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
Is the product of a rational number and an irrational number always irrational?
Answers
Answer:
The product of any rational number and any irrational number will always be an irrational number.
Step-by-step explanation:
2. Which type of variation is represented by the following equation?
Answers
indirect variation
Verification
[tex]\\ \rm\Rrightarrow s\propto \dfrac{1}{y}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{s_1}{t_2}=\dfrac{s_2}{t_1}[/tex]
[tex]\\ \rm\Rrightarrow s_1t_1=s_2t_2[/tex]
What is the sum of -14
and -15?
Answers
Hope it helps !!! Brainliest if it is!!
Answer:
-29
Step-by-step explanation:
(-14) + (-15) =
-14 - 15 =
-29
find the value of x
x = [?]
Answers
Answer
4
Step by step explanation
2/3=x/10-x
cross multiply
2(10-x)=3x
20-2x=3x
20=3x+2x
20=5x
x=4
3 3/5 divided by 1/6?
Answers
Answer:
10.8
Step-by-step explanation:
3 3/5 divided by 1/6
(3(3/5))/(1/6)
(9/5)/(1/6)
54/5
10(4/5)
10.8
Find any domain restrictions on the given rational equation x/x+4 + 12/x^2+5x+4 = 8x/5x-15
Answers
Answer:
x ≠ -4, -1, 3
Step-by-step explanation:
12/(x^2+5x+4) = 8x/(5x-15)
12/((x+1)(x+4)) = 8x/(5(x-3))
division by zero is undefined
There are 3 answers to your question. x= -1 x= 3 x= -4
Point V is on line segment UW. Given VW = 5x - 4, UV = 2x, and UW = 5x, determine the numerical length of VW
Answers
Answer:
VW = 6
Step-by-step explanation:
To find x, set up the following equation:
(5x - 4) + (2x) = 5x
Solve out left side
7x - 4 = 5x
Subtract 5x from both sides
2x - 4 = 0
Add 4 to both sides
2x = 4
Divide both sides by 2
x = 2
Plug into 5x - 4
5(2) - 4
10 - 4
6
Answer:
6
Step-by-step explanation:
5x - 4 + 2x = 5x
7x - 4 = 5x
-4 = -2x
2 = x
5(2) - 4
10 - 4
6
12.5 x 0.8
(without rounding and pls help me)
Answers
So answer
12.5* 0.8 = 10
Answer:
That’s simple 10
Step-by-step explanation
Let me know if I’m correct
hope this helps
The arithmetic mean of ten numbers is 36. if one of the numbers is 18,What is the mean of the other nine?
Answers
My answer is in the picture
7-15
Dividing decimals
Answers
Answer:
7-15= -8
Step-by-step explanation:
so therefore the other person is correct
Find all points on the x-axis that are 14 units from the point (5, -7)
Answers
Answer:
can you submit the coordinet plane?
Step-by-step explanation:
After subtracting 20% of a number from a number we get 12975.what was the number?
Answers
Answer:
Step-by-step explanation:
d
Consider the set A with n(A) = 20. How many subsets could be formed from this set?
Answers
Answer:
There are [tex]2^{20}[/tex] subsets of [tex]A[/tex]
Step-by-step explanation:
Using the formula for the number of subsets of a given (finite) set, the number of subsets of [tex]A[/tex] is
[tex]2^{n(A)}=2^{20}[/tex]
what is
[tex] \frac{28}{141} [/tex]
as a percentage
Answers
Answer:
19,86%
Step-by-step explanation:
[tex] \frac{28}{141} = 0.198582[/tex]
➡️ [tex]0.1986[/tex]
➡️ [tex]0.1986 \times \frac{100}{100} [/tex]
➡️ [tex] \frac{0.1986 \times 100}{100} [/tex]
➡️ [tex] \frac{19.86}{100} [/tex]
➡️ [tex] = 19.86\%[/tex] ✅
help plsss . 10 points !
Answers
Answer:
B
Step-by-step explanation:
(7×2)⁶
when we put a number to the power of something, we need to include the whole number. bit just a part of it.
so, it must be 7⁶×2⁶
just think about the simple example 6², which we could write as (2×3)².
would it be sufficient to e.g. square only one of the factors ?
6² = 36
but e.g. 2×3² = 18. so, that's really not it.
or add the two factors and then square them ?
2+3 = 5. 5² = 25. so, that's not it either.
or multiply the exponent in ?
2×3×2 = 12. so that's not it either.
no, it truly is you need to do the operation to all parts.
2²×3² = 4×9 = 36. yes, that fits.
therefore, B is the right answer.
