Answers
Part A
The notation [tex]\lim_{x \to 2^{+}}f(x)[/tex] means that we're approaching x = 2 from the right hand side (aka positive side). This is known as a right hand limit.
So we could start at say x = 2.5 and get closer to 2 by getting to x = 2.4 then to x = 2.3 then 2.2, 2.1, 2.01, 2.001, etc
We don't actually arrive at x = 2 itself. We simply move closer and closer.
Since we're on the positive or right hand side of 2, this means we go with the rule involving x > 2
Therefore f(x) = (x/2) + 1
Plug in x = 2 to find that...
f(x) = (x/2) + 1
f(2) = (2/2) + 1
f(2) = 2
This shows [tex]\lim_{x \to 2^{+}}f(x) = 2[/tex]
Then for the left hand limit [tex]\lim_{x \to 2^{-}}f(x)[/tex], we'll involve x < 2 and we go for the first piece. So,
f(x) = 3-x
f(2) = 3-2
f(2) = 1
Therefore, [tex]\lim_{x \to 2^{-}}f(x) = 1[/tex]
===============================================================
Part B
Because [tex]\lim_{x \to 2^{+}}f(x) \ne \lim_{x \to 2^{-}}f(x)[/tex] this means that the limit [tex]\lim_{x \to 2}f(x)[/tex] does not exist.
If you are a visual learner, check out the graph below of the piecewise function. Notice the gap or disconnect at x = 2. This can be thought of as two roads that are disconnected. There's no way for a car to go from one road to the other. Because of this disconnect, the limit doesn't exist at x = 2.
===============================================================
Part C
You'll follow the same type of steps shown in part A.
However, keep in mind that x = 4 is above x = 2, so we'll deal with x > 2 only.
So you'd only involve the second piece f(x) = (x/2) + 1
You should find that f(4) = 3, and that both left and right hand limits equal this value. The left and right hand limits approach the same y value. The limit does exist here. There are no gaps to worry about when x = 4.
===============================================================
Part D
As mentioned earlier, since [tex]\lim_{x \to 4^{+}}f(x) = \lim_{x \to 4^{-}}f(x) = 3[/tex], this means the limit [tex]\lim_{x \to 4}f(x)[/tex] does exist and it's equal to 3.
As x gets closer and closer to 4, the y values are approaching 3. This applies to both directions.
Related Questions
Which expression is equivalent to the given expression? 2√/20-4√6
A. 32/30
OB. 16√30
OC. 3/20
D. √120
Answers
Answer:
D.
because 2 times 120 reduced.
please help me solve what is on that document
Answers
The most misleading graph is graph B because the blue rectangle and the red rectangle do not have the same width when plotted on the same scale
The ratio of the median weekly earningsFrom the graph, we have the median weekly earnings to be:
High school diploma = $750Bachelor's degree = $1250So, the ratio is:
Ratio = $750 : $1250
Simplify
Ratio = 3 : 5
Hence, the ratio of the median weekly earnings is 3 : 5
The ratio of the area of the red rectangle to the blue rectangle in graph A?
In (a), we have:
Ratio = 3 : 5
The scale on the horizontal axis is given as:
1 unit per grid mark
Both rectangles have a width of 1 unit.
So, we have:
Ratio = 3 * 1: 5 * 1
Simplify
Ratio = 3 : 5
Hence, the ratio of the area of the red rectangle to the blue rectangle in graph A is 3 : 5
The ratio of the area of the red rectangle to the blue rectangle in graph B?In (a), we have:
Ratio = 3 : 5
The scale on the horizontal axis is given as:
1 unit per grid mark
The red rectangle has a width of 3 units, while the blue has 5 units as its width
So, we have:
Ratio = 3 * 3 : 5 * 5
Simplify
Ratio = 9 : 25
Hence, the ratio of the area of the red rectangle to the blue rectangle in graph B is 9 : 25
The ratio of the volume of the red cube to the blue cube in graph C?
In (a), we have:
Ratio = 3 : 5
The scale on the horizontal axis is given as:
1 unit per grid mark
The red rectangle has a width of 3 units, while the blue has 5 units as its width.
Since the base are squares, we have:
Ratio = 3 * 3 * 3 : 5 * 5 * 5
Simplify
Ratio = 27 : 125
Hence, the ratio of the volume of the red cube to the blue cube in graph C is 27 : 125
The most misleading graph
The most misleading graph is graph B.
