using the path method for multivariable limits, demonstrate that the limit doesn't exist: lim ({x, b}->{1, 1}) (x y - x - y + 1) / (x^2 + y^2 - 2 x - 2 y + 2)
Answers
To demonstrate that the limit of a multivariable function as (x,y) approaches (1,1) doesn't exist, we can use the path method and find two different paths with different limits.
To demonstrate that the limit doesn't exist using the path method for multivariable limits, we need to find two different paths that approach the point (1, 1) and give different limits.
Let's consider the paths y = x + b and x = y + b, where b is a constant.
Path 1: y = x + b
As we approach the point (1, 1) along this path, we have:
lim (x,y)->(1,1) (x y - x - y + 1) / (x^2 + y^2 - 2 x - 2 y + 2)
= lim x->1 [(x^2 + bx - x - bx + 1) / (x^2 + (x + b)^2 - 2 x - 2 (x + b) + 2)]
= lim x->1 [(x^2 - x + 1) / (2x^2 - 2x + 2b^2 + 2b)]
Using L'Hopital's rule, we can find that the limit as x approaches 1 is:
lim x->1 [(x^2 - x + 1) / (2x^2 - 2x + 2b^2 + 2b)] = 1 / (4b^2 + 2b)
Path 2: x = y + b
As we approach the point (1, 1) along this path, we have:
lim (x,y)->(1,1) (x y - x - y + 1) / (x^2 + y^2 - 2 x - 2 y + 2)
= lim y->1 [(y^2 + by - y - (y + b) + 1) / ((y + b)^2 + y^2 - 2 (y + b) - 2 y + 2)]
= lim y->1 [(y^2 - 2by + 1) / (2y^2 - 4by + 2b^2 + 2)]
Using L'Hopital's rule, we can find that the limit as y approaches 1 is:
lim y->1 [(y^2 - 2by + 1) / (2y^2 - 4by + 2b^2 + 2)] = 1 / (2b^2 - 2b)
Since the limits along the two paths are different, the limit of the function as (x,y) approaches (1,1) does not exist.
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Related Questions
yeah, i need help solving this and i need help quick please.
Answers
Angle e and c are the same because they are opposite interior angles. Therefore you can set it up in an equation like: 117=5P+27 and solve.
117=5P+27 you need to subtract 27 from both sides
90=5P you need to divide both sides by 5 to get
P=18
At a sale, coats were sold for $10 each. If the coats originally cost $20 each, what percentage of its original price was a coat sold for?
Answers
Answer: 50%
Step-by-step explanation:
10/20×100=50%
Answer:
50
Step-by-step explanation:
which one of the following points lies on the line x = -(t-6), y = 6t 5
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The given points (a) lies on the line, (b) does NOT lie on the line, and (c) lies on the line.
A point lies on a line if and only if it can be written as the sum of a scalar multiple of a direction vector and a position vector of the line.
To determine if each point lies on the line defined by X = -5 + t, y = 6t, and z = 6+ t, we can substitute each point into this equation to see if it is equivalent.
(a) (0, 30, 11):
-5 + t = 0, so t = 5
y = 6t = 6 × 5 = 30
z = 6 + t = 6 + 5 = 11
So, (0, 30, 11) can be written as -5 + 5i + 6 × 5j + 6k = -5i + 30j + 11k, which lies on the line.
(b) (5, 6, 11):
-5 + t = 5, so t = 10
y = 6t = 6 × 10 = 60
z = 6 + t = 6 + 10 = 16
So, (5, 6, 11) does NOT lie on the line.
(c) (-7, -12, 4):
-5 + t = -7, so t = -2
y = 6t = 6 × -2 = -12
z = 6 + t = 6 + -2 = 4
So, (-7, -12, 4) can be composed as -5 - 2i + -12j + 4k = -7i - 12j + 4k, which lies on the line.
Therefore, (a) lies on the line, (b) does NOT lie on the line, and (c) lies on the line.
