Yes b can be written as linear combination of 3 vectors
What is linear combination of vector?A linear combination of vectors is a sum of scaled vectors, where the scaling factors are scalars. In other words, it's a linear combination of the form a_1v_1 + a_2v_2 + ... + a_n*v_n, where v_1, v_2, ..., v_n are vectors and a_1, a_2, ..., a_n are scalars. This is a fundamental concept in linear algebra and vector spaces and is used in many areas of mathematics and physics.
if b is linear combination of [tex]a_1,a_2,a_3[/tex]
then [tex]xa_1+ya_2+za_3=b[/tex]
from here we got :
x+5z = 2
-2x+y-6z = -1
2y + 8z = 6
after solving the above equation we get the solution as
{x,y,x} = {2,3,0} is a solution to the above equation so b is a linear combination of [tex]a_1,a_2,a_3[/tex]
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Complete question:
determine if b is a linear combination of [tex]a_1,a_2,a_3[/tex]
[tex]a_1[/tex] = [tex]\left[\begin{array}{ccc}1\\-2\\0\end{array}\right][/tex] , [tex]a_2 = \left[\begin{array}{ccc}0\\1\\2\end{array}\right][/tex] , [tex]a_3 = \left[\begin{array}{ccc}5\\-6\\8\end{array}\right][/tex] , b= [tex]\left[\begin{array}{ccc}2\\-1\\6\end{array}\right][/tex]
help me please asap!!
The scale factor for the dilation in this problem is given as follows:
2.
What is a dilation?A dilation happens when the coordinates of the vertices of an image are multiplied by the scale factor, changing the side lengths of a figure.
As the side lengths are changed, with a dilation the figure loses it's congruence relative to the original figure.
The equivalent side lengths in this problem are given as follows:
AD, with length 8 - 0 = 8.DE, with length 8 - 4 = 4.The dilated triangle is triangle DAB, hence the scale factor is obtained as follows:
k = 8/4 = 2.
(that is, the side lengths of triangle DEF were multiplied by 2 to generate triangle DAB).
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4 2/3 - 3/4 answer? Please help me anwser this
Answer: 3 11/12
Step-by-step explanation:
Rewrite the equation with the parts seperated:
4 + 2/3 − 3/4
Find the LCD of 2/3 and 3/4
(Which is 12)
Keep going
8/12−9/12= −1/12
Finally, subtract 4 − 1/12 to get 3 11/12
Hope this helps!
Tim+is+attempting+to+bike+a+12+mike+race+he+knows+at+every+third+of+the+race+he+will+need+to+take+a+break+to+stretch+his+leg+how+many+miles+wo
Answer:
he takes a stretch every 4 miles
Step-by-step explanation: 12/3 = 4
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There is a mistake in each of these problems. Please help
Answer:
ii) Reduce :
[tex]$\begin{aligned} 5 \ln x-2 \ln (x y) & =\ln x^5-\ln (x y)^2 \\ & =\ln \left(\frac{x^5}{(x y)^2}\right) \\ & =\ln \left(\frac{x^5}{x^2 y^2}\right) \\ & =\ln \left(\frac{x^3}{y^2}\right)\end{aligned}$[/tex]
iii ) Expand :
[tex]\begin{aligned}\log _3\left(\frac{x^5}{y}\right)^4 & =4 \log _3\left(\frac{x^5}{y}\right) \\& =4 \log _3 x^5-4 \log _3 y \\& =20 \log _3 x-4 \log _3 y\end{aligned}[/tex]
Answer:
[tex]\textsf{ii)} \quad \ln \left(\dfrac{x^3}{y^2}\right)[/tex]
[tex]\textsf{iii)} \quad 20 \log_3 x-4 \log_3y[/tex]
Step-by-step explanation:
Question iiGiven expression:
[tex]5 \ln x-2 \ln (xy)[/tex]
[tex]\textsf{Apply the power law}: \quad n \ln x=\ln x^n[/tex]
[tex]\implies \ln (x^5) - \ln ((xy)^2)[/tex]
[tex]\textsf{Apply the quotient law}: \quad \ln x - \ln y=\ln \left(\dfrac{x}{y}\right)[/tex]
[tex]\implies \ln \left(\dfrac{x^5}{(xy)^2}\right)[/tex]
This is where the error occurred in the original calculation as the log quotient law was applied incorrectly.
