Answer:
- 4x³ - 33x² + 31x - 3------------------------------------------------------------------------------------------
Someone help me please
Answer:
A because it is increasing
Step-by-step explanation:
Answer:
I would say c
Step-by-step explanation:
you think if you go down a hill what happen is you speed up so you would need to be go down or heading in a negative direction.
Which statement describes the graph of f(x) = x² - 6x + 9 ?
a line with an x-intercept of (-3, 0)
a parabola with an x-intercept of (-3, 0)
a line with an x-intercept of (3, 0)
a parabola with an x-intercept of (3, 0)
Answer:
About
step 1 highlight numbers
step 2 do ctrl+f
step 3 type in 9
ENJOY
1051541451431641621571721571511441234567881234567812345678123678326470547 2999999259923478990124999995689902993413269916749953349999914649932724997 2994567809912568990139956799809929936781467998299634699818991169966144990 2999994569970124995699801323459999012615302799995324993243699019923412993 2994567801993569980299356780239999456725634569974326992644399243992369936 2994567801992689901239967899029939945745315319931253399436998011992349950 2998012345299999388352999991039953991232012479934673289999982640499999415l
Step-by-step explanation:
Copy the problems onto your paper, mark the givens and prove the statements asked.
Given ∠E ≅ ∠T, M − midpoint of TE Prove: MI ≅ MR
MI ≅ MR proved by using ASA postulate of congruence
Step-by-step explanation:
Let us revise the cases of congruence
SSS ⇒ 3 sides in the 1st Δ ≅ 3 sides in the 2nd Δ SAS ⇒ 2 sides and including angle in the 1st Δ ≅ 2 sides and including angle in the 2nd Δ ASA ⇒ 2 angles and the side whose joining them in the 1st Δ ≅ 2 angles and the side whose joining them in the 2nd Δ AAS ⇒ 2 angles and one side in the 1st Δ ≅ 2 angles and one side in the 2nd Δ HL ⇒ hypotenuse leg of the 1st right Δ ≅ hypotenuse leg of the 2nd right Δ∵ M is the mid-point of TE
∴ MT = ME
In Δs TMI and EMR
∵ ∠T ≅ ∠E ⇒ given
∵ ∠TMI ≅ EMR ⇒ vertical opposite angles
∵ MT = ME ⇒ proved
∴ Δ TMI ≅ ΔEMR by ASA postulate of congruence
- From congruence, corresponding sides are equal
∴ MI ≅ MR
MI ≅ MR proved by using ASA postulate of congruence
Learn more;
You can learn more about the cases of congruence in brainly.com/question/6108628
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The question is about a geometry proof related to congruent angles and midpoints, but additional information is needed to provide a specific solution.
Explanation:The question appears to be a geometry proof involving congruent angles and midpoints. The goal is to prove that two line segments MI and MR are congruent given ∠E ≅ ∠T, and M is the midpoint of TE.
To prove this, one would need to provide more information about the points I and R, such as their positions relative to point E, T, and M.
Without this additional information, it is difficult to give a specific proof. However, typical proof strategies might involve showing that triangles EMI and TMR are congruent by Side-Angle-Side (SAS), Angle-Side-Angle (ASA), or Angle-Angle-Side (AAS) postulates, depending on the position of points I and R.
A 25-foot flag pole casts a shadow of 40 feet. What is the approximate height from the top of the flag pole to the top of the shadow? 45 47 65
Answer:45
Step-by-step explanation:
it is 45
Step-by-step explanation:
Please help I'm a little confused!
Eight is less than or equal to the quotient of a number and negative four.
I will have "x" represent the unknown number, or you could use a "?", doesn't matter.
[tex]8\leq \frac{x}{-4}[/tex] [8 is less than or equal to (≤) the quotient (÷) of a number and -4, so the number is being divided by -4]
If you need to solve this, you need to isolate/get the variable "x" by itself in the inequality:
[tex]8\leq \frac{x}{-4}[/tex] Multiply -4 on both sides to get rid of the fraction and get "x" by itself
[tex](-4)8\geq \frac{x}{-4} (-4)[/tex] When you multiply/divide by a negative number, you flip the sign (< >)
-32 ≥ x [-32 is greater than or equal to x, or x is less than or equal to -32]
lines AD and BE intersect at point C, as shown. create an expression that represents the measure of the angle DCE in terms if X
part 2 of question: using your expression, solve for the missing value of X if DCE was 120 degrees.
