Let f(x)= ⎩
⎨
⎧
−3−x
+1
1
2x+10
if x<−4
if x=−4
if x>−4
Calculate the following limits. Enter "DNE" if the limit does not exist. lim x→−4 −
f(x)= lim x→−4 +
f(x)= lim x→−4
f(x)=
Answers
The limits are as follows:
lim \(x \to -4^-\) \(f(x) = -2\)
lim \(x \to -4^+\) \(f(x) = 8\)
lim \(x \to -4\) \(f(x) = -2 = 8\)
To calculate the limits of \(f(x)\) at \(x = -4\), we need to evaluate the left-hand limit (lim \(x \to -4^-\)), the right-hand limit (lim \(x \to -4^+\)), and the two-sided limit (lim \(x \to -4\)).
lim \(x \to -4^-\) \(f(x)\):
When \(x\) approaches -4 from the left (\(x < -4\)), we use the function \(f(x) = -3 - x\) because \(x = -4\) does not satisfy the condition \(x < -4\).
Therefore, lim \(x \to -4^-\) \(f(x) = -3 - (-4) + 1 = -2\).
lim \(x \to -4^+\) \(f(x)\):
When \(x\) approaches -4 from the right (\(x > -4\)), we use the function \(f(x) = \frac{1}{2}x + 10\) because \(x = -4\) does not satisfy the condition \(x > -4\).
Therefore, lim \(x \to -4^+\) \(f(x) = \frac{1}{2}(-4) + 10 = 8\).
lim \(x \to -4\) \(f(x)\):
Since both the left-hand limit and the right-hand limit exist and are finite, we can say that the limit as \(x\) approaches -4 exists.
Therefore, lim \(x \to -4\) \(f(x)\) is the same as both lim \(x \to -4^-\) \(f(x)\) and lim \(x \to -4^+\) \(f(x)\).
Thus, lim \(x \to -4\) \(f(x) = -2 = 8\) (both limits are equal).
Therefore, the limits are as follows:
lim \(x \to -4^-\) \(f(x) = -2\)
lim \(x \to -4^+\) \(f(x) = 8\)
lim \(x \to -4\) \(f(x) = -2 = 8\)
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Related Questions
Please help !! 25 points
(No links or ill report you, just reply the answer) thanks
Use the general form of the equation for a circle to help write equations for the circles described below:
a) Center (2, -1) and radius 4
b) center (-2, 2) and radius 3
Answers
Step-by-step explanation:
a. (x-2)^2 + (y+1)^2 = 16
b. (x + 2)^2 + (y-2)^2 = 9
The mean time required to repair breakdowns of a certain copying machine is 93 minutes. The company which manufactures the machines claims that breakdowns of its newer model are easier to fix. To test this claim, a sample of 18 breakdowns of the new model were observed, resulting in a mean repair time of 86.8 minutes with a standard deviation of 14.6 minutes. Using a significance level of a = 0.10, determine if the new copy machines are faster to repair. State clearly what your null and alternative hypotheses are, show your work, and state your conclusion.
Answers
A significance level of 0.10, we have enough evidence to conclude that the new copy machines have a significantly faster mean repair time compared to the older model.
To test if the new copy machines are faster to repair, we can set up the following null and alternative hypotheses:
Null Hypothesis (H₀): The mean repair time for the new copy machines is the same as the mean repair time for the older model.
Alternative Hypothesis (H₁): The mean repair time for the new copy machines is less than the mean repair time for the older model.
Let's perform a one-sample t-test to test these hypotheses. The test statistic is calculated as:
t = (sample mean - population mean) / (sample standard deviation / √(sample size))
Given:
Population mean (μ) = 93 minutes
Sample mean (\(\bar x\)) = 86.8 minutes
Sample standard deviation (s) = 14.6 minutes
Sample size (n) = 18
Significance level (α) = 0.10
Calculating the test statistic:
t = (86.8 - 93) / (14.6 / sqrt(18))
t = -6.2 / (14.6 / 4.24264)
t ≈ -2.677
The degrees of freedom for this test is n - 1 = 18 - 1 = 17.
Now, we need to determine the critical value for the t-distribution with 17 degrees of freedom and a one-tailed test at a significance level of 0.10. Consulting a t-table or using statistical software, the critical value is approximately -1.333.
