Answer:
[tex]\displaystyle \lim_{x\to 1^+}(x-1)^\ln(x)}=1[/tex]
Step-by-step explanation:
We are given:
[tex]\displaystyle \lim_{x\to 1^+}(x-1)^\ln(x)[/tex]
And we want to evaluate it using L'Hopital's Rule.
First, using direct substitution, we will acquire:
[tex]=(1-1)^\ln(1)}=0^0[/tex]
Which is indeterminate.
In order to apply L'Hopital's Rule, we first need to manipulate the expression. We will let:
[tex]y=(x-1)^\ln(x)[/tex]
By taking the natural log of both sides:
[tex]\ln(y)=\ln(x)\ln(x-1)[/tex]
And by taking the limit as x approaches 1 from the right of both functions:
[tex]\displaystyle \lim_{x\to 1^+}\ln(y)=\lim_{x\to 1^+}\ln(x)\ln(x-1)[/tex]
Rewrite:
[tex]\displaystyle \lim_{x\to 1^+}\ln(y)= \lim_{x\to 1^+}\frac{\ln(x-1)}{\ln(x)^{-1}}[/tex]
Using direct substitution on the right will result in 0/0. Hence, we can now apply L'Hopital's Rule:
[tex]\displaystyle \lim_{x\to 1^+}\ln(y)= \lim_{x\to 1^+}\frac{1/(x-1)}{-\ln(x)^{-2}(1/x)}[/tex]
Simplify:
[tex]\displaystyle \lim_{x\to 1^+}\ln(y)= \lim_{x\to 1^+}\frac{1/(x-1)}{-\ln(x)^{-2}(1/x)}\Big(\frac{-x\ln(x)^2}{-x\ln(x)^2}\Big)[/tex]
Simplify:
[tex]\displaystyle \lim_{x\to 1^+}\ln(y)= \lim_{x\to 1^+}-\frac{x\ln(x)^2}{x-1}[/tex]
Now, by using direct substitution, we will acquire:
[tex]\displaystyle \Rightarrow -\frac{1\ln(1)^2}{1-1}=\frac{0}{0}[/tex]
Hence, we will apply L'Hopital's Rule once more. Utilize the product rule:
[tex]\displaystyle \lim_{x\to 1^+}\ln(y)= \lim_{x\to 1^+}-(\ln(x)^2+2x\ln(x))[/tex]
Finally, direct substitution yields:
[tex]\Rightarrow -(\ln(1)^2+2(1)\ln(1))=-(0+0)=0[/tex]
Thus:
[tex]\displaystyle \lim_{x\to 1^+}\ln(y)=0[/tex]
By the Composite Function Property for limits:
[tex]\displaystyle \lim_{x\to 1^+}\ln(y)=\ln( \lim_{x\to 1^+}y)=0[/tex]
Raising both sides to e produces:
[tex]\displaystyle e^{\ln \lim_{x\to 1^+}y}=e^0[/tex]
Therefore:
[tex]\displaystyle \lim_{x\to 1^+}y=1[/tex]
Substitution:
[tex]\displaystyle \lim_{x\to 1^+}(x-1)^\ln(x)}=1[/tex]
To linear functions are shown below. Which function has a greater rate of change?
At a wedding reception, the caterer served fishing chicken. The fish dinner cost $12 per person in the chicken cost $9 per person. The caterer charged 2544 total for the dinner. If 232 people went to the reception, how many people had fish how many had chicken.
Answer:
80 persons haf fish and 152 persons had chicken
Step-by-step explanation:
Given data
let the number of persons that had fish be x and chicken be y
so
12x+9y=2544------------------1
x+y= 232------------------------2
from 2
x= 232-y
substitute the value of x in 1
12(232-y)+9y=2544
2784-12y+9y=2544
2784-3y=2544
3y= 2784-2544
3y=240
y= 240/3
y=80
put y= 80 in 2
x+80= 232
x= 232-80
x=152
Hence 80 persons haf fish and 152 persons had chicken
I really don’t get how to do this. It’s really confusing me! I really need help, if anyone can answer this it’ll be much appreciated
find the length of NO
find the measure of
Answer:4-2+3 Step-by-step explanation:8
At the zoo, Peter saw 9 animals. Some are kangaroos and some are
elephants. There are 30 legs in all. How many of each animal are there?