Answer:
Please, find the image with the diagram of the lines corresponding to this question:
The answers are:
Equation: m ∠ DCE = 180º - x
Value of x = 60º
Explanation:
First question:
You must use the fact that the angles DCE and BCD are adjacent angles, because they share the vertex (C) and have a common side (CD).
Thus, as a first fact, the measure of the angle BCE is equal to the sum of the measures of the angles BCD and DCE.
On the other hand, BE is a straight line, thus the measure of angle BCE is 180º.
Hence, you can write:
m ∠ DCE + m ∠ DCB = 180ºm ∠ DCE + x = 180ºm ∠ DCE = 180º - xSecond question:
Solve for x:
Given: m ∠ DCE = 180º - xAdd x to both sides: m ∠ DCE + x = 180º Subtract m ∠ DCE from both sides: x = 180º - m ∠DCESubstitute m ∠DCE with 120º:
x = 180º - 120º = 60ºHence, for m ∠DCE = 120º, x = 60º.
What is the answer to 13d+25=8d
13d + 25 = 8d
13d - 8d = -25
5d = -25
d = -5
so the answer is (-5)
Answer:
Step-by-step explanation:
13d + 25 = 8d
13d = 8d -25
13d - 8d = -25
5d = -25
d = -5
In the triangle below, what is the sine of 30°?
Answer:
There's no picture. Is it D.√3 / 2
Step-by-step explanation:
Subtracting 1 from which digit in the number 12,345 will decrease the value of the number by 1,000
Match the following items by evaluating the expression for x = -2.
x-2
x-1
x0
x1
x2
By substituting x = -2 in each expression, we find that x-2 equals -4, x-1 equals -3, x₀ equals 1, x₁ equals -2, and x₂ equals 4.
Explanation:To solve this, we need to substitute x = -2 into each expression. Here are the results:
For x-2, substituting -2 would give us -2 - 2 which is -4.In the expression x-1, substituting -2 would provide us with -2 - 1, which equals -3.For x₀, any non-zero number to the power of zero is 1.When we put -2 into x₁, we simply get -2.Finally, for x₂, -2 squared equals to 4.Learn more about Evaluating expressions here:https://brainly.com/question/21469837
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Can someone please answer this question please answer it correctly and show work please
Answer:
42) 122.72 square inches
43) 212.83 additional feet of fabric
Step-by-step explanation:
42)
Diameter = [tex]12\frac{1}{2}[/tex] = 12.5 inches
Radius = (1/2) * Diameter = (1/2) * 12.5 = 6.25 inches
The formula for calculating the area of a circle is π * Radius²
π * Radius²
Substitute radius and pi into formula
3.14159265359 * 6.25²
Solve exponent
3.14159265359 * 39.0625
Multiply
122.7184630309
Round to the nearest hundredth
122.72 square inches
43) Clarissa needs 500 feet of fabric
She has these pieces
3 pieces of fabric that are each 18 yards long
1 piece of fabric that is [tex]16\frac{1}{2}[/tex] yards long
1 piece of fabric that is [tex]75\frac{2}{3}[/tex] feet long
Convert all yard measurements to feet and all fractions to decimals
(1 yard = 3 feet)
3 pieces of fabric that are each 54 feet long
1 piece of fabric that is 49.5 yards long
1 piece of fabric that is 75.67 feet long
Add all lengths
3(54) + 49.5 + 75.67
162 + 49.5 + 75.67 = 287.17
Subtract from 500 to find missing length
500 - 287.17 = 212.83
212.83 additional feet of fabric
Hope this helps :)
Estimate by rounding the largest number to hundreds and then multiplying.
691 x 4 is approximately
Answer:
approximately 2,800
Step-by-step explanation:
you would round 691 to 700 then multiply by 4
Answer:
2,800
Step-by-step explanation:
In saying "largest number," I believe the question means that you should round 691 up to 700 (nearest hundreds). Multiple 700 by 4, and you will get 2,800! Hope this helps! :)
Please answer this question.
Answer:
2
Step-by-step explanation:
Solve for N , I can’t get this , I’ll give brainliest to who gets it!!!
Answer:
55
Step-by-step explanation:
The two shortest sides added up need to be longer then the longest angle so 48 + 55 = 103>73
There are 26 students in Mrs. Augello’s math class. The number of boys is three less than the number of girls. Write a system of equations that represents the number of boys and girls.