Since the test statistic (t = -2.677) is less than the critical value (-1.333), we reject the null hypothesis.
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A cylinder has a radius of 6 meters. Its volume is 180 pi. cubic meters.
Find the height of the cylinder. *
Answers
Answer:
h = 5
Step-by-step explanation:
Formula for volume of a cylinder:
V = πr²h
r = 6
V = 180π
V = πr²h
180π = π6²h
180π = 36πh
(180π)/(36π) = h
h = (180)/(36)
h = 5
Answer:
yeet
Step-by-step explanation:
yeet thanks for the points scrub
Help please guys if you don’t mind
Answers
Answer:
slope = -2
equation y= -2x -1
y - intercept (0,-1)
Answer:
y-int: -1
Slope: -2/1
Equation: Y=-2/1x-1
Step-by-step explanation:
HELP DUE IN 10 MINUTES!!!!!
Two step equations word problems
Tim answered all the questions on his math test but got 10 answers wrong. He received 4 points for every correct answer, and there was no penalty for wrong answers. His score was 76 points.
1. Write an equation to determine the total number of questions (q) on Tim's math test.
2. Find the total number of questions on his math test.
Answers
Let the number of accurate answers provided by Tom equal x.
The number of incorrect responses is ten.
Tim receives 4 marks for each right answer.
Marks deducted for wrong answers = 0.
Total Points = 76
When all of the above statements are combined, the equation
76 = 4 x + 10 * 0
4 x = Equals 76.
19 = x
The number of correct answers is 19
The number of incorrect responses equals ten.
Total number of questions = 19 + 10 = 29
If the coefficient matrix A in a homogeneous system in 20 variables of 16 equations is known to have rank 9, how many parameters are there in the general solution? (1)
cross (X) the correct answer:
A 11
B 10
C 6
D 21
E 17
F 4
Answers
The number of parameters in the general solution of a homogeneous system with 20 variables and a coefficient matrix of rank 9 is 10.
The number of parameters in the general solution is determined by subtracting the rank of the coefficient matrix from the number of variables. In this case, the number of variables is 20 and the rank is 9. Therefore, the number of parameters is 20 - 9 = 11.
However, among the given options, the closest answer is (B) 10. While the actual number of parameters is 11, the option 10 is the best approximation available. It is important to note that the number of parameters represents the degrees of freedom in the solution and indicates the number of variables that can be chosen arbitrarily to satisfy the system of equations.
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hi! please help in math!
i need the solution/explanation on how you got the answer
(y + 3) = -8(x - 4)
what is the slope?
Answers
Answer:
slope m = - 8
Step-by-step explanation:
the equation of a line in point- slope form is
y - b = m(x - a)
where m is the slope and (a, b ) a point on the line
y + 3 = - 8(x - 4) ← is in point- slope form
with slope m = - 8
The slope is :
↬ -8Solution:
Given: \(\bf{y+3=-8(x-4)}\)
To determine the slope, it's important to know the form of the equation first.
There are 3 forms that you should be familiar with.
The three forms of equations of a straight line are:
Slope Intercept (y = mx + b)Point slope (y-y₁) = m(x - x₁)Standard form (ax + by = c)This equation matches point slope perfectly.
The question becomes, how do you work with point slope to find slope?
Point slopeIn point slope, m is the slope and (x₁, y₁) is a point on the line.
Similarly, the slope of \(\bf{y+3=-8(x-4)}\) is -8.
Hence, the slope is -8.someone pls help me!
Answers
Answer:
the answer is b or the second blank
Step-by-step explanation:
Sierra's ice cream shop made $45 before she had to close early when the power went out. She realized she lost Two-thirds of a dollar each hour when the shop was closed. The expression that models this situation is 45 minus two-thirds h. What was Sierra's profit if the shop was closed for 6 hours? $24 $36 $41 $49
Answers
Answer:
the profit in the case when the shop is closed for 6 hours is $41
Step-by-step explanation:
The computation of the profit in the case when the shop is closed for 6 hours is shown below:
= Expression of the situation × number of hours in which the shop is closed
= 45 - 2 ÷ 3 × 6 hours
= $41
hence, the profit in the case when the shop is closed for 6 hours is $41
Therefore the third option is correct
Answer:
c
Step-by-step explanation:
Please help I do not understand
Answers
Answer:
-3/7
Step-by-step explanation:
Hi,
First of all, we know that the slope is not undefine because both the x-coordinates are not the same.
