determine the value of x​

A suitable delta (δ) for ε = 0.01 is any positive value smaller than √6.

To find a suitable delta (δ) for the given limit, we need to consider the epsilon-delta definition of a limit.

The definition states that for a given epsilon (ε) greater than zero, there exists a delta (δ) greater than zero such that if the distance between x and the limit point (2, in this case) is less than delta (|x - 2| < δ), then the distance between the function (√x + 7) and the limit (3) is less than epsilon (|√x + 7 - 3| < ε).

Let's solve the inequality |√x + 7 - 3| < ε:

|√x + 7 - 3| < ε

|√x + 4| < ε

-ε < √x + 4 < ε

To remove the square root, we square both sides:

(-ε)^2 < (√x + 4)^2 < ε^2

ε^2 > x + 4 > -ε^2

Since we're interested in the interval around x = 2, we substitute x = 2 into the inequality:

ε^2 > 2 + 4 > -ε^2

ε^2 > 6 > -ε^2

Since ε > 0, we can drop the negative term and solve for ε:

ε^2 > 6

ε > √6

Please note that this solution assumes the function √x + 7 approaches the limit 3 as x approaches 2. To verify the solution, you can substitute different values of δ and check if the conditions of the epsilon-delta definition are satisfied.

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Answer:

d ≈ 8.1

Step-by-step explanation:

calculate the distance d using the distance formula

d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]

with (x₁, y₁ ) = R (5, 7 ) and (x₂, y₂ ) = S (- 2, 3 )

d = [tex]\sqrt{(-2-5)^2+(3-7)^2}[/tex]

  = [tex]\sqrt{(-7)^2+(-4)^2}[/tex]

  = [tex]\sqrt{49+16}[/tex]

  = [tex]\sqrt{65}[/tex]

  ≈ 8.1 ( to 1 decimal place )

a) The mean of the data-set is of 2.

b) The range of the data-set is of 4 units, which is of around 4.3 MADs.

The mean of a data-set is obtained as the sum of all observations in the data-set divided by the number of observations in the data-set, which is also called the cardinality of the data-set.

The dot plot shows how often each observation appears in the data-set, hence the mean of the data-set is obtained as follows:

Mean = (1 x 0 + 5 x 1 + 3 x 2 + 5 x 3 + 1 x 4)/(1 + 5 + 3 + 5 + 1)

Mean = 2.

The range is the difference between the largest observation and the smallest, hence:

4 - 0 = 4.

4/0.93 = 4.3 MADs.

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Answer:

89

Step-by-step explanation:

Take half of 177 and round up, which is 177/2 = 88.5 = 89

This is because 89+88=177 and 89>88, so there will be a majority.

The system B is gotten from system A by operation (d)

From the question, we have the following parameters that can be used in our computation:

x + y = 8

4x - 6y = 2

Also, we have the solution to be (5, 3)

Recall that

x + y = 8

4x - 6y = 2

Multiply the first equation by 6

So, we have

6x + 6y = 48

4x - 6y = 2

Add the equations

10x = 50

This means that the system B from system A is (d)

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Answer:

Step-by-step explanation:

$28 for 1st hr and $20per hr after that:

t = 28 +  20(h-1)

$30 for 1st hr and $20per hr after that:

t = 30 + 20(h-1)

t - 30 - 20(h-1)

The solution for k in the equation 10 - 10|-8k + 4| = 10, expressed in set notation, is {1/2}.

1. Start with the equation: 10 - 10|-8k + 4| = 10.

2. Simplify the expression inside the absolute value brackets: -8k + 4.

3. Remove the absolute value brackets by considering two cases:

  Case 1: -8k + 4 ≥ 0 (positive case):

    -8k + 4 = -(-8k + 4)  [Removing the absolute value]

    -8k + 4 = 8k - 4     [Distributive property]

    -8k - 8k = -4 + 4    [Group like terms]

    -16k = 0             [Combine like terms]

    k = 0               [Divide both sides by -16]

  Case 2: -8k + 4 < 0 (negative case):