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--The given question is incomplete; the complete question is
"Determine whether each point lies on the line.
X = -5 + t, y = 6t, z = 6+ t (a) (0, 30, 11) (b) (5, 6, 11) (c) (-7, -12, 4)"--
What must be a factor of the polynomial function f(x) graphed on the coordinate plane below?
Answers
The factor of the polynomial f(x) graphed on the coordinate plane is
(x - 1).
What is factor of the polynomial?
A polynomial with coefficients in a certain field or in integers is expressed as the product of irreducible factors with coefficients in the same domain in mathematics and computer algebra, which is known as polynomial factorization.
A polynomial can be factored by expressing it as the product of two or more components; this is somewhat the opposite of multiplying.
From the given graph we can see that, graph crosses x - axis at points
x = 1 and x = 8.
That means x = 1, 8 are the zero of the polynomial.
Since, if x = a is a zero of the polynomial, then (x - a) is the factor of the polynomial.
As x = 1, 8 are zero of the polynomial so (x - 1) and (x - 8) are the factors of the polynomial.
Hence, from the given options (x - 1) is factor of the polynomial.
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How do you write a linear inequality?
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Mathematical expressions known as linear inequalities compare two expressions and used the inequality symbol. The expression may be either a numerical or an algebraic expression, or it may be both.
Define the term linear inequality?Expressions that compare two linear expressions using inequality symbols are referred to as linear inequalities.
It's important to remember that if p < q, then p must be a number that is unambiguously less than q. If p ≤ q, then p is a number that is strictly smaller than q or exactly equal to q. The same is true for the final two inequality signs > (greater than) and (greater than or equal to).When resolving multi-step linear inequalities, we must follow the laws of inequality.
Step 1: Simplify an inequality on both sides, both the LHS and the RHS.Step 2: If indeed the inequality is just a strict inequality, when the value is acquired, the solution for x is either less than or larger than the calculated value as stated in the question.To know more about the linear inequality, here
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y = x-2
x-2y=-6
solve the equation step by step?
Answers
Answer:
need
Step-by-step explanation:
-x+\(5-x=-3\)
Answers
Answer:
x = 4
Step-by-step explanation:
-x + 5 - x = -3 Combine like terms
-2x + 5 = -3 Subtract 5 from both sides
-2x = -8 Divide both sides by -2
x = 4
-5 -5
-2x=-8
dived both sides by -2
X=4
2 miles in 3/4 of an hour
Answers
Answer:
2
Step-by-step explanation:
What is an equation in point-slope form of the line that passes through (-3,-1) and has a slope of 2?
A. y + 1 = 2(x + 3)
B. y - 1 = 2(x + 3)
С. y-1 = 2(x - 3)
D. y + 1 = 2(x - 3)
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Consider a clock with an hour hand and a minute hand. What is the measure of the angle the minute hand traces in 49 minutes. Your answer should be in radians.
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The measure of the angle the minute hand traces in 49 minutes on a clock is 4.084 radians A radian is an angle measure defined by the ratio of an arc length to a radius.
A radian is a measure of angle equal to the radius of a circle, and is defined as the amount of angle subtended by an arc that is equal in length to the radius of the circle. The radian measure of an angle is defined as the ratio of the length of the arc s to the radius r of the arc s of a circle:\(θ = s/r\)
Where,θ is the measure of the angle in radians.s is the arc length. r is the radius of the circle.
One revolution of the clock is equal to 2π radians.
The minute hand of the clock makes one full revolution in 60 minutes. Therefore, in 49 minutes, the minute hand of the clock moves through an angle equal to:
θ = 49/60 × 2π
= 4.084 radians
Thus, the measure of the angle the minute hand traces in 49 minutes is 4.084 radians.