Answer:
[tex]b+g=26[/tex]
[tex]b=g-3[/tex]
Step-by-step explanation:
Let the number of boys be reprented by [tex]b[/tex]
Let the number of girls be represented by [tex]g\\[/tex]
↓ Total number of boys + girls must be 26
[tex]b+g=26[/tex]
↓ Number of boys is 3 less than the number of girls
[tex]b=g-3[/tex]
The system of equations that represents the number of boys and girls are [tex]x+y=26[/tex] and [tex]x = y-3[/tex] and this can be determined by forming the linear equation.
Given :
There are 26 students in Mrs. Augello’s math class. The number of boys is three less than the number of girls.The following steps can be used in order to determine the system of equations that represents the number of boys and girls:
Step 1 - Let the total number of boys be 'x' and the total number of girls be 'y'.
Step 2 - So, the linear equation that represents the total number of students in Mrs. Augello's math class is:
[tex]x+y=26[/tex] --- (1)
Step 3 - The linear equation that represents the situation "number of boys is three less than the number of girls" is:
[tex]x = y-3[/tex] --- (2)
For more information, refer to the link given below:
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question for class 4
5. A hummingbird feeder holds 2/3 of a cup of hummingbird solution. A bottle of solution will fill the feeder 8 times. How much hummingbird solution is in the bottle.
Answer:
1.26 liters.
Step-by-step explanation:
Given that, a hummingbird feeder holds [tex]\frac{2}{3}[/tex] of a cup of hummingbird solution.
Again, a bottle of solution will fill the feeder 8 times.
Therefore, a bottle of solution will measure [tex](\frac{2}{3}\times 8) = \frac{16}{3} = 5.33[/tex] cups of hummingbird solution.
Now, one cup is equivalent to 0.236588 liters.
So, 5.33 cups is equal to (0.236588 × 5.33) = 1.26 liters. (Answer)
Afon read pages in 30 minutes.
Find the number of minutes read per page and the number of pages read per minute.
I need help im bad at 7th grade math!!!!!!!!
Answer:
Therefore Afon can read 0.75 pages per minute.
Step-by-step explanation:
Afon read 22 1/2 pages in 30 minutes
Therefore Afon read 22.5 pages in 30 minutes.
Therefore the number of pages Afon can read in 1 minute
[tex]= \dfrac{22.5}{30} = \dfrac{45}{60} = \dfrac{9}{12} = \dfrac{3}{4} = 0.75 pages[/tex]
Therefore Afon can read 0.75 pages per minute.
A frame around a rectangular family portrait has a perimeter of 150 inches. The length is ten more than four times the width. Find the length and width of the frame.
Answer: The length is 62 inches and the width is 13 inches
Step-by-step explanation: The perimeter of the rectangular portrait has been given as 150 inches. We also know that the perimeter of a rectangle is given as
Perimeter= 2(L + W)
However we don't have the measurements for the length and width. What we do have are descriptions of both. The length is given as W, while the length is ten more than four times the width. That is, the length equals
10 + 4W
Therefore we have the length and the width as
L = 4W + 10 and
W = W
If the perimeter is 150, and
Perimeter = 2(L + W) then,
150 = 2(4W + 10 + W)
150 = 2(5W + 10)
150 = 10W + 20
Subtract 20 from both sides of the equation
130 = 10W
Divide both sides of the equation by 10
13 = W
With that in mind we can now calculate the length as
L = 4W + 10
Substitute for the value of W
L = 4(13) + 10
L = 52 + 10
L = 62
Therefore, the length is 62 inches and the width is 13 inches
Answer:
Length = 62 inches, Width = 13 inches
Step-by-step explanation:
Let L represent the length of the frame, W, the width and P, the perimeter
Perimeter of a rectangle, P = 2 (L + W) .....eq 1
Also, P = 150 inches
and
L = 10 + 4W .....eq 2
Slotting in the respective values of P and L in eq 1
150 = 2 {(10 + 4W) + W}
Expanding the bracket
150 + 2 (10 + 5W)
150 = 20 + 10W
Subtracting 20 from both sides of the equation
150 - 20 = 20 - 20 + 10W
130 = 10W
Dividing both sides by the coefficient of W which is 10
13 = W
Therefore, W = 13 inches
Slotting in the value of W in eq 2
L = 10 + 4 (13)
L = 10 + 52
L = 62 inches
Lets ensure that the values of L and W are correct
P = 2 (L + W)
150 = 2 (13 + 62)
150 = 2(75)
150 = 150
Hence, L = 62 inches, W = 13 inches
Let y=f(x) be the particular solution to the differential equation dy/dx=(x^2+1)/e^y with the initial condition f(1)=0. What is the value of f(2) ?