This is so because the blue rectangle and the red rectangle do not have the same width when plotted on the same scale
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Two trains leave station 416 miles apart. One train is traveling 90 miles per hour while the other is traveling 70 miles per hour. How long will it take the two trains to meet
Answers
Answer:
2.6 hours
Step-by-step explanation:
Let t be the time that the trains meet.
[tex]90t = 416 - 70t[/tex]
[tex]160t = 416[/tex]
[tex]t = 2.6[/tex]
)The width of a rectangle is shown below:
A coordinate plane with a point A at negative 3, 3 and C at negative 3, negative 2.
If the area of the rectangle to be drawn is 20 square units, where should points B and D be located if they lie to the right of points A and C?
B(1, 3) and D(1, −2)
B(3, 3) and D(3, −2)
B(2, 3) and D(2, −2)
B(1, 2) and D(1, −3)
Answers
Answer: B(1, 3) and D(1, −2)
Step-by-step explanation:
The distance from A to C is 5, meaning we need the length to be 4.
If they lie to the right, we need to add 4 to the x coordinates, so the x coordinates should be 1, and the y coordinates should be 3 and -2.
The locations of B and D are given by (A) B(1,3) and D(1,-2).
The area of rectangle is given by the product of the length and width of that rectangle.
Area = Length*Width
Distance between two points (a,b) and (c,d) is given by,
[tex]d=\sqrt{(c-a)^2+(d-b)^2}[/tex]
Here in the given question it is mentioned that AC forms width of a rectangle and A is at (-3,3) and C is at (-3,-2)
Now the width is [tex]=\sqrt{(-3-(-3))^2+(-2-3)^2}=\sqrt{0^2+(-5)^2}=\sqrt{25}=5[/tex] unit.
Also it is given that, the area of the rectangle is 20 square units.
So the length is = 20/5 = 4 units
B and D lie to the right of points A and C respectively.
We have to go 4 units right to the A to find B and 4 units right to C to find D.
A is at (-3,3)
then B is at (-3+4,3) = (1,3)
C is at (-3,-2)
So D is at (-3+4,-2) = (1,-2)
Hence correct option is (A) B(1,3) and D(1,-2)
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What 2 numbers add up to 37 and multiply to 210?
Answers
Answer: 30 and 7
Step-by-step explanation: 30 plus 7 is 37, 30 times 7 is 210
A linear function contains the following points: 0,-3 and 6,15.
WHat is the slope and y-intercept
Answers
The slope of the linear function is 3 and the y-intercept is -3 if the linear function contains the following points: (0,-3) and (6,15)
What is the slope?The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).
[tex]\rm m =\dfrac{y_2-y_1}{x_2-x_1}[/tex]
We have:
A linear function contains the following points: (0,-3) and (6,15)
The slope:
[tex]\rm m =\dfrac{15-(-3)}{6-0}[/tex]
m = 18/6
m = 3
The equation of a line:
y = mx + c
Plug x = 0, y = -3
-3 = c
Thus, the slope of the linear function is 3 and the y-intercept is -3 if the linear function contains the following points: (0,-3) and (6,15)
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The sum of a number and 3 is less than 10 if x denotes the number what would be the possible values of x
x>13
x<13
x>7
x<7
Answers
Answer:
x<7
Step-by-step explanation:
Consider the number x (as mentioned in the question). Now carefully pay attention to the given condition
⇒"The sum of a number and 3 is less than 10"
Remember the notation of the term "less than" in mathematical context is indicated by "<" sign. Hence our inequality is
⇒ x+3<10
adding or subtracting any number from the both sides of an inequality DOESN'T change its direction. Thus, if we cancel out 3 from both sides it won't swap the inequality. Therefore
⇒ x+3-3<10-3
⇒ [tex]x<7[/tex]
In terms of interval notation
[tex]\implies x\in( -\infty,7)[/tex]
In conclusion
The answer is D) [tex]x < 7[/tex]
Let the number be x
The sum of number and 3
x+3Is less than 10
x+3<10Lets solve it
x<10-3x<7Hence
Option D is correct
Find the solutions to the equation below. check all that apply. x^2-2x-15=0
Answers
The student did not make any mistake, the solution is correct. Then the correct option is A.
What is a factorization?It is a method for dividing a polynomial into pieces that will be multiplied together. At this moment, the polynomial's value will be zero.
A student perform the following steps to find the solution of the equation x² – 2x – 15 = 0.
The factor of the equation will be
Step 1 : x² – 2x – 15 = (x – 5)(x + 3)
Step 2 : (x – 5) = 0 and (x + 3) = 0
Step 3 : x = 5 and – 3
The student did not make any mistake, the solution is correct.