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Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = e−2t cos(3t), y = e−2t sin(3t), z = e−2t; (1, 0, 1)
Answers
The parametric equations for the tangent line to the curve with the given parametric equations at the specified point is \(e^{(-2t)\)cos(3t), y = \(e^{(-2t)\)sin(3t), z = \(e^{(-2t)\)
To find the parametric equations of the tangent line to a curve with given parametric equations at a point, we need to follow the following steps:
Find the derivatives dx/dt, dy/dt, and dz/dt of the given parametric equations.Plug in the value of t that corresponds to the given point to find the corresponding values of dx/dt, dy/dt, and dz/dt.Use the point and the values of dx/dt, dy/dt, and dz/dt at that point to write the equation of the tangent line in parametric form.Let's apply these steps to the given problem:
Find the derivatives dx/dt, dy/dt, and dz/dt of the given parametric equations:
\(dx/dt = -2e^{(-2t)}cos(3t) - 3e^{(-2t)}sin(3t)\)
\(dy/dt = -2e^{(-2t)}sin(3t) + 3e^{(-2t)}cos(3t)\)
dz/dt = -2e^(-2t)
Plug in the value of t that corresponds to the given point (1,0,1) to find the corresponding values of dx/dt, dy/dt, and dz/dt:
dx/dt(1) = -2e⁻² ¹ cos(3(1)) - 3e⁻² ¹ sin(3(1)) = -5.348
dy/dt(1) = -2e⁻² ¹ sin(3(1)) + 3e⁻² ¹)cos(3(1)) = 2.317
dz/dt(1) = -2e⁻² ¹ = -1.471
Use the point (1,0,1) and the values of dx/dt, dy/dt, and dz/dt at that point to write the equation of the tangent line in parametric form. Let's call the parametric equation of the tangent line (x(t), y(t), z(t)):
x(t) = 1 - 5.348(t-1)
y(t) = 0 + 2.317(t-1)
z(t) = 1 - 1.471(t-1)
Therefore, the parametric equations of the tangent line to the curve x = \(e^{(-2t)\)cos(3t), y = \(e^{(-2t)\)sin(3t), z = \(e^{(-2t)\) at the point (1,0,1) are x(t) = 1 - 5.348(t-1), y(t) = 2.317(t-1), z(t) = 1 - 1.471(t-1).
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if the probability that a baby born in a certain hospital will speak in the next day is $1/4$, what is the probability that at least $2$ babies out of a cluster of $5$ babies will speak tomorrow?
Answers
The probability that at least 2 babies out of a cluster of 5 babies will speak tomorrow is 47/128.
To calculate the probability that at least 2 babies out of a cluster of 5 babies will speak tomorrow, we can use the binomial probability formula. The binomial probability formula is given by:
P(X ≥ k) = 1 - P(X < k) = 1 - ∑(i=k-1)^(n) [C(n, i) * p^i * (1-p)^(n-i)]
Where:
P(X ≥ k) is the probability that X is greater than or equal to k
P(X < k) is the probability that X is less than k
n is the number of trials (number of babies in the cluster) = 5
p is the probability of success (probability that a baby speaks) = 1/4
k is the number of successes we are interested in (at least 2 babies speaking)
Let's calculate the probability using this formula:
P(X ≥ 2) = 1 - P(X < 2) = 1 - [P(X = 0) + P(X = 1)]
P(X = 0) = C(5, 0) * (1/4)^0 * (3/4)^(5-0) = 1 * 1 * (3/4)^5 = 243/1024
P(X = 1) = C(5, 1) * (1/4)^1 * (3/4)^(5-1) = 5 * (1/4) * (3/4)^4 = 405/1024
P(X ≥ 2) = 1 - (243/1024 + 405/1024) = 376/1024 = 47/128
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Given that f(x) = 4x + 5 and g(x) = x2 - 2x - 2, find (f + g)(x).
A) (f + g)(x) = x2 + 2x+3
B) (f + g)(x) = x2 + 6x+ 3
C) (f + g)(x) = -x2 + 6x + 7
D) (f + g)(x) = – x2 + 2x - 7
Answers
Answer:
A
Step-by-step explanation:
You just add the two equations
(4x+5)+(x²-2x-2)
combine like terms
x²+2x+3
A
Please help fast. Need soon!