Answer:
6 elephants and 3 kangaroos in the zoo.
Step-by-step explanation:
Number of animals = 9
Number of legs seen = 30
But an elephant has 4 legs, while a kangaroo has 2 legs.
Let the number of elephant be represented by E, and the number of kangaroo by K.
So that,
E + K = 9 .................. 1
4E + 2K = 30 ............. 2
Multiply equation 1 by 2 to have
2E + 2K = 18 ............ 3
Subtract equation 3 from 1
2E = 12
E = 6
Substitute the value of E in equation 1
E + K = 9
6 + K = 9
k = 3
Thus, there were 6 elephants and 3 kangaroos in the zoo.
Which fraction is equivalent to -3/2
Answer:
Any fraction that will reduce to -3/2 will be equivalent to -3/2.
Examples:
-6/4
-24/16
Answer:
-6/4
Step-by-step explanation:
You can just multiply them by any factor. Remeber that what you do to the bottom you do to the top.
So,
-3/2 x 2
-6/4 (this is a fraction equivalent to it.
hope this heped :)
Michael practiced his harmonica 3 times as long as Timothy did. Michael practiced for 126 minutes. Write an equation that represents the relationship between the amount of time Timothy practiced x and the amount of time Michael practiced.
Answer:
126 = 3x
Step-by-step explanation
1) Find the slope and the y-byintercept of the line that passes through the points (1,5) and (3,9)
Answer: slope= 2, y-intercept= 3
Find the area...
HURRY!!!!!
Answer: 38
Step-by-step explanation:
The volume of the rectangular prism to the left is represented by the equation V = x ^ 4 + 11x ^ 3 + 34x ^ 2 + 5x - 75 Using what you know about these shapes, whose volumes are calculated using the formula V = lwh find the expression that represents the height (h). Length = x+5, Width = x+3
Answer:
[tex]h = x^2+3x-5[/tex]
Step-by-step explanation:
Given
[tex]V = x^4 + 11x^3 + 34x^2 + 5x - 75[/tex]
[tex]L = x + 5[/tex]
[tex]W = x + 3[/tex]
Required
Determine the height (h) of the prism
Volume is calculated as:
[tex]V =lwh[/tex]
Substitute values for V, l and w
[tex]x^4 + 11x^3 + 34x^2 + 5x - 75 = (x + 5) * (x + 3) * h[/tex]
Factorize the expression on the left-hand side
[tex](x+3)(x^3+8x^2+10x-25)= (x + 5) * (x + 3) * h[/tex]
Further, factorize:
[tex](x+3)(x+5)(x^2+3x-5)= (x + 5) * (x + 3) * h[/tex]
Divide both sides by (x+3)(x+5)
[tex]\frac{(x+3)(x+5)(x^2+3x-5)}{(x+3)(x+5)}= \frac{(x + 5) * (x + 3) * h}{(x+3)(x+5)}[/tex]
[tex]\frac{(x+3)(x+5)(x^2+3x-5)}{(x+3)(x+5)}= h[/tex]
[tex](x^2+3x-5)= h[/tex]
[tex]x^2+3x-5= h[/tex]
[tex]h = x^2+3x-5[/tex]
The height of the prism is [tex]x^2+3x-5[/tex]
HELP PLS use the original price and markup to find the retail price. original price $68 markup 20%
Answer: $81.60
Step-by-step explanation:
When we have an original price OP, and a markup of X% (this is how much increases the price with respect to the original price)
The retail price will be:
RP = OP + (X%/100%)*OP
RP = OP*(1 + X%/100%)
In this case, we have:
Original price = OP = $68
markup = X% = 20%
Replacing these in the above equation, we get:
RP= $68*(1 + 20%/100%) = $68*(1 + 0.20)
RP = $68*(1.20) = $81.60
Then the retail price of this particular item is $81.60
Pls help I have no idea what the answer is
Answer:
Step-by-step explanation:
x/5=5/2
x=25/2
x=12.5
Solve the system of linear equations by elimination.
