Answers
Answer:
[tex]\displaystyle \lim_{x \to 2} \frac{\sqrt{x + 7} - 3 \sqrt{2x - 3}}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} = \boxed{ \frac{34}{23} }[/tex]
General Formulas and Concepts:
Calculus
Limits
Limit Rule [Variable Direct Substitution]:
[tex]\displaystyle \lim_{x \to c} x = c[/tex]
Limit Property [Multiplied Constant]:
[tex]\displaystyle \lim_{x \to c} bf(x) = b \lim_{x \to c} f(x)[/tex]
Limit Property [Multiplication]:
[tex]\displaystyle \lim_{x \to c} f(x)g(x) = \lim_{x \to c} f(x) \lim_{x \to c} g(x)[/tex]
Step-by-step explanation:
*Note:
The problem is too big to fit all work. I will assume that you know how to do basic algebra and calculus.
Step 1: Define
Identify given limit.
[tex]\displaystyle \lim_{x \to 2} \frac{\sqrt{x + 7} - 3 \sqrt{2x - 3}}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}}[/tex]
Step 2: Find Limit (Algebriaclly)
Rationalize the function and apply basic limit techniques listed under "Calculus":
[tex]\displaystyle\begin{aligned}\lim_{x \to 2} \frac{\sqrt{x + 7} - 3 \sqrt{2x - 3}}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} & = \lim_{x \to 2} - \frac{17(x - 2)}{\big( \sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5} \big) \big( \sqrt{x + 7} + 3 \sqrt{2x - 3} \big)} \\& = -17 \lim_{x \to 2} \frac{1}{\sqrt{x + 7} + 3 \sqrt{2x - 3}} \lim_{x \to 2} \frac{x - 2}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}}\end{aligned}[/tex]
[tex]\displaystyle\begin{aligned}\lim_{x \to 2} \frac{\sqrt{x + 7} - 3 \sqrt{2x - 3}}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} & = - \frac{17}{6} \lim_{x \to 2} \frac{x - 2}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} \\\end{aligned}[/tex]
Let's now focus on just the rationalization of the function within the limit:
[tex]\displaystyle\begin{aligned}\frac{x - 2}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} & = \frac{(x - 2) \bigg[ (x + 6)^\Big{\frac{2}{3}} + 2 \sqrt[3]{x + 6} \sqrt[3]{3x - 5} + 4(3x - 5)^\Big{\frac{2}{3}} \bigg] }{\Big( \sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5} \Big) \bigg[ (x + 6)^\Big{\frac{2}{3}} + 2 \sqrt[3]{x + 6} \sqrt[3]{3x - 5} + 4(3x - 5)^\Big{\frac{2}{3}} \bigg] } \\& = - \frac{1}{23} \bigg[ (x + 6)^\Big{\frac{2}{3}} + 2 \sqrt[3]{x + 6} \sqrt[3]{3x - 5} + 4(3x - 5)^\Big{\frac{2}{3}} \bigg] \\\end{aligned}[/tex]
Substitute in the simplified function and continue:
[tex]\displaystyle\begin{aligned}\lim_{x \to 2} \frac{\sqrt{x + 7} - 3 \sqrt{2x - 3}}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} & = - \frac{17}{6} \lim_{x \to 2} \frac{x - 2}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} \\& = \frac{17}{138} \lim_{x \to 2} \bigg[ (x + 6)^\Big{\frac{2}{3}} + 2 \sqrt[3]{x + 6} \sqrt[3]{3x - 5} + 4(3x - 5)^\Big{\frac{2}{3}} \bigg] \\& = \frac{17}{138} (12) \\& = \boxed{ \frac{34}{23} } \\\end{aligned}[/tex]
∴ we have evaluated the following limit without L' Hopital's Rule.
Step 3: Geometrical Interpretation
We can also graph the given function and approximate the limit. Please refer to the attachment for visualization.
___
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___
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Limits
Answer:
34/23
Step-by-step explanation:
The limit can be evaluated by simplifying the expression in such a way as to cancel the offending zeros.