[tex]\( f(2) = \ln \left( \frac{14}{3} \right) \)[/tex]. You can calculate this value to get a numerical result.
To find the particular solution [tex]\( y = f(x) \)[/tex] to the given differential equation [tex]\( \frac{dy}{dx} = \frac{x^2 + 1}{e^y} \)[/tex] with the initial condition [tex]\( f(1) = 0 \)[/tex] and then evaluate [tex]\( f(2) \),[/tex] we can follow these steps:
1. Separate variables:
[tex]\[ e^y dy = (x^2 + 1) dx \][/tex]
2. Integrate both sides:
[tex]\[ \int e^y dy = \int (x^2 + 1) dx \][/tex]
[tex]\[ e^y = \frac{1}{3} x^3 + x + C \][/tex]
3. Apply the initial condition [tex]\( f(1) = 0 \):[/tex]
[tex]\[ e^0 = \frac{1}{3} (1)^3 + 1 + C \][/tex]
[tex]\[ 1 = \frac{1}{3} + 1 + C \][/tex]
[tex]\[ C = \frac{2}{3} \][/tex]
So, the particular solution is:
[tex]\[ e^y = \frac{1}{3} x^3 + x + \frac{2}{3} \][/tex]
4. Solve for y :
[tex]\[ y = \ln \left( \frac{1}{3} x^3 + x + \frac{2}{3} \right) \][/tex]
Now, we need to find f(2), which means finding the value of y when x = 2:
[tex]\[ y = \ln \left( \frac{1}{3} (2)^3 + 2 + \frac{2}{3} \right) \][/tex]
[tex]\[ y = \ln \left( \frac{8}{3} + 2 + \frac{2}{3} \right) \][/tex]
[tex]\[ y = \ln \left( \frac{14}{3} \right) \][/tex]
buzz makes 75% of her free throws.If she whats to set the record of making 48 free throws in 5 minutes,how many free throws does she have to make?
Answer:
63
Step-by-step explanation:
Answer:
65
Step-by-step explanation:
75% of 65 = 48
Suppose Ellie starts making a lamb stew using a 6.5-quart pot. She decides to make a bigger stew. She pours everything into a pot that is 40 percent bigger. How big is the bigger pot?
Answer:
The bigger pot is a 9.1-quart pot
Step-by-step explanation:
Initial pot size = 6.5-quart
New pot is 40% bigger than the initial pot size
Bigger pot size = 6.5-quart + (0.4 × 6.5-quart) = 6.5-quart + 2.6-quart = 9.1-quart
which plane is closer to the base of the airplane tower? explain. The distance for Plane A, to the nearest tenth, is ____ kilometers. The distance for Plane B, to the nearest tenth, is _____ kilometers. help!!
Answer:
Plane B is closer.
The distance for Plane A is 7.9 km.
The distance for Plane B is 7.4 km.
Step-by-step explanation:
To find the distance to the tower, altitude must be converted to kilometers. Each 1000 ft is 0.3048 km, so the heights of the planes are ...
-- Plane A: (20 thousand ft)*(0.3048 km/thousand ft) = 6.096 km
-- Plane B: (8 thousand ft)*(0.3048 km/thousand ft) = 2.4384 km
The Pythagorean theorem can be used to find the distance from each plane to the tower:
-- distance = √((ground distance)² + (height)²)
-- Plane A distance = √(5² +6.096²) ≈ √62.16 ≈ 7.9 . . . km
-- Plane B distance = √(7² +2.4384²) ≈ √54.95 ≈ 7.4 . . . km
The distance for Plane B is shorter, so Plane B is closer to the tower.
PLEASE HELP ME ASAP IT"S VERY URGENT WILL MARK THE BRAINLEST
The interior angles formed by the sides of a quadrilateral have measures that sum to 360°.
What is the value of x?
Enter your answer in the box.
x =
Answer:
Step-by-step explanation:
3x - 6 + 2x + 108 + 88 = 360 Gather like terms
5x + 108 + 88 - 6 = 360
5x = 190 Divide by 5
x = 190/5
x = 38
Please solve for me!!
During a trip , Jonathan had driven a total of 75 miles by 6:20 PM and a total of 85 miles by 6:40 Pm. He drove at the same rate for the entire trip. At what time had he driven a total of 140 miles?