    -8k + 4 = -(-8k + 4)  [Removing the absolute value and changing the sign]

    -8k + 4 = -8k + 4     [Simplifying the expression]

    0 = 0                [True statement]

4. Combine the solutions from both cases: {0}.

5. Check if the solution satisfies the original equation:

  For k = 0: 10 - 10|-8(0) + 4| = 10

             10 - 10|4| = 10

             10 - 10(4) = 10

             10 - 40 = 10

             -30 = 10 [False statement]

6. Since k = 0 does not satisfy the equation, it is not a valid solution.

7. Therefore, the final solution expressed in set notation is {1/2}.

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4/21 of the books in the library are both fiction and paperbacks.

To determine the fraction of books in the library that are both fiction and paperback, we need to multiply the fractions representing each condition.

Let's start with the fraction of books in the library that are fiction. If 4/7 of the books are fiction, then this fraction represents the number of fiction books.

Next, we want to find the fraction of fiction books that are also paperbacks. Since 1/3 of the fiction books are paperbacks, we multiply 4/7 (fiction books) by 1/3 (paperback fraction).

Multiplying fractions is done by multiplying the numerators together to get the new numerator and multiplying the denominators together to get the new denominator.

Thus, the fraction of books in the library that are both fiction and paperbacks is:

(4/7) * (1/3) = (4 * 1) / (7 * 3) = 4/21

Therefore, 4/21 of the books in the library are both fiction and paperbacks.

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Todd should expect approximately 624 people at his gym during the week of July 15.

A: To find the quadratic equation that models the data, we can use the vertex form of a quadratic equation:

[tex]y = a(x - h)^2 + k[/tex] where (h, k) represents the vertex of the parabola.

Let's analyze the data to determine the vertex. We observe that the number of people is highest during the first week and gradually decreases over the following weeks.

This suggests a downward-opening parabola.

From the data, the highest point occurs during the week of 6/3 to 6/9 with 624 people.

Therefore, the vertex is located at (6/3 to 6/9, 624).

Using the vertex form, we have:

[tex]y = a(x - 6/3 to 6/9)^2 + 624[/tex]

Now, we need to find the value of 'a.'

To do this, we can substitute any other point and solve for 'a.' Let's use the data from the week of 5/27 to 6/2:

[tex]618 = a(5/27 to 6/2 - 6/3 to 6/9)^2 + 624[/tex]

Simplifying the equation and solving for 'a,' we find:

[tex]618 - 624 = a(-6/3)^2[/tex]

-6 = 4a

a = -3/2

Therefore, the quadratic equation in vertex form that models the data is:

[tex]y = (-3/2)(x - 6/3 to 6/9)^2 + 624[/tex]

B: To predict the number of people Todd should expect during the week of July 15, we substitute x = 7/15 into the equation and solve for y:

[tex]y = (-3/2)(7/15 - 6/3 to 6/9)^2 + 624[/tex]

Simplifying the equation, we find:

[tex]y = (-3/2)(1/15)^2 + 624[/tex]

y = (-3/2)(1/225) + 624

y = -3/450 + 624

y = -1/150 + 624

y = 623.993

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Answer:

8

Step-by-step explanation:

i took the test mde 100

Answer:

27

Step-by-step explanation:

The graph in this problem is the graph of y = 2sin(x), not y = x, as it has a amplitude of 2.

The standard definition of the sine function is given as follows:

y = Asin(B(x - C)) + D.

For which the parameters are given as follows:

The function in this problem has an amplitude of 2, with no phase shift, no vertical shift and period of 2π, hence it is defined as follows:

y = 2sin(x)

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Answer:

|5x - 1| < 1

-1 < 5x - 1 < 1

0 < 5x < 2

0 < x < 2/5

The solution for p=2l+2w, the value of w= 3 units.

To solve the equation p = 2l + 2w for w, we will follow the steps below:

Step 1: Start with the given equation: p = 2l + 2w.

Step 2: To isolate the variable w, we need to get rid of the terms involving l. We can do this by subtracting 2l from both sides of the equation:

p - 2l = 2w.