__
Define the following (to help keep the algebra cleaner):
[tex]a=\sqrt{x+7},\ b=3\sqrt{2x-3}\\\\c=\sqrt[3]{x+6}.\ d=2\sqrt[3]{3x-5}[/tex]
Then the desired limit is ...
[tex]\displaystyle \lim_{x\to 2}\dfrac{a-b}{c-d}[/tex]
We observe that ...
[tex](a-b)(a+b)=\underline{a^2-b^2}=(x+7)-(3^2(2x-3))=-17x+34=\underline{-17(x-2)}\\\\(c-d)(c^2+cd+d^2)=\underline{c^3-d^3}=(x+6)-(2^3(3x-5))=-23x+46=\underline{-23(x-2)}\\\\\displaystyle\lim_{x\to2}{(a+b)}=3+3=6\\\\\lim_{x\to2}{(c^2+cd+d^2)}=4+4+4=12[/tex]
In the following expression, the factors (x-2) cancel, so, our limit is ...
[tex]\displaystyle\lim_{x\to2}\dfrac{a-b}{c-d}=\lim_{x\to2}\dfrac{(a^2-b^2)(c^2+cd+d^2)}{(c^3-d^3)(a+b)}=\lim_{x\to2}\dfrac{-17(x-2)(12)}{-23(x-2)(6)}=\boxed{\dfrac{34}{23}}[/tex]
Related Questions
As of 2016, Illinois has a population of 12, 801,539 people and has a land area of 55,519 square miles. Texas has a population of 27,862,596 people and a land area of 261, 232 square miles. Based on this data, which state has the greater population density? Move the correct state to the first blank, and move numbers into the remaining blanks to justify the answer. The state with the greater population density is because Illinois Texas 107 231 55,519 261, 232 12,801, 539 27,862, 596
Answers
Using it's concept, it is found that the state with greater population density is Illinois.
What is population density?Population density is the number of habitants divided by the area.
Considering the data given, for Illinois, the density is given by:
Di = 12,801,539/55,519 = 230.58 habitants per square mile.
For Texas, the density is given by:
Dt = 27,862,596/261,232 = 106.66 habitans per square mile.
Di > Dt, hence the state with greater population density is Illinois.
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The list 11, 20, 31, 51, 82 is an example of a list of five strictly positive integers
arranged in ascending order and of which we obtain the third integer by adding the
first and second integers, the fourth by adding the second and third
integers, and the fifth by adding the third and fourth integers. How much
such lists of five positive integers have 124 as the fifth integer?
Answers
Write and solve an equation to find the value of x.
Answers
Answer:
x = 5
Step-by-step explanation:
By the property intersecting chords:
x(16) = (10)(8)
-> x (16) = 80
-> x = 80/16
-> x = 5
Find the area of the composite figure below
Answers
Answer:
the answer is 25 hope it helps
Can someone please help
Answers
Answer:
48
Step-by-step explanation:
The sum of the measures of the angles of a triangle is 180°.
We know one angle measures 42°.
Another angle is a right angle, so it measures 90°.
Only the angle labeled x is unknown.
42 + 90 + x = 180
132 + x = 180
x = 48
Answer: 48
A circle with a radius of 9 millimeters.
Answers
Answer:
You didnt give us a question to answer, just a statement.
Step-by-step explanation:
solve the system using elimination. show all work. write your answer in the form (x,y) 2x-3y=18 x+y=14
Answers
Answer:
x=12 y=2
Step-by-step explanation:
Due to fractions have screen shorted the steps in the image above
find four consecutive whole numbers whose sum is 74
Answers
Answer:
17, 18, 19, 20
Explanation:
Let the numbers be x, x + 1, x + 2, x + 3
Their sum is equals to 74
x + x + 1 + x + 2 + x + 3 = 74 (group the variables)
x + x + x + x + 1 + 2 + 3 = 74 (simplify)
4(x) + 6 = 74 (subtract both sides by 6)
4x = 68 (divide both sides by 4)
x = 17
Other numbers:
x + 1 = 17 + 1 = 18
x + 2 = 17 + 2 = 19
x + 3 = 17 + 3 = 20
Answer:
17, 18, 19, 20
Step-by-step explanation:
Let four consecutive whole numbers be x, (x + 1), (x + 2) and (x + 3)According to the given condition:x + (x + 1) + (x + 2) + (x + 3) = 74-> 4x + 6 = 74-> 4x = 74 - 6-> 4x = 68-> x = 68/-> x = 17-> x + 1 = 17 + 1 = 18-> x + 2 = 17 + 2 = 19-> x + 3 = 17 + 3 = 20Thus, the four consecutive whole numbers are 17, 18, 19 and 20Rachael mixes 750 mL of mango juice 650 mL of orange juice and 325 mL pineapple juice she add sparkling water to fill the punch bowl that has a capacity of 2.5 L how many liters of juice does Rachel use show all work
Answers
Answer:
A. 1.725 L
B. 0.0775 L or 775 mL
Step-by-step explanation:
Part A
↓ Determine the liters of juice Rachel used ↓
750 + 650 +325 = 1,725
Rachel used 1,725 mL of juice. Convert the given mililiters to liters.
