement) is 1/64The probability that the two cellphones are green (with replacement) is 9/16The probability that one green and one yellow is selected (with replacement) is 3/8Given that:
[tex]n = 8[/tex]
[tex]G = 6[/tex] --- Green
[tex]Y = 2[/tex] --- Yellow
(a) Probability that the two cellphones are green (without replacement).
Since the cellphone is not replaced, the probability is calculated as follows:
[tex]Pr = \frac Gn \times \frac{G - 1}{n-1}[/tex]
So, we have:
[tex]Pr = \frac 68 \times \frac{6 - 1}{8-1}[/tex]
[tex]Pr = \frac 68 \times \frac 57[/tex]
[tex]Pr = \frac{30}{56}[/tex]
[tex]Pr = \frac{15}{28}[/tex]
Hence, the probability that the two cellphones are green (without replacement) is 15/28
(b) Probability that one green and one yellow is selected (without replacement).
Since the cellphone is not replaced, the probability is calculated as follows:
[tex]Pr = \frac Gn \times \frac{Y}{n-1} + \frac Yn \times \frac{G}{n-1}[/tex] ---- The subtraction means the cellphones are not replaced
This gives
[tex]Pr = \frac 68 \times \frac{2}{8-1} + \frac 28 \times \frac{6}{8-1}[/tex]
[tex]Pr = \frac 34 \times \frac{2}{7} + \frac 14 \times \frac{6}{7}[/tex]
[tex]Pr = \frac 32 \times \frac17 + \frac 12 \times \frac 37[/tex]
[tex]Pr = \frac{3}{14} + \frac{3}{14}[/tex]
Take LCM
[tex]Pr = \frac{3+3}{14}[/tex]
[tex]Pr = \frac{6}{14}[/tex]
[tex]Pr = \frac{3}{7}[/tex]
Hence, the probability that one green and one yellow is selected (without replacement) is 3/7
(c) Probability that the all three cellphones are yellow (with replacement).
Since the cellphone is replaced, the probability is calculated as follows:
[tex]Pr = \frac Yn \times \frac Yn \times \frac Yn[/tex]
So, we have:
[tex]Pr = \frac 28 \times \frac 28 \times \frac 28[/tex]
[tex]Pr = \frac 14 \times \frac 14 \times \frac 14[/tex]
[tex]Pr = \frac 1{64}[/tex]
Hence, the probability that all three cellphones are yellow (with replacement) is 1/64
(d1) Probability that the two cellphones are green (with replacement).
Since the cellphone is replaced, the probability is calculated as follows:
[tex]Pr = \frac Gn \times \frac{G}{n}[/tex]
So, we have:
[tex]Pr = \frac 68 \times \frac{6}{8}[/tex]
[tex]Pr = \frac 34 \times \frac{3}{4}[/tex]
[tex]Pr = \frac{9}{16}[/tex]
Hence, the probability that the two cellphones are green (with replacement) is 9/16
(d2) Probability that one green and one yellow is selected (with replacement).
Since the cellphone is replaced, the probability is calculated as follows:
So, we have:
[tex]Pr = \frac Gn \times \frac{Y}{n} + \frac Yn \times \frac{G}{n}[/tex]
This gives
[tex]Pr = \frac 68 \times \frac{2}{8} + \frac 28 \times \frac{6}{8}[/tex]
[tex]Pr = \frac 34 \times \frac{1}{4} + \frac 14 \times \frac{3}{4}[/tex]
[tex]Pr = \frac 3{16} + \frac{3}{16}[/tex]
Take LCM
[tex]Pr = \frac {3+3}{16}[/tex]
[tex]Pr = \frac {6}{16}[/tex]
[tex]Pr = \frac {3}{8}[/tex]
Hence, the probability that one green and one yellow is selected (with replacement) is 3/8
Read more about probabilities at:
https://brainly.com/question/795909
please help me it's for today
Answers
Check the pictures attached for the answers.
the selling price of an article is 20% less than its marked price and the marked price is 30% above the cost price. find the profit percent.
Answers
Step-by-step explanation:
here,
let MP be x then,
SP=x-20%of X
=x - 20x/100
=x- 0.2x
=0.8x
CP= x- 30% of x
= x- 30x/100
=x- 0.3x
=0.7x
now,profit (P) = SP -CP
= 0.8x-0.7x
=0.1x
so,
P%=( P\CP ) ×100%
= (0.1x/0.7x) ×100%
= 14.28 %
3 Which of the scientists responsible for cell theory would be the most likely to write a book titled Cells Come from Cells?
Hooke
Schleiden
Virchow
Schwann
Answers
Answer:
Virchow
Step-by-step explanation:
He was the one who discovered that all cells come from pre-existing cells, the third part of the cell theory.
refer to image given and please answer!!!!!
Answers
Step-by-step explanation:
According to the quotient rule of differentiation, if [tex]y(x) = u(x)/v(x),[/tex] then its derivative is given by
[tex]\dfrac{dy}{dx} = \dfrac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}}{v^2}\:\:\:\:\:\:\:\:\:(1)[/tex]
We can see that
[tex]u(x) = \ln (4x^2 + 1)[/tex]
[tex]\dfrac{du}{dx} = \dfrac{8x}{4x^2 + 1}[/tex]
[tex]v(x) = 2x - 3[/tex]
[tex]\dfrac{dv}{dx} = 2[/tex]
Plugging the above expressions into Eqn(1), we find that the derivative is
[tex]\dfrac{dy}{dx} = \dfrac{\dfrac{8x(2x - 3)}{(4x^2 + 1)} - 2\ln (4x^2 + 1)}{(2x - 3)^2}[/tex]
[tex]\:\:\:\:\:\:\:= \dfrac{8x}{(4x^2 + 1)(2x - 3)} - \dfrac{2\ln (4x^2 + 1)}{(2x - 3)^2}[/tex]