[tex]\textsf{Apply exponent rule} \quad (ab)^c=a^{c}b^c\quad \textsf{to the denominator}:[/tex]
[tex]\implies \ln \left(\dfrac{x^5}{x^2y^2}\right)[/tex]
Simplify the fraction:
[tex]\implies \ln \left(\dfrac{x^3}{y^2}\right)[/tex]
Question iiiGiven expression:
[tex]\log_3\left(\dfrac{x^5}{y}\right)^4[/tex]
[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]
[tex]\implies 4\log_3\left(\dfrac{x^5}{y}\right)[/tex]
[tex]\textsf{Apply the log quotient law}: \quad n\log_a \left(\dfrac{x}{y}\right)=n\log_ax - n\log_ay[/tex]
[tex]\implies 4 \log_3 x^5-4 \log_3y[/tex]
This is where the error occurred in the original calculation as the coefficient 4 was not applied to the both logarithms when applying the log quotient law.
[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]
[tex]\implies 5 \cdot 4 \log_3 x-4 \log_3y[/tex]
[tex]\implies 20 \log_3 x-4 \log_3y[/tex]
4(5x-3)-3(2x-5)=17 solve for x.
Answer:
x=1
Step-by-step explanation:
4(5x-3)-3(2x-5)=17
20x-12-6x+15=17
14x+3=17
-3. -3
14x=14
/14. /14
x=1
carrie is flying on two different airlines with a heavy bag both airlines have the same weight limit by charge different fees the first airline charges $25 checked bag fee plus an additional four dollars for every pound that the bag is over the weight limit with the second airline and the price is three dollars per pound over the weight limit and addition to a $28 fee for checking the bag Carrie ended up paying the exact amount and baggage fees at the two airlines how many pounds over the limit was Carrie’s bag how much did she have to pay in fees for each airline
Answer: Let x be the weight of the bag over the limit in pounds.
We know that both airlines have the same weight limit and Carrie paid the same baggage fee at each airline, so we can set up the following equation:
25 + 4x = 28 + 3x
Solving for x:
x = 25
So, Carrie's bag was 25 pounds over the weight limit.
To find the fee for each airline, we can substitute x into the equations for the fees:
Fee for the first airline: 25 + 4(25) = $125
Fee for the second airline: 28 + 3(25) = $103
Therefore, Carrie paid $125 in fees for the first airline and $103 in fees for the second airline.
Step-by-step explanation:
12 Juma, Ali and Hassan share the profit of their business in the ratios 3:7:9 respectively. If Juma receives sh. 6000. How much profit did the business yield. (3mks)
If the ratio of profit shares is 3:7:9, then the total profit yield is 38000.
What is ratio?
Comparing two amounts of the same units and determining the ratio tells us how much of one quantity is in the other. Two categories can be used to categorise ratios. Part to whole ratio is one, and part to part ratio is the other. The part-to-part ratio shows the relationship between two separate entities or groupings.
The share of Juma's profit is = 3
The share of Ali's profit is = 7
The share of Hassan's profit is = 9
The ratios can be written as 3x : 7x : 9x
The amount of money Juma received is = 6000
The equation for Juma's profit is -
3x = 6000
x = 6000/3
x = 2000
So, the value for Ali's profit -
7x = 7 × 2000
= 14000
The value for Hassan's profit -
9x = 9 × 2000
= 18000
Now, the total profit is represented by the equation -
= 3x + 7x + 9x
= 6000 + 14000 + 18000
= 38000
Therefore, the total profit is 38000.
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6 1/5 multiplied by 2 2/3 equals
Answer:1325
Step-by-step explanation:
A youth group and their leaders visited a museum
Answer:
What is the question please respond so I may give a complete response
Francisco added a new sidewalk to his front yard. The
diagram represents Francisco's sidewalk. Determine
the area of the sidewalk to the nearest square foot.
A 10 ft²
B 16.28 ft²
C 3.14 ft²
D 13.14 ft²
Is this correct?
The required of the sidewalk to the nearest square foot is 13.14 ft².
What is area of rectangle?
The area a rectangle occupies is the space it takes up inside the limitations of its four sides. The dimensions of a rectangle determine its area. In essence, the area of a rectangle is equal to the sum of its length and breadth.
Explain are of circle.A circle's area is the area that it takes up in a two-dimensional plane. It can be simply calculated using the formula A = r2, (Pi r-squared), where r is the circle's radius. The square unit, such as m2, cm2, etc., is the unit of area.