[tex]1725 \: ml \: \times \frac{1liter}{1000 \: ml} = 1.725 \: liters[/tex]
→ Rachel used 1.725 L of juice
Part B.
Determine the amount of sparkling water added to the punch bowl.
2.5 - 1.725 = 0.775 L
0.775 L was added tot he punch bowl. Convert liters to milliliters. ↓
[tex]0.775 \: \times \frac{1000 \: ml}{1 \: liters} = 775 \: ml[/tex]
Therefore, the answer is:
1.725 L & 775 mL
A solid object was sliced to form two new objects. Each of the two new objects had a circular base. Which of the following solids may have been the original object? Select all that apply.
A: trapezoidal prism
B: cylinder
C: hexagonal prism
D: triangular pyramid
E: cone
Answers
What is the volume of a right circular cylinder with a base radius of 6 feet and a height of 10 feet? 900 cubic meters 120 cubic meters 180 cubic meters 360n cubic meters
Answers
Step-by-step explanation:
the volume of a cylinder is
base area × height = pi×r² × height
with r being the radius.
so, we get
pi×6² × 10 = pi×36 + 10 = 360pi =
= 1,130.973355... ft³
now we need to look at the conversion rate between feet and meters :
1 ft = 0.3048 m
1 ft² = 0.3048² m² = 0.09290304 m²
1 f³ = 0.3048³ m³ = 0.028316847... m³
1,130.973355... ft³ = 1,130.973355... × 0.028316847... =
= 32.025599... m³
so, something is not right with your answer options. it looks like you put maybe the answer options of a different question in here.
but none of them get even close to the actual solution of the given question.
fred buys a shipment of fruit that consists of only apples and oranges. for every 3 apples, there are 2 oranges. if there are 5 dozen pieces of fruit total, how many are oranges?
Answers
Answer:
24
Step-by-step explanation: 5 dozens = 60 so 3 +2=5 . 60 / 5 = 12 12 x 2 = 24
The swim team is sharing 6 pizzas. There are 36 players on the team. If each member has 1 slice, and each pizza has 8 slices, how much pizza in fraction form is left over?
Answers
Then take 36 and subtract that from 48 getting 12.
So there is 12 slices of pizza left over.
The amount of pizza left over in the fraction form will be 1/4.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that the swim team is sharing 6 pizzas. There are 36 players on the team. If each member has 1 slice, and each pizza has 8 slices.
Total slices = 8 x 6 = 48
Total players = 36
Left slices = 48 - 36 = 12
The portion in fraction left is,
L = 12 / 48
L = 1 / 4
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Identify the Pre-Image in the diagram below. EXPLAIN your answer.
Answers
A transformation's preimage is the shape before the change. The image that is a pre-image is A'B'C', which is in blue colour.
What are an image and pre-image?We have a term for the shape before the transformation and another one for the shape after the transformation. The shape after a transformation is the image of the transformation. A transformation's preimage is the shape before the change.
In maths, the pre-image is the original figure. Therefore, the image that is a pre-image is ABC, which is in red colour.
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3. Given the geometric series 300 +360+432 +518.4+, what is the sum of the first 10 terms?
Answers
The sum of the first 10 terms of the given geometric series 300 +360+432 +518.4+..... is 7787.60.