Then the correct option is A.
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How do I get this answer?
Answers
Answer:
Step-by-step explanation:
Several trig identities are involved in the proof of this. This is the order in which they are used.
cos(2x) = cos²(x) -sin²(x)cos²(x) +sin²(x) = 1cos(x) = 1/sec(x)ProofStarting with the left side, we can transform it into the right side.
[tex]2\cos(2x)=2(\cos^2(x)-\sin^2(x)) = 2(\cos^2(x)-(1-\cos^2(x)))\\\\=2(2\cos^2(x)-1)=2\left(\dfrac{2}{\sec^2(x)}-\dfrac{\sec^2(x)}{\sec^2(x)}\right)\\\\=\boxed{\dfrac{4-2\sec^2(x)}{\sec^2(x)}}[/tex]
Which equation describes the same line as y-5=-2(x+4)
Answers
The circle below is centered at the point (-1, 2) and has a radius of length 3.
What is its equation?
O A. (*+ 1)2 + (y- 2)2 = 9
O B. (x- 2)2 + (y + 1)? =
32
O c. (x- 1)2 + (y + 2)2 = 9
l
O D. (x+ 2)2 + (y- 1)2 =
32
Answers
Answer:
it should be option A
and in option there should be "x" instead of "*"
Equation of circle is,
⇒ (x + 1)² + (y - 2)² = 9
What is mean by Circle?The circle is a closed two dimensional figure , in which the set of all points is equidistance from the center.
Given that;
The circle below is centered at the point (-1, 2) and has a radius of length 3.
Now, We get;
The equation of circle is,
⇒ (x - (- 1))² + (y- 2)² = 3²
⇒ (x + 1)² + (y - 2)² = 9
Thus, Equation of circle is,
⇒ (x + 1)² + (y - 2)² = 9
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. It costs C(x) = [tex]\sqrt{x}[/tex] dollars to produce x golf balls. What is the marginal production cost to make a golf ball? What is the marginal production cost when x = 25? when x= 100? (Include
units.)
Answers
The marginal cost when x = 25 and when x = 100 are $0.1 and $0.05 respectively.
What is a Marginal Production Cost?A marginal production cost is a derivative of a cost function. From the information given, the total cost is:
C(x) = [tex]\mathbf{\sqrt{x}}[/tex]The marginal production cost can be expressed as:
[tex]\mathbf{C'(x) = \dfrac{d}{dx}(\sqrt{x})}[/tex]
[tex]\mathbf{C'(x) = \dfrac{1}{2\sqrt{x}}}[/tex]
However, when the cost is 25, the marginal production is:
[tex]\mathbf{C'(25) = \dfrac{1}{2\sqrt{25} }}[/tex]
[tex]\mathbf{C'(25) = \dfrac{1}{2\times5}}[/tex]
[tex]\mathbf{C'(25) = \dfrac{1}{10}}[/tex]
C' (25) = $0.1
Also, when the cost is 100, the marginal production is:
[tex]\mathbf{C'(100) = \dfrac{1}{2\sqrt{100} }}[/tex]
[tex]\mathbf{C'(100) = \dfrac{1}{2\times10}}[/tex]
[tex]\mathbf{C'(100) = \dfrac{1}{20}}[/tex]
C' (100) = $0.05
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the graph shows the value of a certain model of car compared with it's age. which statement is true?
A. The data show a negative linear relationship.
B. The correlation coefficient is close to 0.
C. The Car's age is the response variable.
D. The Car's age is not correlated to it value.
Answers
The only true statement is A:
"The data show a negative linear relationship."
Which statement is true?On the graph, we can see how the car's vale decreases almost linearly with the age of the car.
Where the response variable would be the one on the y-axis, which is the car's value.
For that linear behavior, we know that there is a correlation coefficient different than zero. So options B, C, and D are false.
Finally, we already saw the linear behavior (decreasing, so the slope is negative). Then we conclude that the only true statement is A.
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which matrix equation has the soul ut ion X
Answers
Step-by-step explanation:
option 2 is correct
here the question ask for value of X
to find it we multiple the matrix X with the given matrix in question and equate the answer.
factorize of 2a³-a²+a-2
Answers
=========================================================
Explanation:
Use the rational root theorem to determine this list of possible rational roots: 1, -1, 1/2, -1/2
Plug each possible root one at a time into the original expression given. If the simplified result is 0, then that possible root is an actual root.