8x-5y=11
4x-3x=5
Answer:
x=5
y=-5.8
Step-by-step explanation:
4x-3x=1x
1x=5
x=5
8(5)-5y=11
40-5y=11
40-11=29
-5y=29
29/-5=-5.8
y=-5.8
I need help with this
PLEASE HURRY !!!
B
c
Which statement is correct for AABC and AA'B'C'?
SABC - AA'B'C', because AA'B'C' is obtained by dilating AABC by a scale factor of 1 and then rotating it about the origin by 90 degrees
6
AABC - AA'B'C', because AA'B'C' is obtained by dilating AABC by a scale factor of and then rotating it about the origin by 180 degrees
AABC - AA'BºC, because AA'B'C' is obtained by dilating AABC by a scale factor of and then rotating it about the origin by 180 degrees
AABC - AA'B'C', because AA'B'C' is obtained by dilating AABC by a scale factor of and then rotating it about the origin by 90 degrees
3
Answer:
2nd choice
Step-by-step explanation:
1. dilation factor: 1/2
2. rotation: about origin anti-clockwise 180°
Δ ABC - Δ A'B'C', because AA'B'C' is obtained by dilating AABC by a scale factor of 1/2 and then rotating it about the origin by 180 degrees.
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
We have to find the statement which is correct for triangle ABC and triangle A'B'C'.
Let us find the coordinates of triangle ABC
A=(2, 2), B=(2, 10) and C=(8, 12)
After transformation the triangle A'B'C'.
Coordinates A' = (-1, -1), B(-1, 5) and C'(-4, 6)
The scale factor is 1/2.
A dilation is a transformation which preserves the shape and orientation of the figure, but changes its size.
Δ ABC - Δ A'B'C', because AA'B'C' is obtained by dilating AABC by a scale factor of and then rotating it about the origin by 180 degrees.
Hence, Δ ABC - Δ A'B'C', because AA'B'C' is obtained by dilating AABC by a scale factor of 1/2 and then rotating it about the origin by 180 degrees.
To learn more on Graph click:
https://brainly.com/question/17267403
#SPJ7
PLEASE HELP WILL GIVE BRAINLIEST
Please explain too just so I can believe you. If someone else answers I will give you brainliest (if you are correct only)
Answer:
B. y = 3x + 1
Step-by-step explanation:
We can look at the table like a bunch of values, where (x) is what is inserted into the equation, and (y) is what comes out.
Now, simply insert the (x) values into each equation and see if they work out or not.
y = (1/3)(-1) + 1 does not equal -2, meaning this equation is defunct.
y = 3(-1) + 1 = -2; y = 3(0) + 1 = 1. This must be the equation as it works for the (x) values given. You can try the other equations, but a simple look at them is all you need to prove them false.
Can anybody help me please?
Answer:
Well BStep-by-step explanation:
Just make the graph and compare them. Graph B. can be seen below
Answer: Well B
Step-by-step explanation:
Calculating, the rate was found out to be
y= 2.5x
Let y1= 0, y2= 10
x1= 0, x2= 4
(y2-y1)/(x2-x1) = (y- y1)/(x- x1)
(10-0)/(4-0) = (y-0)/(x-0)
10/4 = y/x
y= (10/4)x= 2.5x
PLS HELP ME pls???!! I’m
Answer:
13.5 centimeters
Step-by-step explanation:
Solution #1:
15 * 0.9 = 13.5
Solution #2:
15 * 0.1 = 1.5
15 - 1.5 = 13.5
Question 1 (1 point)
What is the solution to the equation
below?
5 + 6x = 2x - 7
Answer:
x = negative 3
Step-by-step explanation:
i just know lol
Can I please have help with this one
Answer:
The second line cuts the x-axis at x = -5/2.
Step-by-step explanation:
The given line:
[tex]3x-2y=12[/tex]
Intersects a second line at x = 2.