Step 3: Next, we want to solve for w. To do this, we divide both sides of the equation by 2:

(p - 2l) / 2 = w.

Step 4: Simplify the expression on the right side:

w = (p - 2l) / 2.

Now, let's apply this formula to a specific example. Suppose we have a rectangle with a perimeter of 16 units (p = 16) and a length of 5 units (l = 5). We can find the width (w) using the formula:

w = (16 - 2(5)) / 2

w = (16 - 10) / 2

w = 6 / 2

w = 3.

By following the steps outlined above and substituting the given values of the perimeter (p) and length (l) into the formula w = (p - 2l) / 2, you can determine the value of the width (w) for any given rectangle.

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Answer:

Number of Females in Class = x Given: Ratio of Males to Females = 2:3 Given: Number of Males in Class = 12 Assume the total number of people in class = y 2/3 of y = x 2x = 3y 12 + x = y y - 12 = x y - 12 = 2x 3y - 36 = 2x 3y = 2x + 36 y = (2x + 36) / 3 y = (2(12) + 36)/3 y = 24 x = y - 12 x = 24 - 12 x = 12 Answer: There are 12 females in the class.

Step-by-step explanation:

Angle C can be found by subtracting the sum of angles A and B from 180 degrees:

b. A = 34.1°; B = 44.4°; C = 101.5°

To solve a triangle with side lengths a = 4, b = 5, and c = 7, we can use the law of cosines and the law of sines.

First, let's find angle A using the law of cosines:

[tex]cos(A) = (b^2 + c^2 - a^2) / (2\times b \times c)[/tex]

[tex]cos(A) = (5^2 + 7^2 - 4^2) / (2 \times 5 \times 7)[/tex]

cos(A) = (25 + 49 - 16) / 70

cos(A) = 58 / 70

cos(A) ≈ 0.829

A ≈ arccos(0.829)

A ≈ 34.1°

Next, let's find angle B using the law of sines:

sin(B) / b = sin(A) / a

sin(B) = (sin(A) [tex]\times[/tex] b) / a

sin(B) = (sin(34.1°) [tex]\times[/tex] 5) / 4

sin(B) ≈ 0.822

B ≈ arcsin(0.822)

B ≈ 53.4°

Finally, angle C can be found by subtracting the sum of angles A and B from 180 degrees:

C = 180° - A - B

C = 180° - 34.1° - 53.4°

C ≈ 92.5°.

b. A = 34.1°; B = 44.4°; C = 101.5°

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The money raised by Lucas for the fundraiser is $30.

The bars sold on Saturday are 12 bars. Each bar costs $1.50.

So, the amount will be:  12 * 1.50 = 18

The bars sold on Sunday are  8 bars.

So, the amount will be:  8 * 1.50 = 12

Hence, the total amount: is 18+12= 30

Answer:

Step-by-step explanation:

1. The system of inequalities that models this scenario can be represented as:

Let x be the number of servings of dry food.

Let y be the number of servings of wet food.

The cost constraint:

1x + 5y ≤ 15

The minimum number of dogs constraint:

x + y ≥ 4

2. The graph of the system of inequalities would be a shaded region in the coordinate plane.

To graph the inequality 1x + 5y ≤ 15, we can first graph the equation 1x + 5y = 15 (the corresponding boundary line) by finding two points on the line and connecting them. For example, when x = 0, y = 3, and when y = 0, x = 15. Plotting these points and drawing a line through them will represent the equation 1x + 5y = 15.

Next, we need to shade the region below the line because the inequality is less than or equal to (≤). This shaded region represents the solutions that satisfy the cost constraint.

To graph the inequality x + y ≥ 4, we can again find two points on the line x + y = 4 (the corresponding boundary line). For example, when x = 0, y = 4, and when y = 0, x = 4. Plotting these points and drawing a line through them will represent the equation x + y = 4.

Lastly, we shade the region above the line x + y = 4 because the inequality is greater than or equal to (≥). This shaded region represents the solutions that satisfy the minimum number of dogs constraint.