What is the sum of terms of a geometric sequence?Let's suppose its initial term is a , the multiplication factor is r
and let it has total n terms, then, its sum is given as:
[tex]S_n = \dfrac{a(r^n-1)}{r-1}[/tex]
(sum till nth term)
For the given geometric series, the common ratio is,
r = (360/300) = 1.2
Also, the first term of the series is 300. Therefore, the sum of the first ten terms of the series are:
Sum = a(rⁿ-1)/(r-1) = 300(1.2¹⁰-1)/(1.2-1) = 300(5.1917)/0.2 = 7787.60
Hence, the sum of the first 10 terms of the given geometric series is 7787.60.
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The graph of F(x), shown below, has the same shape as the graph of
G(x) = x4, but it is shifted to the right 3 units. What is its equation?
A. F(x) = - 3
O B. F(x) = (x-3)4
O C. F(x) = (x+3)4
O D. Fx) = 44+ 3
Answers
Answer: F(x) = 4x - 16
Step-by-step explanation: G(x) = 4x
if F(x) is the graph G(x) shifted over to the right by 4 then
F(x) = G(x-4) = 4(x-4) = 4x - 16
The general equation is f(x - amount_shifted)
The equation of the graph of the function F(x) is F(x) = (x - 3)⁴.
What is Translation of Functions?Translation of functions is defined as the when each point in the original graph is moved by a fixed units in the same direction.
There are horizontal translation and vertical translation of functions.
Given a function F(x) and G(x).
Given that,
F(x) has the same shape as the graph of G(x) = x⁴.
The difference is that F(x) is shifted to the right 3 units.
A function G(x) when shifted right to k units becomes G(x - k).
So the function F(x) is,
F(x) = G(x - 3)
= (x - 3)⁴
Hence the correct option is B.
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select the option for ? 3, 5, 12, 55, 648, ?
Answers
Answer:
What option are you talking about for, is it for your quiz or for your classwork
If f(x) =1/x and g(x) = x - 2 what is f(g(x)) when X = 1
Answers
Answer:
f(g(x)) = -1
Step-by-step explanation:
[tex]f(x) = \frac{1}{x} \\ \\ g(x) = x - 2 \\ \\ \implies \: f(g(x)) = \frac{1}{x - 2} \\ \\ \implies \: f(g(1)) = \frac{1}{1 - 2} \\ \\ \implies \: f(g(1)) = \frac{1}{ - 1} \\ \\ \implies \: f(g(1)) = - 1[/tex]
what's the answer?? please help me
Answers
-5/2
explanation:
can anyone help without using reciprocal?
Answers
Answer:
x = -5/3
Step-by-step explanation:
-4 = [tex] \dfrac{20}{3}x[/tex]
-4×3 = 20 x
-12 = 20 x
[tex]x = \dfrac{-5}{3}[/tex]
Write a second degree polynomial with real coefficients and the given root; -2i
Answers
Answer: x^2+4
Work Shown:
x = -2i
x^2 = (-2i)^2
x^2 = 4i^2
x^2 = 4(-1)
x^2 = -4
x^2+4 = 0
Side note: Since -2i is one root, this means 2i is the other conjugate root.
[tex]\text{For quadratic equations, if one root is complex, the other root will be its conjugate.}\\\\\text{So, the roots of the second degree polynomial are}~ \alpha = -2i ~~ \text{and}~~ \beta =2i \\\\\text{The equation is,}\\\\~~~~~~~x^2-(\alpha + \beta) x +\alpha \beta = 0\\ \\\implies x^2 -(-2i+2i)x +(2i)(-2i)=0\\\\\implies x^2 -0\cdot x -4i^2 =0\\ \\\implies x^2 -4(-1)=0\\\\\implies x^2 +4=0[/tex]
Grace collected data from twenty of her friends about the number of pets they have. The median number of pets is two.
What does the median say about Grace's data set?
Answer options with 4 options
A.
Everyone Grace surveyed has two pets.
B.
More of Grace's friends have two pets than any other number.
C.
Two is the middle number when the number of pets in the data set is ordered from least to greatest.
D.
Two is the difference between the greatest and least number of pets Grace's friends have.