According to question:We have,
Length = 5 ft, width = 2 ft, Radius of circle = 2 ft
Now,
Area of sidewalk = area of rectangle + area quadrant
Area of sidewalk = (5×2) + π(2)^2/4
Area of sidewalk = 10 + 3.14
Area of sidewalk = 13.14 ft²
Thus, required area of the sidewalk is 13.14 ft².
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find the bake if x and y
The value of x is 76°. The solution has been obtained using the properties of hexagon.
What is a hexagon?
A closed, two-dimensional, six-sided polygon in geometry is called a hexagon. It has six vertices, which together create six internal angles, and six line segments. The total of a hexagon's internal angles is 720°.
We know that sum of angles on a straight line is 180°.
So, on the first line with angle 60°, the other angle will be
180° - 60° = 120°
Similarly, on the second line with angle 64°, the other angle will be
180° - 64° = 116°
Similarly, on the third line with angle 36°, the other angle will be
180° - 36° = 144°
Similarly, on the fourth line with angle 48°, the other angle will be
180° - 48° = 132°
Similarly, on the fifth and sixth lines with angle x°, the other angles will be
180° - x° = (180-x)°
We know that sum of angles of interior angles of a hexagon is 720°
Therefore,
⇒ 120° + 116° + 144° + 132° + (180-x)° + (180-x)° = 720°
⇒ 512° + 180° - x° + 180° - x° = 720°
⇒ 872° - 2x° = 720°
⇒ 2x° = 152°
⇒ x = 76°
Hence, the value of x is 76°.
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Humber Practice Skills - Estimating REVISION AVAILABLE 2.29 + 9.11 15.381-5.414 7.631.11 + 5.99 5.734 + 6.893 18.355-4.411 Answer the following Estimate the sum or difference of the following decimal numbers by rounding to the nearest tenth. ? CALCULATOR AMI RIGHT E Criteria O Search hp Learning 33 NCFE M DON'T KNOW RESET ROU 11:21 24/01/2023
The solution to the expression with decimals rounded to the nearest tenth is: -2
Examine the hundredth number to round the decimal number to the nearest tenth. If the number is more than 5, multiply it by one. If it is less than 5, leave the tenth place value alone and eliminate any numbers after the tenth. 4.624, for example, is 4.6 rounded to the closest tenth.
Thus, given the expression:
2.29 + 9.11 + 15.381 - 5.414 - 7.631.11 + 5.99 - 5.734 + 6.893 - 18.355 -4.411
We first round them up to the nearest tenths is:
2.1 + 9.1 + 15.4 - 5.4 - 7.6 + 6.0 - 5.7 + 6.9 - 18.4 - 4.4 = -2
Thus, it is correct to state that the version of the above-given expression resolved to the nearest tenths (2.1 + 9.1 + 15.4 - 5.4 - 7.6 + 6.0 - 5.7 + 6.9 - 18.4 - 4.4) = -2
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Please help me on this
Here we are interested in finding the slope of the line. From the given graph we can see that that the line passes through points (6,0) and (0,-8) . We know that slope can be calculated by ,
[tex]\longrightarrow slope \ (m)=\dfrac{y_2-y_1}{x_2-x_1} =\tan\theta [/tex]
[tex]\longrightarrow m =\dfrac{y_2-y_1}{x_2-x_1}\\[/tex]
[tex]\longrightarrow m =\dfrac{0-(-8)}{6-0}\\[/tex]
[tex]\longrightarrow m =\dfrac{8}{6}\\ [/tex]
[tex]\longrightarrow m =\dfrac{4}{3}\\ [/tex]
[tex]\longrightarrow \underline{\underline{ m = 1.\bar{3}}}[/tex]
And we are done!
Select the figure with an angle of 300°
on a circle in standard position.
(a)
An angle in standard position with terminal side at five-sixths of a full revolution counterclockwise.
(b)
An angle in standard position with terminal side at one-twelfth of a full revolution counterclockwise.
(c)
An angle in standard position with terminal side at one-third of a full revolution counterclockwise.
(d)
An angle in standard position with terminal side at two-thirds of a full revolution counterclockwise.
Note that the figure with an angle of 300° on a circle in standard position is: "An angle in standard position with the terminal side at two-thirds of a full revolution counterclockwise." (Option D)
What is an Angle in Standard Position?A standard position angle is one measured counterclockwise from the positive x-axis in a coordinate plane.