The solution set is the overlapping region where the shaded areas of both inequalities intersect. This region represents the combination of servings of dry food and wet food that Michelle can purchase within her budget ($15) to feed at least four dogs at the animal shelter.

The inequalities D + W > 4 and D + 5W ≤ 15 model the problem. The graph represents these inequalities, with the overlap of shaded regions showing possible food serving combinations.

Let's define D as the number of servings of dry food and W as the number of servings of wet food. The system of inequalities that models this scenario is:

The graph will show the solution sets to the inequalities. D and W must both be non-negative, hence the graphed area is in the first quadrant. The first inequality requires shading above a line that connects (0,4) and (4,0). This line is solid since numbers equal to 4 are included. The second inequality requires shading below a line that connects (0,3) and (15,0). This is also a solid line because Michelle can spend exactly $15. The overlapping region of the graph is the solution set, quantifying the combinations of dry and wet food servings that Michelle can buy.

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A set S is T-finite if it satisfies Tarski's finite set condition, which states that for every nonempty subset X of P(S), there exists a maximal element u in X such that there is no v in X with u as a proper subset of v and u is distinct from v. If a set does not satisfy this condition, it is considered T-infinite.

In set theory, a set S is said to be T-finite if it satisfies a particular property called Tarski's finite set condition. This condition states that for every nonempty subset X of the power set of S (denoted as P(S)), there exists a maximal element u in X such that there is no element v in X that properly contains u (i.e., u is not a proper subset of v) and u is distinct from v.

To understand this concept, let's break it down further:

T-finite set: A set S is T-finite if, for any nonempty subset X of P(S), there exists an element u in X that is maximal. This means that u is not properly contained in any other element in X.

Maximal element: In the context of Tarski's finite set condition, a maximal element refers to an element u in X that is not a proper subset of any other element in X. In other words, there is no v in X such that u is a proper subset of v.

Distinct elements: This means that u and v are not the same element. In the context of Tarski's finite set condition, u and v cannot be equal to each other.

T-infinite set: A set S is T-infinite if it does not satisfy Tarski's finite set condition. This means that there exists a nonempty subset X of P(S) for which no maximal element u can be found, or there exists an element v in X that properly contains another element u.

In conclusion, a set S is T-finite if it meets Tarski's finite set condition, which asserts that there exists a maximal element u in X such that there is no v in X with v as a proper subset of u and u is different from v. A set is regarded as T-infinite if it does not meet this requirement.

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The value of the given expression is 1370.

To calculate the given expression, we can group the terms in pairs and simplify them.

We have the following pattern:

1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 + ... + 97 + 98 - 99

Grouping the terms in pairs, we can see that each pair consists of a positive and a negative term. The positive term increases by 1 each time, and the negative term decreases by 1 each time. Therefore, we can rewrite the expression as:

(1 - 3) + (2 + 4) + (5 - 6) + (7 + 8) + ... + (97 + 98) - 99

The sum of each pair in parentheses simplifies to a single term:

-2 + 6 - 1 + 15 + ... + 195 - 99

Now, we can add up all the terms:

-2 + 6 - 1 + 15 + ... + 195 - 99 = 1370

As a result, the supplied expression has a value of 1370.

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5 whole numbers are written in order. 5,8,x,y,12 The mean and median of the five numbers are the same then the values of x and y are:[tex]$$\boxed{x=8, \ y=3}$$[/tex] OR [tex]$$\boxed{x=12, \ y=53}$$[/tex].

let's first calculate the median of the given numbers.

Median of the given numbers is the middle number of the ordered set.

As there are five numbers in the ordered set, the median will be the third number.

Thus, the median of the numbers = x.

The mean of a set of numbers is the sum of all the numbers in the set divided by the total number of items in the set.

Let the mean of the given set be 'm'.

Then,[tex]$$m = \frac{5+8+x+y+12}{5}$$$$\Rightarrow 5m = 5+8+x+y+12$$$$\Rightarrow 5m = x+y+35$$[/tex]

As per the given statement, the median of the given set is the same as the mean.