To find the slope, do change in y/change in x. Here, it would be ((2/7)-(5/7))/(2-1) or (-3/7)/1. Hence, the slope is -3/7.
what are the roots of the quadratic equation below 2x ^ 2 - 12x + 13=0 PLEASE HELP
Answers
Answer:
messy... but hope it helps you.
the triangles have the following coordinates: A(2,4), B(2,10), C(10,10), F(14,13), G(14,16). Find the coordinates of H.
Answers
Answer: A
Step-by-step explanation: because I got the correct answer
The image of the point B(8,-3) under the translation T6,0?
Answers
Answer:
(14, - 3 )
Step-by-step explanation:
Under the translation < 6, 0 >
Add 6 to the x- coordinate and 0 to the y- coordinate, that is
B(8, - 3 ) → B'(8 + 6, - 3 + 0 ) → B'(14, - 3 )
can someone pls help me
Answers
Answer:
B
Step-by-step explanation:
This ones pretty simple, all you need to do is write the two varible equations for how many fruits they bought and how much it cost.
In the first one there are 3 apples and 4 oranges, x represents the price of the apples and y represents price of the number of oranges, so the equation would be written like this:
3x+4y=
Then after the equal sign you put how much the total price was in total, which is 4.25 in the first one so
3x+4y=4.25
On the second equation there are 5 apples and 2 oranges, again x will be the price of the apples, and y will be the price of the oranges. The total price of both apples and oranges is 4.75 for this one, so the equation would be written like this:
5x+2y=4.75
Both equations together would be
3x+4y+4.25
5x+2y=4.75
This is the anwser
AB formed by (2, 3) and (-1, 4)
CD formed by (-5, 3) and (-4, 6)
Answers
Justin is designing a new board game, and is trying to figure out all the possible outcomes. How many different possible outcomes are there if he spins a spinner with 5 equal-sized sections labeled Monday, Tuesday, Wednesday, Thursday, Friday and rolls a fair die in the shape of a cube that has six sides labeled 1 to 6?
Answers
For the die, there are 6 different outcomes since there are 6 different sides you can land on.
To find the possible outcomes of them combined, you multiply the number of outcomes of the spinner and the die by eachother.
You get 5•6 = 30 possible outcomes
index exceeds the number of array elements. index must not exceed 1.
Answers
This error occurs when you try to extract more number of elements then present in the array.
What do you mean by Array?An array is a collection of similar type of data type stored at linear memory locations. Array is a linear data structure that store the elements in the linear order in the memory allocations .
Array is of three types:-
1- one - D array
2- two - D array
3- multi - D array
To create an array,
type of data type (like int ) and name of the array with square brackets [].
To insert values to it, use a comma-separated list, inside curly braces:
data-type array-name[index];
for example-
int Numbers[] = {25, 50, 75, 100};
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simplify the following expression -2a- (-3+a^2)
Answers
Answer is in the image
Step-by-step explanation:
Sorry if wrong or blurry
Which of the following shows the correct first step to solve x^2-18x=-45
A x^2 - 18x + 18= -45 + 18
B. x^2 - 18x + 9 = -45 + 18
C. x^2 -18x + 81 = -45
D. X^2 -18 + 81 = -45 + 81
Answers
Answer:
D, X^2 -18 + 81 = -45 + 81
Step-by-step explanation:
it is complating squer method that used for solving x in a quadratic equetion . in this step you will add
(y/2)^2 if y is the cofitient of x .
is 8.141141114 a rational or irrational and why
Answers
Answer:
It's an irrational number
Step-by-step explanation:
Yes, 8.141141114 is not a rational number . it is a irrational number .generally we can find out a number is rational or irrational by seeing the number. If mentioned number were 8.141141141 then It would be a rational number because after point number 141 is repeated. Hence,Mentioned number is not look like that. so, it is a irrational number
On a map, the scale is 1 inch=125 miles. What is the actual distance between two cities if the map distance is 4 inches?
Answers
(e) Find all values of Z for which z5 + 32 = 0 for and expressed the roots in polar form. Plot all the roots on an Argand diagram. (7 marks)
Answers
The values of z that satisfy the equation z^5 + 32 = 0 can be expressed in polar form. The roots are plotted on an Argand diagram.