This signifies that the angle's starting side is on the positive x-axis, while the angle's terminal side is in the first or fourth quadrant. The angle's vertex is placed at the origin.
Angles are commonly measured in degrees or radians, with a full rotation being comparable to 360 degrees or 2 radians. Angles may now be readily graphed and compared to one another in a coordinate plane.
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a basketball player made 62 out of 80 free throw shots. what percent of the free throws did she make
Julio is buying a car. He makes a down payment of 20%
and he makes arrangements to finance the remaining
$14,000
What is the dollar amount of Julio's down payment?
Julio's down payment would be 20% of the total cost of the car. If he is financing $14,000 of the remaining cost, the total cost of the car would be $14,000 / (1 - 0.20) = $14,000 / 0.80 = $17,500. Therefore, Julio's down payment would be $17,500 * 0.20 = $3,500.
What is a percentage in plain English?
A percentage is a portion of a total stated as a number between 0 and 100 as opposed to being expressed as a fraction. The percentages of everything are 100%, nothing is 0%, 50% of something is 50%, and nothing is 0%. To calculate the percentage, divide the component by the total and multiply the result by 100.
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Using percentage, Julio's down payment would be $3,500.
What does a % mean in everyday terms?In contrast to being expressed as a fraction, a percentage is a portion of a total expressed as a number between 0 and 100. The percentages are as follows: 100% of everything, 0% of nothing, 50% of something, and 0% of nothing. Divide the component by the total, then multiply the result by 100 to get the percentage.
Julio's down payment would be 20% of the total cost of the car.
If he is financing $14,000 of the remaining cost,
the total cost of the car would be
$14,000 / (1 - 0.20)
= $14,000 / 0.80
= $17,500.
Therefore, Julio's down payment would be $17,500 * 0.20 = $3,500.
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Drag the tiles to the correct boxes to complete the pairs.
Match the units of measurement to their International System (SI) equivalents.
1 foot
1 pound
1 fluid ounce
29.5 milliliters-
30 centimeters or 0.30 meters-
450 grams or 0.45 kilograms-
Answer:
see below
Step-by-step explanation:
1 foot = 30 centimeters or 0.30 meters-
1 pound =450 grams or 0.45 kilograms-
1 fluid ounce = 29.5 milliliters-
14. Calculate the x-intercepts of the equation by factoring:
8x²-16x = 0
The corner store sells pencils for $0.05, $0.10, and $0.15. List all the ways Kay can spend exactly $0.45 on pencils.
Nine pencils at $0.05 each, 2. Five pencils at $0.10 each and three pencils at $0.05 each 3.Six pencils costing $0.05 apiece and three pencils costing $0.15 each. Pencils: four for $0.15 each, and five for $0.05 each.
What is combination theory?This question uses the combination theory, which states that the total number of combinations of a set of items is equal to the number of variations in one item multiplied by the number of variations in the second item, and so on.
in this question
1. Nine pencils at $0.05 each: Kay can buy nine pencils for $0.45. Nine pencils times $0.05 each equal $0.45.
2. Five pencils at $0.10 each and three pencils at $0.05 each: Kay can buy five pencils for $0.50 and three pencils for $0.15.
Five pencils times $0.10 each equal $0.50 and three pencils times $0.05 each equal $0.15.
Adding the two totals together equals $0.45.
3. Three pencils at $0.15 each and six pencils at $0.05 each: Kay can buy three pencils for $0.45 and six pencils for $0.30.
Three pencils times $0.15 each equals $0.45 and six pencils times $0.05 each equal $0.30.
Adding the two totals together equals $0.45.
4. Four pencils at $0.15 each and five pencils at $0.05 each:
Kay can buy four pencils for $0.60 and five pencils for $0.25.
Therefore, Four pencils times $0.15 each equals $0.60 and five pencils times $0.05 each equal $0.25. Subtracting the two totals together equals $0.45.
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9b² – 100
Factor it
Which of the following arrow diagrams represent a mapping? If that is the case give the domain and range.
The arrow diagram that represents a mapping include the following: arrow diagram C.
The domain of this arrow diagram is equal to {a, c, d}
The range of this arrow diagram is equal to {q, r, s}.
What is a domain?In Mathematics, a domain can be defined as the set of all real numbers for which a particular function is defined.
This ultimately implies that, a domain is the set of all possible input numerical values (x-values) to a function and the domain of a graph comprises all the input numerical values (real numbers) which are located on the x-coordinate.