If we tried say a = -1, then,
2a³-a²+a-2 = 2(-1)³-(-1)²+(-1)-2 = -6
The result is not zero, so a = -1 is not an actual root.
But if we tried say a = 1, then,
2a³-a²+a-2 = 2(1)³-1²+1-2 = 0
We get 0 so a = 1 is an actual root. I'll let you try the other values, but you should find that a = 1 is the only rational root.
Since a = 1 is a root, this makes (a-1) to be a factor.
From here, use either synthetic or polynomial long division to determine the other factor. Refer to the diagram below for each method.
Regardless of which method you pick, the quotient is 2a² + a + 2 which is the other factor needed. The remainder of 0 tells us we have (a-1) as a factor. For more information, check out the remainder theorem.
Select the correct answer from each drop-down menu.
The function f is given by the table of values as shown below.
x 1 2 3 4 5
f(x) 13 19 37 91 253
Use the given table to complete the statements.
If function f was translated down 4 units, the _
values would be _
A point in the table for the transformed function would be _
Answers
Using translation concepts, the correct statement is given by:
If function f was translated down 4 units, the f(x) values would be subtracted by 4. A point in the table for the transformed function would be (1,9).
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
A translation down 4 units means that the output values are subtracted by 4, hence the table would be given by:
x 1 2 3 4 5
f(x) 9 15 33 87 249
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Answer:
The CORRECT answer is:
1. f(x)
2. decreased by 4
3. (4, 87)
Step-by-step explanation:
Took the test
Given:
PLEASE HELP ASAP!
Part I: Use the given information to determine . Show all of your work.
Part II: Use complete sentences to explain the steps necessary in solving for . Justify your explanation by including all theorems, postulates, or definitions used.
Answers
The value of angle ∠DEC = 60 degree.
The complete question is
In the given the figure above, m∠BAC = 64° and m∠CBA = 56°.
Part I: Find the m∠DEC.
Part II: Explain the steps you took to arrive at your answer. Make sure to justify your answer by identifying any theorems, postulates, or definitions used.
What is a Triangle ?A triangle is a polygon with three sides and three vertices and three angles.
As it is given that
AB || CD
Let BC be the transversal , then
∠CBA = ∠DCB (alternate opposite angles)
∠DCB = 56 degree
As the sum of opposite interior angle sis equal to exterior angle therefore ∠ ECB = 120 degree
∠DCE = 120-56 = 64 degree
BC || DE
∠EDC = ∠DCB = 56 degree (alternate opposite angles)
Sum of all the interior angles of a triangle is equal to 180 degree
56 +64+∠DEC = 180
∠DEC = 60 degree.
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Select the solution(s) of the original equation.
0 x = √√√2
x = 1
X=1
x = -1
0 *=-√√2
X = -1
DONE ✔
0000
Answers
Answer:
the first,second,forth and fifth.
Step-by-step explanation:
I got it off of edg.
which equation has the solution x=6
Answers
Answer:
need more info
Step-by-step explanation:
what I can tell you is that when you change all the x's in the equation to 6s, both sides of the equation will equal eachother.
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PLEASE LOOK AT PICTURE
What are the foci of this ellipse?
ANSWER CHOICES: are in picture
Answers
Choose the system of inequalities that best match the graph below.
Answers
The system of the linear inequality that fulfill the condition of the graph will be y ≥ 2x – 1 and y ≤ 1/3 x + 1. Then the correct option is A.
The complete question is attached below.
What is inequality?Inequality is defined as an equation that does not contain an equal sign. Inequality is a term that describes a statement's relative size and can be used to compare these two claims.
The system of the linear inequality that fulfill the condition of the graph will be
y ≥ 2x – 1
y ≤ 1/3 x + 1
Then the correct option is A.
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Can someone answer this
Answers
The order of the graphs from largest to lowest correlation coefficients is:
Graph D, Graph A, graph C, graph B.
Which graph has the largest correlation coefficient?The correlation coefficient between two variables is a coefficient that tells us "how much" these variables relate.
So, in the case for linear correlation, as "more linear" the data appears to be, a large correlation is between the two variables.
With that in mind, we conclude that the order of the graphs (going for larger correlation coefficient to smaller correlation coefficient) is:
Graph D, Graph A, graph C, graph B.
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Find the average rate of change of f ( x ) = 9 x 2 − 7 on the interval [ 2 , a ] . Your answer will be an expression involving a
Answers
Answer:
9a + 18
Step-by-step explanation:
so the first step is to evaluate the value of f(2).