Therefore, we have,[tex]$$m = \text{median} = x$$[/tex]

Substituting this value of 'm' in the above equation, we get:[tex]$$x= \frac{x+y+35}{5}$$$$\Rightarrow 5x = x+y+35$$$$\Rightarrow 4x = y+35$$[/tex]

Also, as x is the median of the given numbers, it lies in between 8 and y.

Thus, we have:[tex]$$8 \leq x \leq y$$[/tex]

Substituting x = y - 4x in the above inequality, we get:[tex]$$8 \leq y - 4x \leq y$$[/tex]

Simplifying the above inequality, we get:[tex]$$4x \geq y - 8$$ $$(5/4) y \geq x+35$$[/tex]

As x and y are both whole numbers, the minimum value that y can take is 9.

Substituting this value in the above inequality, we get:[tex]$$11.25 \geq x + 35$$[/tex]

This is not possible.

Therefore, the minimum value that y can take is 10.

Substituting y = 10 in the above inequality, we get:[tex]$$12.5 \geq x+35$$[/tex]

Thus, x can take a value of 22 or less.

As x is the median of the given numbers, it is a whole number.

Therefore, the maximum value of x can be 12.

Thus, the possible values of x are:[tex]$$\boxed{x = 8} \text{ or } \boxed{x = 12}$$[/tex]

Now, we can use the equation 4x = y + 35 to find the value of y.

Putting x = 8, we get:

[tex]$$y = 4x-35$$$$\Rightarrow y = 4 \times 8 - 35$$$$\Rightarrow y = 3$$[/tex]

Therefore, the values of x and y are:[tex]$$\boxed{x=8, \ y=3}$$[/tex]  OR [tex]$$\boxed{x=12, \ y=53}$$[/tex]

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Answer:

A) The y-intercept(s) is/are 2

Step-by-step explanation:

Y-intercepts are where the graph of a function cross over the y-axis. In this case, the line passes through y=2, which is the y-intercept.

Answer:

Assuming Miguel rolled up his sleeping bag tightly and neatly, the length and circumference of the sleeping bag can help us estimate the length of string needed to tie it up.

Let's say the length of the sleeping bag is 6 feet and the circumference (distance around) is 3 feet. To tie it up, Miguel would need to wrap the string around it 2-3 times, depending on how long the string is and how tight he ties the knot.

So, we can estimate that he used about 12-18 feet of string (i.e. 2-3 times the circumference). In inches, that would be about 144-216 inches of string (i.e. 12-18 feet * 12 inches/foot).

Keep in mind that this is just an estimate and the actual amount of string used may vary depending on the factors mentioned above.

Step-by-step explanation:

Answer:

π(2^2 - .75^2) = 55π/16 m² = 10.8 m²

D is the correct answer.

Answer:

[tex]\textsf{a)} \quad \textsf{Radius = $\overline{HG}$}[/tex]

[tex]\textsf{b)} \quad \textsf{Chord = $\overline{GF}$}[/tex]

[tex]\textsf{c)} \quad \textsf{Diameter = $\overline{JF}$}[/tex]

[tex]\textsf{d)} \quad \textsf{Secant = $\overleftrightarrow{GF}$}[/tex]

[tex]\textsf{e)} \quad \textsf{Tangent = $\overleftrightarrow{GK}$}[/tex]

[tex]\textsf{f)} \quad \textsf{Point of tangency = $\overset{\bullet}{G}$}[/tex]

[tex]\textsf{g)} \quad \textsf{Circle $H$}[/tex]

Step-by-step explanation:

The radius is the distance from the center of a circle to any point on its circumference. The center of the circle is point H. Therefore, the radius of the given circle is line segment HG.

A chord is a straight line joining two points on the circumference of the circle. There are two chords in the given circle:  line segments GF and JF. Therefore, a chord of the given circle is line segment GF.

The diameter of a circle is a straight line segment passing through the center of a circle, connecting two points on its circumference.

Therefore, the diameter of the given circle is line segment JF.

A secant is a straight line that intersects a circle at two points.