In this scenario, we can reasonably infer and logically deduce that only the above mentioned relation (arrow diagram C) represents a function because each input variable (domain) has exactly an output variable (range) as illustrated in the diagram.
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A company is expanding its team at a rate of 10% per year. 17 a) Calculate how many members of staff there will be after 1 year if there are initially 200 members of staff.
Step-by-step explanation:
If a company is expanding its team at a rate of 10% per year, and it currently has 200 members of staff, we can calculate the number of staff after 1 year using the formula:
number of staff after 1 year = 200 + (10/100) * 200
number of staff after 1 year = 200 + 20
number of staff after 1 year = 220
So, there will be 220 members of staff after 1 year if there are initially 200 members of staff.
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Review the Monthly Principal & Interest Factor chart to answer the question:
APR 30-Year Term 20-Year Term 15-Year Term
5.5 $5.68 $6.88 $8.17
6.0 $6.00 $7.16 $8.44
6.5 $6.32 $7.46 $8.71
7.0 $6.65 $7.75 $8.99
7.5 $6.99 $8.06 $9.27
The Apolaro household will be financing a principal balance of $115,200.00 for their property. They are trying to decide whether to take a 30-year or 15-year term mortgage. Based on their credit score, they qualify for an APR of 7.0%. Calculate the difference in total interest they would pay between the two terms. (2 points)
$81,012.83
$84,353.72
$89,372.16
$92,416.64
Answer: c
Step-by-step explanation:
Answer:
Step-by-step explanation:
its $89,372 i did the math took the test
A shipping container will be used to transport several 100-kilogram crates across the country by rail. The greatest weight that can be loaded into the container is 27500 kilograms. Other shipments weighing 5400 kilograms have already been loaded into the container. Write and solve an inequality which can be used to determine xx, the number of 100-kilogram crates that can be loaded into the shipping container.
The inequality to determine the number of 100-kilogram crates that can be loaded into the shipping container is 27500 ≤ 5400 + 100x
How to represent a situation with inequality?A shipping container will be used to transport several 100-kilogram crates across the country by rail. The greatest weight that can be loaded into the container is 27500 kilograms. Other shipments weighing 5400 kilograms have already been loaded into the container.
Therefore, the inequality that can be used to determine x, the number of 100-kg crates that can be loaded into the shipping container can be calculated as follows:
27500 ≤ 5400 + 100x
where
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A bank representative studies compound interest, so she can better serve customers. She analyzes what happens when $2,000 earns interest at a rate of 4% for 3 years. Part a) Find the ending balance if it is compounded continuously. Part b) Find the interest earned if it is compounded continuously
The ending balance if the amount is compounded continusoly is $6,244.84
The interest earned is $4,244.84.
What is the ending balance and the interest?If an amount is compounded continuously, it means that both the amount invested and the interest earned increases in value infinitely over the investment period.
The formula for calculating future value when there is continuous compounding is : A x e^r x N
Where:
A= amount e = 2.7182818 N = number of years r = interest rateInterest = future value - amount invested
2000 x e^0.04 x 3 = $6,244.84
Interest = $6,244.84 - $2000 = $4,244.84
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The area of a triangle is 48 square inches. If the base of the triangle is 8 inches, what is the height?
Answer:
height = 12
Step-by-step explanation:
The formula for the area of a triangle is: [tex]\frac{1}{2}(b*h)[/tex] = A
A = area of a triangle
b = base
h = height
The problem states that the area (A) of triangle is 48, and the base (b) is 8, so we can plug those values into the equation above:
48 = [tex]\frac{1}{2}(8*h)[/tex]
Let's solve for h now:
48 = [tex]\frac{1}{2}8h[/tex]
48 = 4h
h = 48/4
h = 12
Answer:
12 inches
Step-by-step explanation:
Formulae (Area of a triangle): 1/2 × Measure of Base × Measure of Height
Given Information:
Area of triangle = 48 in²Measure of Base = 8 inSubstitute the information into the formula to create an equation;
⇒ 1/2 × Measure of Base × Measure of Height = Area of a triangle⇒ 1/2 × 8 in × Measure of Height = 48 in²Solve the equation to find out the measure of the height;
⇒ 4 in × Measure of Height = 48 in²⇒ Measure of Height = (48 in²)/(4 in)⇒ Measure of Height = (48 in)/(4) = 12 inchesTherefore, the measure of the height is 12 inches.