Answer:
8:30 PM
Step-by-step explanation:
Let's say that x is minutes since 6:20 PM, and y is position in miles.
Since Jonathan drives at a constant rate, the position vs time graph is linear. Two points on the line are (0, 75) and (20, 85). The slope of the line is:
m = Δy / Δx
m = (85 − 75) / (20 − 0)
m = ½
So the equation of the line is:
y = ½ x + 75
We want to find x when y is 140.
140 = ½ x + 75
65 = ½ x
x = 130
So 130 minutes after 6:20 PM is when Jonathan's position will be 140 miles. 130 minutes is 2 hours and 10 minutes. So the time will be 8:30 PM.
By calculating the speed at which Jonathan was driving and applying that rate to the total distance, it can be concluded that Jonathan would have driven a total of 140 miles by about 8:30 PM.
Explanation:Jonathan drove 10 miles from 6:20 PM to 6:40 PM, giving a 20-minute duration. We can say the rate of speed he was driving at is 10 miles/20 minutes. We can simplify this rate down to 1 mile/2 minutes to make it easier to work with.
To calculate when he would've driven a total of 140 miles we follow these steps:
Subtract the 75 miles he had already driven from the total of 140 miles we want to find out about, which gives us 65 miles.
Divide these 65 miles by the rate he's driving at (1 mile/2 min), which gives us 130 minutes.Add these 130 minutes to the original time of 6.20 PM. The result is approximately 8.30 PM.Therefore, Jonathan would've driven a total of 140 miles by about 8:30 PM.
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It takes 1 1/4 minutes to fill a 3 gallon bucket at this rate how long will it take to fill a 50 gallon tub
It will take [tex]20\frac{5}{6}[/tex] minutes to fill a 50 gallon tub.
Step-by-step explanation:
Time taken to fill 3 gallons bucket = [tex]1\frac{1}{4}=\frac{5}{4}\ minutes[/tex]
We have to find time required to fill 50 gallons tub.
3 gallons = [tex]\frac{5}{4}\ minutes[/tex]
1 gallon = [tex]\frac{5}{4}*\frac{1}{3}\ minutes[/tex]
1 gallon = [tex]\frac{5}{12}\ minutes[/tex]
To find the time required for 50 gallons;
50 gallons = 50 * Time per gallon
50 gallons = [tex]50*\frac{5}{12}[/tex]
50 gallons = [tex]\frac{250}{12}[/tex] = [tex]\frac{125}{6}[/tex]
50 gallons = [tex]20\frac{5}{6}\ minutes[/tex]
It will take [tex]20\frac{5}{6}[/tex] minutes to fill a 50 gallon tub.
Keywords: fraction, division
Learn more about fractions at:
brainly.com/question/11007026brainly.com/question/11207748#LearnwithBrainly
Final answer:
It takes approximately 20.83 minutes to fill a 50 gallon tub if it takes 1 1/4 minutes to fill a 3 gallon bucket, considering a constant rate of filling.
Explanation:
The question asks us to determine how long it will take to fill a 50 gallon tub if it takes 1 1/4 minutes to fill a 3 gallon bucket. To solve this, we can set up a proportion since the rates of gallons per minute will be the same for both the bucket and the tub.
First, let's convert the time from minutes to seconds to match the unit for seconds provided in the example. There are 60 seconds in a minute, so 1 1/4 minutes is equal to 75 seconds.
Now, we know it takes 75 seconds to fill 3 gallons. Let's calculate the number of seconds it takes to fill 1 gallon, and then we will use that rate to find out how many seconds it will take to fill 50 gallons.
We divide 75 seconds by 3 gallons, to get 25 seconds per gallon. Then we multiply this rate by 50 gallons to find the total time needed to fill the tub:
25 seconds/gallon × 50 gallons = 1250 seconds.
Finally, to convert seconds back to minutes, we divide 1250 seconds by 60:
1250 seconds ÷ 60 seconds/minute = 20 5/6 minutes or approximately 20.83 minutes.
So, it takes approximately 20.83 minutes to fill a 50 gallon tub at the given rate.
What do you know about two different integers that are opposites?
Only if they are each the same distance away from zero, but on opposite sides of the number line.
Hope this help! :)
Answer:If you add a negative and a positive you do the the integers step by changing the add operation to a subtraction.
Step-by-step explanation:-4+3
integers property -4 - 3 = 1
negative plus a positive is a positive.