To find the values of z that satisfy the equation z^5 + 32 = 0, we can rewrite it as z^5 = -32. By expressing -32 in polar form, we have -32 = 32(cos(pi) + i sin(pi)).
Now, we can express z in polar form as z = r(cos(theta) + i sin(theta)), where r is the magnitude and theta is the argument. Substituting this into the equation z^5 = -32, we get r^5(cos(5theta) + i sin(5theta)) = 32(cos(pi) + i sin(pi)).
From this equation, we can equate the magnitudes and arguments separately to find the values of r and theta. The magnitude equation gives us r^5 = 32, so r = 2.
For the argument equation, we have 5theta = pi + 2kpi, where k is an integer. Solving for theta, we get theta = (pi/5) + (2kpi/5).
Therefore, the roots in polar form are z = 2(cos((pi/5) + (2kpi/5)) + i sin((pi/5) + (2kpi/5))), where k is an integer.
To plot these roots on an Argand diagram, we can substitute different values of k into the equation and find the corresponding complex numbers. Plotting these complex numbers will give us the roots on the Argand diagram.
In summary, the values of z that satisfy the equation z^5 + 32 = 0 can be expressed in polar form. The roots are given by z = 2(cos((pi/5) + (2kpi/5)) + i sin((pi/5) + (2kpi/5))), where k is an integer. These roots can be plotted on an Argand diagram to visualize their locations.
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find the component form of v given its magnitude and the angle it makes with the positive x-axis. sketch v.magnitude angle v = 3 v in the direction 4i 3j
Answers
Given information:v.magnitude = 3v makes an angle with positive x-axis.The angle made by v with the positive x-axis is 150°.v is in the direction 4i + 3j.Solution: We have to find the component form of v.
Let's represent the given information on a graph:From the above graph, we can say that v makes an angle of 30° with negative y-axis.The angle between v and the negative y-axis = 180° - 30° = 150°Let's find the magnitude of v:Let v be (x,y).v = x i + y j
Magnitude of v, |v| = √(x² + y²) = 3 Therefore, x² + y² = 3² = 9 ...(1)Angle made by v with negative y-axis, θ = 30°∴ The angle made by v with positive x-axis, α = 150° - 90° = 60°sinθ = y/|v|y = 3 sinθ = 3 sin(30°) = 1.5From equation (1),x² + 1.5² = 9x² = 9 - 2.25 = 6.75x = ± √6.75Since v makes an angle of 150° with the positive x-axis, the vector v lies in the second quadrant. Therefore, x is negative.So, x = - √6.75 and y = 1.5Hence, the component form of v is:(- √6.75, 1.5).Thus, the component form of v is (- √6.75, 1.5).
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A right triangle has side lengths a, b, and c as shown below.
Use these lengths to find sinx, tanx, and cosx.
Help plsss
Answers
Answer:
Step-by-step explanation:
The value of sinx=c/b, cosx=a/b and tanx=c/a.
What are trigonometric ratios?The sides and angles of a right-angled triangle are dealt with in Trigonometry. The ratios of acute angles are called trigonometric ratios of angles. The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).
Given that, a right triangle has side lengths a, b, and c.
We know that, sinθ=Perpendicular/Hypotenuse, cosθ=Base/Hypotenuse and tanθ=Perpendicular/Bse
From the figure,
sinx=c/b, cosx=a/b and tanx=c/a
Therefore, the value of sinx=c/b, cosx=a/b and tanx=c/a.
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The number of users of social media has increased significantly since the year 2001. In fact, the approximate number of users has tripled each year. It was reported that in 2005 there were 3 million users of social media.
a. Assuming that the number of users continues to triple each year, for the next three years, determine the number of users in 2006, 2007, and 2008.
How many users were there in 2006?
Answers
Answer:
9 million
Step-by-step explanation:
3 million × 3 = 9 millionI hope this helps!
The number of users in 2006, 2007, and 2008 will be 9 million, 27 million, and 51 million respectively.
What is a sequence?It is defined as the systematic way of representing the data that follows a certain rule of arithmetic.
Divergent sequences are those in which the terms never stabilize; instead, they constantly increase or decrease as n approaches infinity, approaching either infinity or -infinity.
It is given that, the approximate number of users has tripled each year and in 2005 there were 3 million users of social media.