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given: pr is parralel to vo; ro is parralel to pv; pr is congruent to ro prove: angle 1 and angle 2 are complementary
According to the information ∠1 and ∠2 are complementary to each other.
∠1 ⊥ ∠2
Since PR is parallel to VO and RO is parallel to PV, it follows that PR is congruent to RO. This means that ∠1 and ∠2 are supplementary angles and thus, complementary.
Since PR // VO and RO // PV, then by the Transitive Property of Congruence, VO // PV.
Since PR ≅ RO and VO // PV, then by the Corresponding Angles Postulate, ∠1 ≅ ∠2.
Since ∠1 and ∠2 are congruent, then by the definition of complementary angles, ∠1 and ∠2 are complementary.
Thus, ∠1 and ∠2 are complementary.
∴ ∠1 and ∠2 are complementary
Congruent refers to something that is "absolutely equal" in terms of size and shape. The shapes hold true regardless of how we rotate, flip, or turn them. Draw two circles with the same radius, for instance, cut them out, and stack them on top of one another.
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Suppose the sample space for a continuous random variable is 0 to 100. If
the area under the density curve for the variable from 0 to 15 is 0.20, what is
the area under the density curve from 15 to 100?
The area under the density curve from 15 to 100 is 0.80.
How can you determine a continuous random variable's density?With pdf f and cdf f, consider X to be a continuous random variable.
By definition, integrating the pdf yields the cdf: F(x)=x∫−∞f(t)dt.
By differentiating the cdf, one may find the pdf according to the Fundamental Theorem of Calculus: f(x) = ddx[F(x)]
An interval or group of intervals makes up the sample space of a continuous random variable, X.
1- 0.20 0.80
[tex]\int\limits^[/tex] f(x) dx from 0 to 100 = 1
[tex]\int\limits^[/tex] f(x) dx from 0 to 15 + int 15 ^ 100 =1
[tex]\int\limits^[/tex] d from 15 to 100 =1-0.20=0.80.
The area under the density curve from 15 to 100 is 0.80.
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graph these equations? y = 3x - 5, y = 1/2x + 5
The solution of the given system of equations is (4, 7).
What is a linear system of equations?A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
The given system of equations are y=3x-5 -------(I) and y=1/2 x+5 -------(II)
Substitute equation (I) in (II), we get
3x-5=1/2 x+5
3x-1/2 x=5+5
2.5x=10
x=4
Substitute x=4 in equation (I), we get
y=3(4)-5
y=7
So, solution is (4, 7)
Therefore, the solution of the given system of equations is (4, 7).
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for each limit in exercise 55-64 use graphs and algebra to approximate the largest delta or the smallest magnitude n taht corresponds to the given value of epsilon or m, according to the appropriate formal limit definition
The largest δ value of the given limit [tex]\lim_{x \to 1^+} \frac{1}{x^2-1}=\infty[/tex] and M=1000 is calculated. The required δ value is [tex]\delta=\sqrt{1+10^{-3}}-1[/tex].
When no values are defined, a limit is used to assign values to certain functions. It is possible to assess the limits using algebraic techniques. A collection of guidelines for manipulating limits when using other operators is known as the algebra of limits. Given the expression [tex]\lim_{x \to 1^+} \frac{1}{x^2-1}=\infty[/tex] and M=1000. When x>1, we write this expression as,
[tex]\begin{aligned} \frac{1}{x^2-1}& > 1000\\x^2-1& < 1000^{-1}=10^{-3}\\x^2& < 10^{-3}+1\\x& < \sqrt{1+10^{-3}}\end{aligned}[/tex]
Now, choosing, [tex]\delta=\sqrt{1+10^{-3}}-1[/tex] when we have δ > 0. Therefore, [tex]\sqrt{1+10^{-3}} > 1[/tex]. Then, [tex]x\in(1, 1+\delta)[/tex]. This is given by,
[tex]\begin{array}{ccc}1& < x& < 1+\delta\\1& < x& < \sqrt{1+10^{-3}}\end{array}[/tex]
This implies, [tex]\frac{1}{x^2-1} > 1000=M[/tex].
Therefore, the largest δ value will be [tex]\sqrt{1+10^{-3}}-1[/tex].
The complete question is -
For each limit, use graphs and algebra to approximate the largest δ or the smallest magnitude N that corresponds to the given value of ∈ or M, according to the appropriate formal limit definition.
[tex]\lim_{x\to 1^+} \frac{1}{x^2-1}=\infty[/tex], M=1000, find largest δ > 0.
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