Simplify .
[tex]\sqrt{x} 18[/tex]
Answer:
[tex]3\sqrt{2} \sqrt{x}[/tex]
Jenny had $17 more than Susan, and together they had enough money to buy a game for $93 and to have a pizza for $6. How much money did Susan have?
Answer:
Susan had $41.
Step-by-step explanation:
The amount of money they have is enough for $99 spending. The answer can be found by testing two numbers and seeing whether the amounts need to be increased or decreased to make the distance between them 17, with a sum of 99.
40 and 57: distance of 17, but not enough for $99. $41 and $58 equal 99, with a distance of 17 between them. Jenny, with $17 more, has 58. Susan has 41.
[tex]\lim_{x\to \infty} \frac{\sqrt{9x^{2} +x+1} -\sqrt{4x^{2} +2x+1} }{x+1}[/tex]
Answer:
[tex]\lim _{x\to \infty \:}\left(\frac{\sqrt{9x^2+x+1}-\sqrt{4x^2+2x+1}}{x+1}\right)=1[/tex]
Step-by-step explanation:
Considering the expression
[tex]\lim _{x\to \infty \:}\frac{\sqrt{9x^2+x+1}-\sqrt{4x^2+2x+1}}{x+1}[/tex]
Steps to solve
[tex]\lim _{x\to \infty \:}\frac{\sqrt{9x^2+x+1}-\sqrt{4x^2+2x+1}}{x+1}[/tex]
[tex]\mathrm{Divide\:by\:highest\:denominator\:power:}\:\frac{\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}}{1+\frac{1}{x}}[/tex]
[tex]\lim _{x\to \infty \:}\left(\frac{\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}}{1+\frac{1}{x}}\right)[/tex]
[tex]\lim _{x\to a}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{\lim _{x\to a}f\left(x\right)}{\lim _{x\to a}g\left(x\right)},\:\quad \lim _{x\to a}g\left(x\right)\ne 0[/tex]
[tex]\mathrm{With\:the\:exception\:of\:indeterminate\:form}[/tex]
[tex]\frac{\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)}{\lim _{x\to \infty \:}\left(1+\frac{1}{x}\right)}.....[1][/tex]
As
[tex]\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)=1[/tex]
Solving
[tex]\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)....[A][/tex]
[tex]\lim _{x\to a}\left[f\left(x\right)\pm g\left(x\right)\right]=\lim _{x\to a}f\left(x\right)\pm \lim _{x\to a}g\left(x\right)[/tex]
[tex]\mathrm{With\:the\:exception\:of\:indeterminate\:form}[/tex]
[tex]\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}\right)-\lim _{x\to \infty \:}\left(\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)[/tex]
Also
[tex]\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}\right)=3[/tex]
Solving
[tex]\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}\right)......[B][/tex]
[tex]\lim _{x\to a}\left[f\left(x\right)\right]^b=\left[\lim _{x\to a}f\left(x\right)\right]^b[/tex]
[tex]\mathrm{With\:the\:exception\:of\:indeterminate\:form}[/tex]
[tex]\sqrt{\lim _{x\to \infty \:}\left(9+\lim _{x\to \infty \:}\left(\frac{1}{x}+\lim _{x\to \infty \:}\left(\frac{1}{x^2}\right)\right)\right)}[/tex]
[tex]\lim _{x\to \infty \:}\left(9\right)=9[/tex]
[tex]\lim _{x\to \infty \:}\left(\frac{1}{x}\right)=0[/tex]
[tex]\lim _{x\to \infty \:}\left(\frac{1}{x^2}\right)=0[/tex]
So, Equation [B] becomes
⇒ [tex]\sqrt{9+0+0}[/tex]
⇒ [tex]3[/tex]
Similarly, we can find
[tex]\lim _{x\to \infty \:}\left(\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)=2[/tex]
So, Equation [A] becomes
⇒ [tex]3-2[/tex]
⇒ 1
Also
[tex]\lim _{x\to \infty \:}\left(1+\frac{1}{x}\right)=1[/tex]
Thus, equation becomes
[tex]\frac{\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)}{\lim _{x\to \infty \:}\left(1+\frac{1}{x}\right)}=\frac{1}{1}=1[/tex]
Therefore,
[tex]\lim _{x\to \infty \:}\left(\frac{\sqrt{9x^2+x+1}-\sqrt{4x^2+2x+1}}{x+1}\right)=1[/tex]
Keywords: limit
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