The number of users in 2006 = 3 × number of users in 2005]
The number of users in 2006 = 3 × 3 million
The number of users in 2006 = 9 million
The number of users in 2007 = 3 × number of users in 2006
The number of users in 2007 = 3 × 9 million
The number of users in 2007 = 27 million
Thus, the number of users in 2006, 2007, and 2008 will be 9 million, 27 million, and 51 million respectively.
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3.) What is the constant of proportionality (c.o.p.) in the table below?* number of days food in grams 7 4 30
Answers
According to the given table.
Days Grams
7 4 1/5
30 18
The constant of proportionality is found with the following formula.
\(m=\frac{y_2-y_1}{x_2-x_1}\)First, we transform the mixed number 4 1/5 into a fraction.
\(4\frac{1}{5}=\frac{4\cdot5+1}{5}=\frac{20+1}{5}=\frac{21}{5}\)So, we use the following
\(\begin{gathered} x_1=7 \\ x_2=30 \\ y_1=\frac{21}{5} \\ y_2=18 \end{gathered}\)Replacing these values in the formula, we have
\(m=\frac{18-\frac{21}{5}}{30-7}=\frac{\frac{90-21}{5}}{23}=\frac{\frac{69}{5}}{23}=\frac{69}{23\cdot5}=\frac{69}{115}=\frac{3}{5}\)Therefore, the constant of proportionality is 3/5.What are the vertex and x-intercepts of graph of y=(x+4)(x+2)
Answers
Answer:
(-3,-1)
because its at the bottom of the graph
Alfred begins collecting model cars. He buys 3 new model cars every week. Is the number of cars he owns proportional to the number of weeks that have passed since he started his collection?
Answers
The relationship between the number of weeks and cars are proportional since they are dependent on each other.
ProportionRatio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal. Here, a and b are any two integers. Proportion is an equation that defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or the ratios. In proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other.
In this question, Alfred, buy three car each week. As the number of weeks increase, so is the number of his car increasing at definite proportions.
This implies that we can set an equation to represent the number of cars Alfred will have after certain number of weeks.
Let;
x = number of weeksy = number of carsy = 3x
At week 10, Alfred will have
y = 3(10) = 30 cars
After 10 weeks, Alfred will have 30 cars
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a box contains one 2-inch rod, one 3-inch rod, one 4-inch rod, and one 5-inch rod. what is the maximum number of different triangles that can be made using these rods as sides?
Answers
On solving the provided question, we can say that we excluded example 2 above, a maximum of 3 triangles can be created with those rods.
What is triangle?A triangle is a polygon since it has three sides and three vertices. It is one of the basic geometric shapes. The name given to a triangle containing the vertices A, B, and C is Triangle ABC. A unique plane and triangle in Euclidean geometry are discovered when the three points are not collinear.
\(2-inch, 3 inch, 4-inch\)
triangle, because 2+3 > 4, 2+4 > 3, and 3+4 > 2
\(2-inch, 3 inch, 5-inch\)
This is doesn't make a triangle as \(2+5 > 3, and 3+5 > 2,\)
2+3 is not more than 5,, they cannot make triangle.
\(2-inch, 4 inch, 5-inch\)
triangle, because 2+4 > 5, 2+5 > 4, and 4+5 > 2
\(3-inch, 4 inch, 5-inch\)
a triangle, because 3+4 > 5, 3+5 > 4, and 4+5 > 3
Therefore, since we excluded example 2 above, a maximum of 3 triangles can be created with those rods.
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On a 1/5 scale drawing of a car, one part is 8 inches long. How long will the actual car part be?
- 1/5 inch
-8/5
-5 inches
-40 inches
Answers
On a 1/5 scale drawing of a car, where one part measures 8 inches, the actual length of the car part will be 40 inches.
In a scale drawing, the ratio of the lengths on the drawing to the lengths of the actual objects is maintained. In this case, the scale is 1/5, which means that every inch on the drawing represents five inches in reality.
Given that one part on the 1/5 scale drawing measures 8 inches, we can calculate the actual length of the car part by multiplying the length on the drawing by the scale factor. So, 8 inches on the drawing multiplied by the scale factor of 5 gives us the actual length of the car part, which is 40 inches.
Therefore, the correct answer is 40 inches.
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