Answers
See explanation on picture attached with my method of solving
Find the area of this triangle. Round to
the nearest tenth.
17 m
35°
15 m
[? ]m2
![Find the area of this triangle. Round tothe nearest tenth.17 m3515 m[? ]m2](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/pfLdIWgL64dmRaAHmMcCh7dWelp7bJRC.png)
Answers
Answer:
32m²
Step-by-step explanation:
thanks me later ,correct me if I'm wrong
Answer: 73.1
Step-by-step explanation:
yea
The order of the differential equation d 2 y dx2 4x dy dx = d 3 (cos 2x) dx3 is:_________
Answers
The order of the differential equation d²y/dx² + 4x(dy/dx) = d³(cos(2x))/dx³ is 3.
The order of a differential equation is determined by the highest derivative present in the equation. In the given differential equation, the highest derivative is the third derivative, d³(cos(2x))/dx³. Since this is the highest derivative and there are no higher-order derivatives present, the order of the differential equation is 3. This means that the equation involves up to the third derivative of the dependent variable y with respect to the independent variable x.
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C. $3,006D. $3,135given, compute40 answer3. Which function has the largest value forf(-3)?) puug - 3 functionA f(x) = 2x - 5-786-33-5B. f(x) = 2 - 4x7Q-10-30C. f(x) = 6 - 3*-76-363)D. F(X) = 2* + 10 20-07104. When Mack left the bank, the temperature was 25°C. When he got htomnarature was 730F. If CA=f-32, about how many degrees
Answers
To know which function has the largest value for f(3), we need to substitute x=-3 in each function:
A. f(x)=2x-5
f(-3)=2(-3)-5=-11
B. f(x)=2-4x
f(-3)=2-4(-3)=14
C. f(-3)=6-3^(-3)
f(-3)=6-1/27
f(-3)=161/27
D. f(x)=2^(x)+10
f(-3)=2^(-3)+10
f(-3)=1/8+10=81/8
The largest value for f(-3) is 14, The answer is B f(x)=2-4x
Given f(x) = -2x³ + x² - 2 find f(1)
Answers
Answer:
-3
Step-by-step explanation:
\(f(x)=-2x^3+x^2-2 \\\\f(1)=-2(1)^3+(1)^2-2= \\\\-2+1-2=\\\\-3\)
Hope this helps!
A school sells chips and candy bars for a fundraiser.
· Each bag of chips sells for 75 cents.
· Each candy bar sells for $1.25
• The school sold fifty more bags of chips than candy bars
• The school made a total of $437.50
How many bags of chips did the school sell?
Answers
Answer: 21
Step-by-step explanation: 21 + 21 = 21
The asymmetric cryptography algorithm most commonly used is:
O GPG
O RSA
O ECC
O AES
Answers
Answer
Step-by-step explanation:
Find Tan(ß) in the triangle
Answers
Answer:
the answer is
tan (B)=p/b
=5/12
Answer: C)5/12
Step-by-step explanation:
The tangent is equal to the ratio of the opposite cathet to the adjacent one.Then \(\bf tg \beta = \dfrac{5}{12}\)What are the two types of dilation?
Answers
Two types of dilation are enlargement and reduction.
What is dilation and its type?Resizing an item uses a transition called dilation. Dilation is used to enlarge or contract the items. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size. The initial shape should be stretched or contracted during a dilatation. The phrase "scale factor" describes this transition.
Enlargement is the term used when a dilatation results in a larger image.
Reduction occurs when a dilatation results in a smaller picture.
For Enlargement, scale factor > 1
For Reduction, 0 < scalar factor < 1
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Which table represents a linear function?