Examine the tile pattern at right
Answers
b. The pattern grow by adding 1 tile above the tile and adding 1 tile at the right of the tile.
c. In figure 0, there will be 1 tile. We know this because in each successive figures a tile is added at the above and a tile is added to the right, so ineach preceeding figure the same is reduced. In figure 1, there are e tiles, so in figure 0, there will be 3-2 = 1 tile.
what fraction of the Earth's surface would be covered by the surface of the moon,if the radius of the Earth is 6,378km and the radius of the moon is 1.741km?
Answers
Answer:
3031081 / 40678884
Step-by-step explanation:
To solve this, we can find the surface area of the moon and Earth, and then see how much the moon covers the Earth. The surface area of a sphere is equal to 4πr², so the radius of the Earth is
4πr² = 4 * π * 6378²
and the radius of the moon is
4πr² = 4 * π * 1741²
To figure out how much of the Earth's surface that the moon covers, we can implement a ratio of moons:Earth. This will give us an understanding of how many moons go inside one Earth. We thus have
(4 * π * 1741²) : ( 4 * π * 6378²) = (4 * π * 1741²) / ( 4 * π * 6378²)
cross out the 4 * π in the numerator and denominator
1741²/6378²
Next, we want to make the denominator 1, as that gives us 1 Earth. To do this, we can divide both the numerator and denominator by 6378². Because we are applying the same expression to both the numerator and denominator, this is essentially multiplying the fraction by 1, keeping it the same. We thus have
(1741²/6378²)/(6378²/6378²)
≈0.0745/1
≈ 0.0745
To put this in a fraction, we would have
(1741²/6378²)/1
= (1741²/6378²)
= 3031081 / 40678884
PLssssssss helppppppppppppppppppppppp
Answers
Answer:
6 ft
Step-by-step explanation:
3) What type of correlation would temperature
and ice cubes have?
Answers
PLS HELPP , i tried solving this and i wrote 56 but it was wrong so i don’t know anymore pls pls help
Answers
Answer:
y = 68
A = 44
Step-by-step explanation:
No matter what may be true, that doesn't look like an a to me.
x = y = 68 x and y are opposite equal sides and are therefore =.
68 + 68 + A = 180 all triangles are 180 degrees.
136 + A = 180 Subtract 136 from both sides
A = 180 - 136
A = 44
The shaded rectangle, which is the correct expression of its perimeter
Answers
Answer:
2x + 2y - 4
Step-by-step explanation:
The opposite sides are congruent so perimeter (P) is
P = 2(x + 2) + 2(y - 4)
= 2x + 4 + 2y - 8
= 2x + 2y - 4
Mrs. Diaz had 5/6 gallon of paint to start. When she finished sha had 1/2 gallon. How much paint did Mrs. Diaz use?
Answers
Answer:
1/3
Step-by-step explanation:
To get the amount of paint,you will have to deduct the number of paint she used to start minus the amount of of gallon she finished with.
5/6-1/2
L.C.M
10-6/12
4/12
=1/3
Identify the equation of the circle that has its center at (9, 12) and passes through the origin.
Answers
Answer: [tex](x-9)^2 + (y-12)^2 = 225\\\\[/tex]
This is the same as writing (x-9)^2 + (x-12)^2 = 225
========================================================
Explanation:
Any circle equation fits the template of [tex](x-h)^2 + (y-k)^2 = r^2\\\\[/tex]
The center is (9,12) which tells us the values of h and k in that exact order.
h = 9
k = 12
To find the radius r, we need to find the distance from the center (9,12) to a point on the circle. The only point we know on the circle is the origin (0,0).
Apply the distance formula to find the distance from (9,12) to (0,0)
[tex]d = \sqrt{ (x_1-x_2)^2+(y_1-y_2)^2}\\\\d = \sqrt{ (9-0)^2+(12-0)^2}\\\\d = \sqrt{ (9)^2+(12)^2}\\\\d = \sqrt{ 81+144}\\\\d = \sqrt{ 225}\\\\d = 15\\\\[/tex]
The distance from (9,12) to (0,0) is 15 units. Therefore, r = 15
An alternative to finding this r value is to apply the pythagorean theorem. The distance formula is effectively a modified version of the pythagorean theorem.
---------------------
Since h = 9, k = 12 and r = 15, we can then say:
[tex](x-h)^2 + (y-k)^2 = r^2\\\\(x-9)^2 + (y-12)^2 = 15^2\\\\(x-9)^2 + (y-12)^2 = 225\\\\[/tex]
which is the equation of this circle.