[tex]f(2) = 9(2)^2-7\\f(2)=9(4)-7\\f(2)=36-7\\f(2)=29[/tex]
Now that we have this value we can use the slope formula (y2 - y1) / (x2 - x1) where y2 is f(a), y1 is 29, x2 is a, and x1 is 2
[tex]\frac{f(a)-29}{a-2}\\\\\frac{(9a^2-7)-29}{a-2}\\\frac{9a^2-36}{a-2}[/tex]
Now from here we can use long division to simplify it further
[tex]9a^2/a = 9a\\9a(a-2)=9a^2-18a\\(9a^2-36)-(9a^2-18a)\\18a-36\\\\18a/a=18\\18(a-2)=18a-36\\(18a-36)-(18a-36)\\0\\\\9a+18[/tex]
Find the domain and range of the function graphed below. Write your answers in interval notation.
Answers
Answer:
Domain: [tex][-2, 3)[/tex]Range: [tex](-5, 4][/tex]Is the relation a function? - YesStep-by-step explanation:
The domain is the set of x-values and the range is the set of y-values.
A relation is a function if each value of x maps onto no more than one value of y.
Use the calculator to graph the function y = 2x2 25x + 3. What are the coordinates of the turning point of the graph to the nearest hundredth?
Vertex = (
,
)
Answers
The coordinates of the turning point of the graph of the quadratic equation is given by: (-6.25, -75.13).
What is the vertex of a quadratic equation?A quadratic equation is modeled by:
[tex]y = ax^2 + bx + c[/tex]
The vertex is given by:
[tex](x_v, y_v)[/tex]
In which:
[tex]x_v = -\frac{b}{2a}[/tex][tex]y_v = -\frac{b^2 - 4ac}{4a}[/tex]In this problem, the function is given by:
y = 2x² + 25x + 3.
Hence the coefficients are a = 2, b = 25, c = 3, then the coordinates of the vertex are given as follows:
[tex]x_v = -\frac{25}{2(2)} = -6.25[/tex][tex]y_v = -\frac{25^2 - 4(2)(3)}{4(2)} = -75.13[/tex]The coordinates of the turning point of the graph of the quadratic equation is given by: (-6.25, -75.13).
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Answer:
6.25, -75.13
Step-by-step explanation:
it just is
The pH can be calculated using the equation pH = –log(H+), where H+ is the hydronium ion concentration. Find the hydronium ion concentration of a particular soda if the pH level is 2.3.
Answers
Answer:
Step-by-step explanationp :
ph =lo 4
Photo is attached, please help! Will give brainliest
Answers
Here you go! please make sure you read it all too understand it, and make sure you try work before asking for help :D Have a awesome day/night
Answer:735
Step-by-step explanation:I was hoping you’d have another. Dv/dt=d/dt((7+5t)^3)=3x5x((7+5t)^2)=15x49. Does this work?
Which of the following is the balance for a single $2.520 deposit in an account with an APR of 1.16% that compounds interest monthly and is invested for 37 years? Which of the following is the balance for a single $ 2.520 deposit in an account with an APR of 1.16 % that compounds interest monthly and is invested for 37 years ?
Answers
Answer:
The balance after 37 years is $3869.99
Step-by-step explanation:
Given:
Principal (P) = $2,520Rate of Interest (R) = 1.16%Time (t) = 37 yearsCompounds Interest Monthly (n) = 12The Amount (A),
A = P (1 + [tex]\frac{R}{100n}[/tex][tex])^{nt}[/tex] if compound n times per year
Amount (A) at 37 years,
A = ($2,520) (1 + [tex]\frac{1.16}{100(12)}[/tex][tex])^{12 x 37}[/tex]
A = ($2,520) (1 + [tex]\frac{1.16}{1,200}[/tex][tex])^{444}[/tex]
A = ($2,520) (1.00096667 [tex])^{444}[/tex]
A = ($2,520) (1.5357)
A = $3869.9944
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Question 3 of 10
Mark all the statements that are true.
Answers
By definition a function can only have one y value for each x value which means the other statements are false.
A line / passes through point (-2, 5) and has the same y-intercept as
the line y-3x-4. What is the equation for the line /?
Answers
I'm assuming you meant to write y = 3x-4
We know that the y-intercept of the given line is (0,-4), so we need to find the equation of the line passing through (-2,5) and (0,-4).
The slope is
[tex]\frac{-4-5}{0-(-2)}=-\frac{9}{2}[/tex]
So, this means the equation is [tex]\boxed{y=-\frac{9}{2}x-4}[/tex]
