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]
1) obeserve os numeros e diga:
-15, +6, -1, 0, +54, +12, -93, -8, +23, -72, +72
a) quais os numeros inteiros negativos?
b)quais os numeros inteiros postivos?
Split 108 in the ratio 1:3
Split 24 in the ratio 3:5
Split 65 in the ratio 4:9
In this hexagon, all the angles are equal. What is the measure of angle x°?
Answer:
Option - b: 60
Step-by-step explanation:
Sum of exterior angle = 360
Exterior angle = 60
6x = 360
x = 360 / 6 = 60
-x+5+4x
simply the expression
Answer:
3x+5
Step-by-step explanation:
Need help and fast.
match the following equations with the correct description of the solutions.
equations
(2-3x)=2(5x+3)
x-3(2+3x)=2(5x-3)
x-3(2-3x)=2(5x-3)
soulutions
A infinitely many solutions
B no solutions
C one solution
Answer:
c
Step-by-step explanation:
Step-by-step explanation:
(2-3x)=2(5x+3) has 1 solution
x-3(2+3x)=2(5x-3) has no solution
x-3(2-3x)=2(5x-3) has infinite solution
Hope this helps
Please give me Brainliest!
iTS LEGIT EASY POINTS XD
Answer:
thanks boyz
Step-by-step explanation:
Answer:
7x+78
Step-by-step explanation:
i just know
2r - 1⅔r ＞ -⅓ solve the inequality
Answer:
r > -1.
Step-by-step explanation:
2r - 1⅔r ＞ -⅓
1/3 r > -1/3
r > -1/3 / 1/3
r > -1.
Answer:
r>1/3 ima put the explanation in the comments
Three fifths of the students in the cafeteria bought their lunch from the school. The rest brought their lunch from home. 27 of the students bought their lunch. Let t represent the total number of students in the cafeteria.
Answer:
45 students
Step-by-step explanation:
Three fifths of the students in the cafeteria bought their lunch from the school. The rest brought their lunch from home. 27 of the students bought their lunch. Let t represent the total number of students in the cafeteria.
We are to calculate the total number of students in the cafeteria.
Fraction of the students in the cafeteria that bought their lunch from the school = 3/5
Number of the students that bought their lunch = 27 students
The total number of students can be calculated as:
3/5 Of t = 27 students
3/5 × t = 27
Cross Multiply
3t/5 = 27
3t = 27 × 5
t = 27 × 5/3
t = 45 students
Therefore, the total number of students in the cafeteria = 45 students
Kyle ran the race in three and one hundred three thousandths seconds. Which number shows three and one hundred three thousandths in standard form.
Answer:
Ran =3,103
Seconds = 1330
Step-by-step explanation:
I got this answer hope its right hope this helps: )
Find the product -10 x (-12)
x²-6x-72
Step-by-step explanation:
(x+6)(x-12)
(x×x)-(x×12)+(6×x)-(6×12)
x²-12x+6x-72
x²-6x-72
Answer:
120
Step-by-step explanation:
The rules of multiplication state, that when there are two negatives being multiplied by each other, they become positive. You will multiply -10 x -12, and since there are two negatives, they cancel out.
Now it becomes simple multiplication. Simply multiply 10 x 12, without the negatives since they have been canceled out.
-10 x (-12) = 120
Akira is folding a piece of paper that is 0.2 millimeters thick in half. The thickness of the paper doubles with each fold. Let n represent the fold number and t(n) represent the thickness of the paper. Which exponential function represents this scenario?
Answer:
t(n)= 0.2 * 2ⁿ
Step-by-step explanation:
Thickness= 0.2 millimeters
paper = 1
n= number of folds
The exponential function will be given by
t(n)= 0.2 * 2ⁿ
so if there's a single fold
t(1)= 0.2 * 2¹
t(1)= 0.4 mm
Now for 2 folds the thickness will be
t(2) = 0.2 *2 ²
t(2)= 0.2 *4
t(2) = 0.8mm
The exponential that represents this scenario is
t(n)= 0.2 * 2ⁿ
And for 3 folds the thickness will be
t(3) = 0.2 *2 ³
t(3)= 0.2 *8
t(3) = 1.6 mm
Please help due tonight
Tom buys 3 burgers and 5 drinks for $34. Jane buys 2 burgers and 6 drinks for $36. How much is each burger and each drink?
Please i need help with this please
Answer:
Your answer would be 9.
Step-by-step explanation:
You always do parenthesis first, 9-6 is 3, then you would do division 3÷3 is 1 then you multiply 4×2 and that would be 8, finally you would at the 8 and the 1 and that would give you 9 as your answer. Hope this helps
What is a reasonable height for a 5-story building?
16 mm
16 km
16 cm
16 m
Answer:
16 km !
Step-by-step explanation:
all the others are FAR too small.. they wouldnt even be one story up ! and 16m isnt 5 stories either... km would be right !
i hope this helps !!
Find the Volume of a hemisphere that has a radius of 8 centimeters. Round to the nearest tenth
Answer:1072.3
Step by step:
use the hemisphere formula which is 2/3*pie*r^3
substitute 8 centimeters into the equation.