Therefore, the secant of the given circle is line GF.

A tangent is a straight line that touches a circle at only one point.

Therefore, the tangent line of the given circle is line GK.

The point of tangency is the point where the line touches the circle.

Therefore, the point of tangency of the given circle is point G.

A circle is named by its center point. Therefore, as the center point of the circle is point H, the name of the circle is "Circle H".

Answer:

c

Step-by-step explanation:

add 360 to 265 to get the first number and subtract 360 from 265 to get the second number

The equation of the conic section in conic form is (x - 1) = (y + 6)²/4.

To write the equation of the conic section in conic form, we can complete the square to transform the equation into its standard form. Let's start with the given equation:

y² - 4x + 12y + 32 = 0

Rearranging the terms, we have:

y² + 12y - 4x + 32 = 0

To complete the square for the y-terms, we add and subtract the square of half the coefficient of y (which is 6 in this case):

y² + 12y + 36 - 36 - 4x + 32 = 0

Simplifying this, we get:

(y + 6)² - 4x + 4 = 0

Now, rearranging the terms, we have:

(y + 6)² = 4x - 4

Dividing both sides of the equation by 4, we get:

(y + 6)²/4 = x - 1

Finally, we can write the equation in conic form:

(x - 1) = (y + 6)²/4

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The Probable question may be:
Which type of conic section is defined by the equation y²-4x+12y + 32 = 0?

This is an equation of a parabola

Write the equation of this conic section in conic form:

After solving by formula the second expression is y =  [tex](x^2 + 1)[/tex].

We know that the product of the HCF and LCM of two numbers is equal to the product of the numbers themselves. In this case, we can apply the same principle to expressions:

HCF * LCM = (x + 1) *  [tex](x^3+ x^2 - x - 1)[/tex]

the first number is [tex]x^{2} -1\\[/tex] and  let the second number is y

Therefore, we can set up the equation:

(x + 1) *  [tex](x^3+ x^2 - x - 1)[/tex] =  [tex]x^{2} -1\\[/tex]  * y

[tex]x^4 + x^3 + x^2 - x^3 - x^2 + x - x - 1 = x^2 - 1 * y[/tex]

Simplifying:

[tex]x^4 - 1 = (x^2 - 1) * y[/tex]

Now, we can divide both sides by [tex](x^2 - 1)[/tex]:

[tex](x^4 - 1) / (x^2 - 1) = y[/tex]

Notice that [tex](x^2 - 1)[/tex]can be factored as (x + 1)(x - 1). Therefore, we can simplify further:

[tex](x^4 - 1) / ((x + 1)(x - 1)) = y[/tex]

The expression [tex](x^4 - 1)[/tex] can be factored using the difference of squares:

[tex](x^4 - 1) = (x^2 + 1)(x^2 - 1)[/tex]

[tex][(x^2 + 1)(x^2 - 1)] / ((x + 1)(x - 1)) = y[/tex]

Now, we can cancel out the common factor  [tex](x^2 - 1)[/tex] from the numerator and denominator:

[tex]y =(x^2 + 1)[/tex]

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Answer:

Proving that the limit of the equation 10 - 2x as x approaches -3 is 16 involves using the definition of a limit.

Here's how you would approach it:

Let epsilon be a small positive number. We want to find a value of delta such that if x is within a distance of delta from -3, then 10 - 2x is within a distance of epsilon from 16.

So, we start with:

|10 - 2x - 16| < epsilon

Simplifying,

|-2x - 6| < epsilon

And using the reverse triangle inequality,

|2x + 6| > ||2x| - |6||

Now, we can choose a value for delta such that if x is within delta of -3, then |2x + 6| is within delta + 6 of |-6| = 6.

So,

||2x| - |6|| < epsilon

and therefore:

|2x - 6| < epsilon

Choosing delta = epsilon/2, we can prove that:

0 < |x + 3| < delta -> |2x - 6| < epsilon

Therefore, we have proved that the limit of 10 - 2x as x approaches -3 is 16 using the definition of a limit.

Step-by-step explanation:

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