This second line is perpendicular to the first.
And we want to find the second line's x-intercept.
First, rearrange the first line into slope intercept form:
[tex]-2y=-3x+12[/tex]
So:
[tex]\displaystyle y=\frac{3}{2}x-6[/tex]
We can see that the slope of the first line is 3/2.
It intersects the second line at x = 2.
So, the y-value of both the first and second line at x = 2 is the same.
Find the y-value by using the first line:
[tex]\displaystyle y=\frac{3}{2}(2)-6=3-6=-3[/tex]
So, a point on both lines is (2, -3).
Since the second line is perpendicular to the first, its slope it the first's negative reciprocal.
The negative reciprocal of 3/2 is -2/3.
So, the slope of the second line is -2/3.
And we determined that it passes through (2, -3).
Then by the point-slope form:
[tex]y-y_1=m(x-x_1)[/tex]
Substitute:
[tex]\displaystyle y-(-3)=-\frac{2}{3}(x-2)[/tex]
Distribute:
[tex]\displaystyle y+3=-\frac{2}{3}x+\frac{4}{3}[/tex]
So:
[tex]\displaystyle y=-\frac{2}{3}x-\frac{5}{3}[/tex]
The second line will cut the x-axis when y = 0.
So:
[tex]\displaystyle 0 =-\frac{2}{3}x-\frac{5}{3}[/tex]
Multiply both sides by -3:
[tex]0=2x+5[/tex]
Solve for x:
[tex]\displaystyle x=-\frac{5}{2}[/tex]
The second line cuts the x-axis at x = -5/2.
when johns age is doubled, the number is 5 more than kaylas age. if john is x years old, what is kayalas age?
Answer:
Ang mga Babae sa Relihiyon at Pilosopiya sa Asya. Limang Ugnayan. emperador-mamamayan. ama-anak na lalaki.
i have to solve for H
Answer:
h = [tex]\frac{3v}{\pi r^{2} }[/tex]
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
Willow class has been practicing typing in the computer lab she can type 23 words per minute write an equation to find the W the number of words she will be after 9 minutes then solve
Answer:
(23)(9) = 207
Step-by-step explanation:
Hope this helped
Answer: 207
Step-by-step explanation:
in 1 min she can type 23 words
1 min = 23 words
9 min = x
x = 23 * 9
=> 207
What is the degree of this Polynomial!!!!
Answer:
4
because it is the greatest power in the polynomial
PLEASE i need help
Select all of the expressions that are equivalent to: 8 x 8 x 8 x 8 x 8
8 x 8^4
8^3 x 8^2
5(2^3)
(2^3)^5
8^5
8^3 / 8^2
8^4 / 8^2
Select all the expressions that are equivalent to
6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6
6^9
216 + 6^6
216 x 6^6
6^5 + 6^4
6^7 + 6^2
6^5 x 6^4
9^6
house a and house b are very similar houses located in different parts of the city.The graph shows the value of each house for the first part of the year?
The answer is 17,000 it is simple addition. Ur name says Ur in highschool. I am in sixth grade. I just solved it. Just do me a favor and try next time.
A lion's heart beats 6 times in 8 seconds. How many heartbeats will it have in 40 seconds?
Answer:
30 times
Step-by-step explanation
Answer:
30
Step-by-step explanation:
Asap
Each part needs fourteen small holes drilled in its outer surface and four larger holes drilled in its inner surface. Write an expression that shows how many holes need to be drilled for c parts.
Answer:
(14 + 4)c, or 18c
Step-by-step explanation:
Answer:
(14 + 4)c, or 18c
Step-by-step explanation:
Find the slope of the line that passes through (0, -5) and (3, -4)
Answer: I believe it is 2
slope=y2-y1/x2-x1
y2-y1= (0,-3)
x2-x1= (4,5)
m=5-(-3)/4-0
m=2
What is the equation of the function that is graphed as line a? y= 2x - 1 y= -x y = 3x y= - x - 1
Answer:
y= - x - 1
Step-by-step explanation:
You can see it goes through -1 on both axis' and it's the opposite way, making it -x