The Volume of a hemisphere is,
⇒ Volume of hemisphere = 1071.78 cm³
What is Multiplication?To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.
We have to given that;
Radius of hemisphere = 8 centimeters
We know that;
Volume of hemisphere = 2/3 (πr³)
Hence, We get;
⇒ Volume of hemisphere = 2/3 × 3.14 × 8³
⇒ Volume of hemisphere = 1071.78 cm³
Thus, The Volume of a hemisphere is,
⇒ Volume of hemisphere = 1071.78 cm³
Learn more about the multiplication visit:
https://brainly.com/question/10873737
#SPJ3
7
9
3
10
Simplify the answer if possible.)
i don't really get what the question is could you write it again?
Complete the equation of the line whose y-intercept is (0,5) and slope is -9.
y=
Answer: y=-9x+5
Step-by-step explanation:
slope intercept form is y=mx+b
simply plug in the values.
m= the slope
the slope you gave was -9, so plug that in to m.
b= the y intercept aka when the number touches the y axis. in this case, it would be 5.
y=-9x+5
The area of the triangle is
108 m². What is the value
of x?
xm
(x+6) m
Answer:
x = 12
Step-by-step explanation:
Formula for area of a triangke is;
A = ½bh
Where;
b is base
h is height
From the triangle diagram given;
b = x + 6
h = x
Thus;
A = ½(x + 6)x
A = ½x² + 3x
Now we are told that the area is 108 m², Thus;
½x² + 3x = 108
Multiply through by 2 to get;
x² + 6x = 216
x² + 6x - 216 = 0
Using quadratic formula, we arrive at;
x = -18 or 12
x can't be negative, thus; x = 12
Use the figure shown below to find the value of x
what i don't understand why you are not answer that question
PLEASE ANSWER ASAP!!!!
Sofia is making quarter circle felt pieces for a sewing project. If the diameter of one full circle is 6.2 inches. What is the area of each quarter circle rounded to the nearest hundredth? Use 3.14 for .
A. 7.54
B. 30.01
C. 15.05
Answer:
b
Step-by-step explanation:
Which angle has the same measurement as angle DEA, and how do you know that the measurement is the same? Will mark Brainly or how ever you call it.
Answer:
∠DEA=∠BAC
Step-by-step explanation:
∠DAE=∠BCA (corresponding angles)
Angles in a triangle add up to 180:
Angles in ΔDAE=Angles in ΔABC
∠DAE+∠EDA+∠DEA=∠BCA+∠ABC+∠BAC
∠EDA+∠DEA=∠ABC+∠BAC (since ∠DAE=∠BCA, we can cancel them out)
∠DEA=∠BAC (∠EDA=∠ABC=90)
Write 62/100 as a decimal
Answer:
.62
Step-by-step explanation:
62 ÷ 100 = .62
:)
Answer: 0.62
Step-by-step explanation: plz this is the right answer because we have to always put like this
62/100 -> 62/100 you have to always go backward when it 0 then u can go one time when it 00 then you have to go twice 62/100 count 62 oh sorry I couldn't show very well but that is how it is
[tex]2x^{2} +5x+2[/tex]
Answer:
(2x+1)(x+2)
Step-by-step explanation:
Ez pz
On circle O below, SE is a diameter. What is the measure of ZISE?
Answer: 34
Step-by-step explanation:
HELP MEEEE!!!!!!!!!!!!!!!!!!!! I will give brainliest to whoever answers first.
1) x-1+5[tex]\sqrt{x-1\\}[/tex]+3=0
4) [tex]x^{4}[/tex] -64=0
Answer:
See explanation below
Step-by-step explanation:
Find the value of x in each of the following
1) x-1+5√x-1+3=0
5√x-1 = -x -2
Square both sides
25(x-1) = (-x-2)^2
25(x-1) = x^2+4x + 4
25x-25 = x^2+4x + 4
x^2 +4x + 4 - 25x + 25 = 0
x^2 - 21x + 29 = 0
x = 21±√21²-4(29)/2
x = 21±√441-116/2
x = 21±√325/2
x = 21±18.03/2
x = 21+18.03/2 and 21-18.03/2
x = 19.515 and 1.485
4) x^4-64=0
x^4 = 64
[tex]x = \sqrt[4]{64}[/tex]
x = 2.83
Multiple Choice The Astrodome-in Houston, Texas, is circular and has a diameter of 710 feet. What is the approximate circumference of the Astrodom
Answer:
Step-by-step explanation:
[tex]C=\pi d\\ \\ C=710\pi ft\\ \\ C\approx 2230.5 ft[/tex]
Answer:
2230.53
Step-by-step explanation:
PLEASE HELP!! view attachment!!
What is the answer to 1 ft x 1 1/2 ft = ?
Answer:
1 1/2 is yuh ansuh
Kenny wants to take his dog to obedience school. At Sit and Stay School, 12 lessons cost $96. At Canine Community Center, 25 lessons cost $125. Which is the better deal?
Answer:
25 lessons that cost $125
Step-by-step explanation:
If 8 lessons cost $96
then, 96/12 =8
therefore a lesson costs $8
If 25 lessons costs $125
then, 125/25= 5
therefore a lesson costs $5 which is cheaper.