Answers
The statement which median describe about Grace's data set is 2 is the middle number when the number of pets in the data set is arranged from least to greatest.
What is the median of a data set?The median of the data is the middle value of the data set when it arrange in ascending or descending order.
Grace collected data from twenty of her friends about the number of pets they have. The median number of pets is two.
Median=2
By the definition of median of data set, the median two tells that the middle value of the pet is 2 when arranged in high to low point.
Thus, the statement which median describe about Grace's data set is 2 is the middle number when the number of pets in data set is arranged from least to greatest.
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First round 0.676 to the nearest tenth.
Answers
Answer:
0.7
Step-by-step explanation:
Since 7 is above 5, you round 0.676 to 0.7
Answer: 0.7
Step-by-step explanation:
The nearest tenth is one decimal space after the dot. ex: 0.1234
Therefore, rounding 0.676, the 7 is ≥ 5, therefore we round the 6 up one number. This means the 0.6 becomes 0.7.
Eric has 9 large pitchers for a sport drink at the soccer game. If he pours 4/7 of the sports drink into each pitcher will the total amount he lours into those pitchers be more or less than9 gallons?
Answers
Answer:
more because it is 9 pitchers and 4/7 of energy drink in each
pls help i need a good grade pls
Answers
Answer:
[tex]4*\frac{1}{6}[/tex]
Step-by-step explanation:
The number lines end points are 0 and 1 so the whole number line has a value of 1.
The number line is also broken up into 6 sections.
Then we have the pink rectangles that cover up each section. There are 4 of them and each cover 1/6 of the number line
So the expression that would represent the model would be 4 x 1/6
Answer:
4 * 1/6
Step-by-step explanation:
So to begin, I would find out what all the answers are equal to.
4 * 1/8 = 4/8 = 1/2
4 * 1/4 = 4/4 = 1
4 * 1/6 = 4/6 = 2/3
4 * 1/3 = 4/3
Next I would rule out all the answers bigger or equal to one, so 4 * 1/4 and 4 * 1/3.
Last, this fraction splits one into six parts, and 4 parts are highlighted. 4/6 is equal to 4/6. Your answer is 4 * 1/6
Margo borrows $200, agreeing to pay it back with 8% annual interest after 7 months. How much interest will she pay?
Answers
Answer:
$9.33
Step-by-step explanation:
Interest = Principal × Rate × Time by 100
Therefore, I = 200×8×7/12 by 100
=16×7/12
=9.33
Please i really need help, i really need this i will give a brainliest to the people that answer correctly
Answers
Answer:
Graph B
Step-by-step explanation:
Replace y with x and vice versa.
y=1/2x+2 becomes x=1/2y+2
Then solve for y. Subtract 2 from both sides, then multiply by 2 which results in:
y=2x-4
Graph B is the one that has a y-intercept at -4 and a rise/run of 2.
PLEASE HELP ME I GIVE BRAINLIEST
Answers
Answer:
[tex]\frac{2\pi }{3} \ ,\ \pi \ ,\ \frac{4\pi }{3}[/tex]
Step-by-step explanation:
suppose cos(x) = y
Then
[tex]2\cos^{2} \left( x\right) +3\cos x+1=0\Longleftrightarrow \begin{cases}2y^{2}+3y+1=0&\\ y=\cos \left( x\right) \ \ \text{and} &0\leqslant x\leqslant 2\pi \end{cases}[/tex]
2y² + 3y + 1 = 0 ⇒ y = -1 or y = -1/2 (because 2-3+1=0)
y = -1 ⇔ cosx = -1 ⇒ x = π (because x∈[0,2π] )
y = -1/2 ⇔ cosx = -1/2 ⇒ x = 2π/3 or x = 4π/3
Final answer :
x = π or x = 2π/3 or x = 4π/3
Can someone please help me with 4, 5, 6, 7, 8
Answers
5) 2,1
6)2,1
7) 3,10
8)-3,8
5) x=2, y=1
6) x=2, y=1
7) x=3,y=10
8) x= -3, y= 8
Yahaya, a Tailor started sewing an agabada dress at 9:45am. he completed the dress at 12:30pm. How long did he take to sew the dress