In the Cartesian coordinate system, the x-akis is horizontal.
A.True
B. False
Answers
Answer:
A. TrueStep-by-step explanation:
In the Cartesian coordinate system, the x-axis is the horizontal line while y-axis is the parallel line.[tex]\tt{ \green{P} \orange{s} \red{y} \blue{x} \pink{c} \purple{h} \green{i} e}[/tex]
suppose an object traveling in a straight line has a velocity function given by v(t)= t^2 -8t+ 15 km/hr. Find the displacement and distance traveled by the object from t=2 to t=4 hours.
Answers
v=t^2-8t+15
It has upper limit 4 and lower limit 2[tex]\boxed{\sf {\displaystyle{\int}^b_a}x^ndx=\left[\dfrac{x^{n+1}}{n+1}\right]^b_a}[/tex]
[tex]\\ \sf\longmapsto s={\displaystyle{\int}}vdt[/tex]
[tex]\\ \sf\longmapsto s={\displaystyle{\int^4_2}}t^2-8t+15[/tex]
[tex]\\ \sf\longmapsto s=\left[\dfrac{t^3}{3}-8\dfrac{t^2}{2}+15t\right]^4_2[/tex]
[tex]\\ \sf\longmapsto s=\left[\dfrac{t^3}{3}-4t^2+15t\right]^4_2[/tex]
[tex]\\ \sf\longmapsto s=\left(\dfrac{4^3}{3}-4(4)^2+15(4)\right)-\left(\dfrac{2^3}{3}-4(2)^2+15(2)\right)[/tex]
[tex]\\ \sf\longmapsto s=\left(\dfrac{64}{3}-64+60\right)-\left(\dfrac{8}{3}-16+30\right)[/tex]
[tex]\\ \sf\longmapsto s=\left(\dfrac{64}{3}-4\right)-\left(\dfrac{8}{3}+14\right)[/tex]
[tex]\\ \sf\longmapsto s=\dfrac{64}{3}-4-\dfrac{8}{3}-14[/tex]
[tex]\\ \sf\longmapsto s=\dfrac{64}{3}-\dfrac{8}{3}-4-14[/tex]
[tex]\\ \sf\longmapsto s=\dfrac{46}{3}-18[/tex]
[tex]\\ \sf\longmapsto s=15.3-18[/tex]
Take it +ve[tex]\\ \sf\longmapsto s=|-2.7|[/tex]
[tex]\\ \sf\longmapsto s=2.7km[/tex]
50 Points to correct answer!!!
what is the average rate of change from 2 to 9 of the function represented by the graph?
Answers
Answer:
-3/7
Step-by-step explanation:
It is asking to find the slope of the secant line going through points (2,f(2)) and (9,f(9)).
We must find f(2) by looking at the curve at x=2. We should see that y=2 there so f(2)=2.
We must find f(9) by looking at the curve at x=9. We should see that y=-1 there so f(9)=-1.
The slope of a line is calculated by finding the ratio of the change of y to the change of x.
(-1-2)/(9-2)
(-3)/(7)
-3/7
x ^ 2 − 17x − 60
Which expression is equivalent to the expression above?
(Please explain in simple terms cause, it's usually hard for me to understand)
Answers
[tex] {x}^{2} - 17x - 60 \\ (x + 3)(x - 20)[/tex]
First we put parentheses and in each bracket we put (X) and then we put the signs x² is positive and the 17X before it is a negative q is positive with negative —> negative, and negative before 17X and negative before the 60 —> positive. And then the number that does not have (x) where did it come from, for example 60 came from 20 x 3 or 30 x 2...etc. We can verify this by multiplying the parentheses together and the same number comes out .
Or it can be checked by multiplying the first bracket 3 with x from the second parenthesis comes out 3X and negative 20 from the second parenthesis with X from the first parenthesis and subtract 3X from –20xcomes out –17X .
I hope I helped you^_^
i’ll give brainliest pls
i need to finish this quickly, i need part a b and c
Answers
C 70
I got it right
The product of 2 more than 5 times a number and 4 less than three times a number
Answers
Sorry I'm not sure if I'm right just my thinking...
set the number to be x so that (5x+2)(3x-4)??
Q16. If the perimeter of a rectangular house is 25 1/3 yards, and the length is 22 feet. What is the
width?
16 feet
O 35 feet
O 37 feet
0 40 feet
42 feet
Answers
Answer:
OPTION A :16 FEET
PERIMETER =26 1/3=76/3yard=76/3×3=76 feet
LENGTH=22feet
Width=76/2-22
WIDTH=38-22=16 FEET