Answers
Answer:
B. Is linear
Step-by-step explanation:
if you graph it it’s a straight line going up
Hope this helps! ;-)
Can someone please help me I really need help please help me thank you
Answers
Answer:
(-1,-1), (2,1), (5,3)
Step-by-step explanation:
We can use the formula: rise/run to solve x and y. Since we know the slope is 2/3, 2 will be rise (going up on y axis), and 3 will be run (moving right on x axis).
We are given the point, (2,1) which we can use to find the 2 points that are needed. By using rise/run, we can find the highest point which is (5,3)
How to get the highest point:
(2 + 3, 1 + 2)
Now we can find the lowest point: (-1,-1)
(2 - 3, 1 - 2)
if x and y satisfy the following equations, what is the value of x+y?
4x + 5y = 9
9x + 1y = 10
Answers
If x and y satisfy the system of equations 4x + 5y = 9 and 9x + 1y = 10, then the value of (x + y) is 2.
Given the system of the linear equations are
4x + 5y = 9 ................. (i)
9x + y = 10 ............... (ii)
Now multiplying 5 with equation (ii) we get,
5(9x + y) = 5*10
45x + 5y = 50 ..................... (iii)
Subtracting equation (i) from equation (iii) we get,
(45x + 5y) - (4x + 5y) = 50 - 9
45x + 5y - 4x - 5y = 41
41x = 41
x = 41/41 = 1
Substituting x = 1 in equation (i) we get,
4 * 1 + 5y = 9
4 + 5y = 9
5y = 9 - 4 = 5
y = 5/5 = 1
So the solutions are x = 1 and y = 1.
Hence the value of x + y = 1 + 1 = 2.
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m∠2=(7x−11)∘ and m∠4=(4x+4)∘
What is m∠1 ?
Answers
Answer:
Step-by-step explanation:
7x-11=4x+4(being vertically opposite abgles)
7x-4x=4+11
3x=15
x=15/3
x=5
angle 2+angle1=180 degree(being straight line)
7x-11+angle 1=180
7*5-11+angle 1=180
35-11+angle1=180
24+angle 1=180
angle 1=180-24
angle 1=156 degree
PLEASE HELP ME!!!!! take a look at the question please
Answers
according to clue no. 1 the number is 760 > x > 740. so, it could be any number between 740 - 760.
according to the 2 & 3 clue we can conclude that thousandths and ones digit are even and thousandth digit is twice the ones digit.
so the ones digit could be, 2 & 4 and the thousandth digit could be 4 & 8.
according to the 4th clue we can say that the tenths digit is 1 and thousandths digit is 4 as, thousandth digit is the 2nd even number after ones digit and also, sum of tenths and thousandth digit is 5 so, 1 + 4 = 5.
so, through this we can conclude that,
• ones digit = 2 as ( thousandth is twice the ones = 2 × 2 = 4 )
• tenths digit = 1 ( as 1 + 4 = 5 )
• thousandths digit = 4 ( as 4 + 1 = 5 )
according to clue no. 5 product of hundredths and hundreds is 7 so, 7 × 1 = 7.
so, according to this we can conclude that hundreds digit is 7 because in the required number the hundreds digit is itself 7
• hundreds digit = 7
&
• hundredths digit is = 1
and also, tens digit is 5 because only ones and thousandths digit are even.
so, the required number is, 752.114
hope this answer helps you dear & may u have a great day ahead!... take care!
the population of a small town in alberta was 2600. the population increased by 5% one year and by 15% the next year. what was the town’s population after the 2 years? Plz answer
Answers
Answer:
3120
Step-by-step explanation:
5% of 2600 is 130, keep that in mind
15% of 2600 is 390, also keep that in mind
If you add 130+390, that equals 520, 520+2600 is 3120, your welcome
~Akmp10
The table shows the number of cruise ships in a harbor on various days.
Monday: x-4
Tuesday: x+9
Wednesday: 2x
Thursday: 3x-7
Friday: 4
Write an expression for the total number of cruise ships in the harbor on all 5 days.
Answers
Answer:
7x + 2
Step-by-step explanation:
Hope this helps,
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I need help with this fast some one pls help plssssss
