If we evaluate at infinity, we have:
[tex]\bf{\displaystyle L = \lim_{x \to \infty}{\frac{\log(x^8 - 5)}{x^2}} = \frac{\infty}{\infty} }[/tex]
However, the infinity of the denominator has a higher order. Therefore, we can conclude that [tex]\boldsymbol{L = 0.}[/tex]
However, proving that the limit is 0 without using L'Hopital or the "order" criterion is complicated. To do so, let us denote:
[tex]\boldsymbol{\displaystyle f(x) = \frac{\log(x^8 - 5)}{x^2} }[/tex]
To find the limit, we must look for two functions h(x) and g(x) such that h(x)≤ f(x)≤ g(x) and
[tex]\boldsymbol{\displaystyle \lim_{x \to \infty}{h(x)} = 0, \qquad \lim_{x \to \infty}{g(x)} = 0}[/tex]
If we find these functions, then we can conclude that [tex]\bf{\lim_{x \to \infty}{f(x)} = 0.}[/tex]
First, let's note that when x⁸ - 5 > 1, then log(x⁸ - 5) > 0 (and this is true when x is large). Likewise, we have that x² > 0 for x > 0. Therefore, we have:
[tex]\boldsymbol{\displaystyle f(x) = \frac{\log(x^8 - 5)}{x^2} \geq 0}[/tex]
when x "is big enough". Thus, we have h(x) = 0 where it is clear that [tex]\bf{\lim_{x \to \infty}{h(x)} = 0.}[/tex]
To find the second function, let's first note that \log is an increasing function, so since x⁸ ≥ x⁸ - 5, then log(x⁸) ≥ log(x⁸ - 5). So we have to
[tex]\boldsymbol{\displaystyle \frac{\log(x^8 - 5)}{x^2} \leq \frac{\log(x^8)}{x^2} }[/tex]
now, if we take y = e^y, then we can write
[tex]\boldsymbol{\displaystyle \frac{\log(x^8)}{x^2} = \frac{\log(e^{8y})}{e^{2y}} = \frac{8y}{e^{2y}}}[/tex]
A very important property about the exponential function is
[tex]\boldsymbol{\displaystyle e^x > \frac{x^n}{n!}}[/tex]
For any n [tex]\bf{n \in \mathbb{N}}[/tex] and x > 0. If we take n = 2, then we have
[tex]\boldsymbol{\displaystyle e^{2y} > \frac{(2y)^2}{2!} = \frac{4y^2}{2} = 2y^2}[/tex]
From this it follows that
[tex]\boldsymbol{\displaystyle \frac{1}{e^{2y}} < \frac{1}{2y^2} }[/tex]
Therefore, we have to
[tex]\boldsymbol{\displaystyle \frac{\log(x^8 - 5)}{x^2} \leq \frac{\log(x^8)}{x^2} < \frac{8y}{2y^2} = \frac{4}{y} = \frac{4}{\log x} }[/tex]
yes, [tex]\bf{g(x) = 4/\log x}[/tex] where [tex]\bf{\lim_{x \to \infty}{g(x)} = 0}[/tex]. Also, [tex]\bf{h(x) \leq f(x) < g(x)}[/tex]. Therefore, [tex]\bf{\lim_{x \to \infty}{f(x)} = 0}[/tex].
[tex]\red{\boxed{\green{\boxed{\boldsymbol{\sf{\purple{Pisces04}}}}}}}[/tex]
Give the name (monomial,
binomial, trinomial etc.) and
degree of the polynomial.
15x²
Name? [?]
Degree? [ ]
Answer:
monomial and degree is 2
Rafael purchased a video game for 35.60.if he gave the cashier 40 how much change will he get
Answer:
4.4
Step by step explanation: If Rafael gave the cashier 40, cashier will return 4.4 by subtracting 35.60 from 40
Use the zero product property to find the solutions to the equation (x 2) (x 3) = 12.
The solution of the given equation is -6 and 1.
What is Quadratic Equation?A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its standard form is ax² + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term.
Here, given equation:
(x+2)(x+3) = 12
x(x+3)+2(x+3) = 12
x² + 3x + 2x + 6 = 12
x² + 5x + 6 - 12 = 0
x² + 5x - 6 = 0
x² + 6x - x - 6 = 0
x(x+6) -1(x+6) = 0
(x+6)(x-1) = 0
Now, x + 6 = 0 or x - 1 = 0
x = -6 or x = 1
Thus, the solution of the given equation is -6 and 1.
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48 girls, 41 boys, and 5 teachers sat in the gym. Write the ratio of girls to boys in three ways.
I took the text last month
Find the area of a circle with radius,
r
= 7.15m.
Give your answer rounded to 2 DP.
[tex] \boxed{\Large {\mathcal {{{\color{orange}{Area \: of \: circle \: = \: πr²}}}}}}[/tex]
r = 7.15m[tex]\large \rm \pink{ \: Area \: of \: circle }\:\large\blue\implies \tt \large πr²\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \large \rm \pink{ \: Area \: of \: circle }\:\large\blue\implies \tt \large \frac{22}{7} \: \times\:{7.15}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\\large \rm \pink{ \: Area \: of \: circle }\:\large\blue\implies \tt \large \frac{22}{7} \: \times\:51.1225 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \large \rm \pink{ \: Area \: of \: circle }\:\large\blue\implies \tt \large \frac{22}{ \cancel7} \: \times\: \cancel{51.1225} \: \: ^{ \red{7.303} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \large \rm \pink{ \:Area \: of \: circle }\:\large\blue\implies \tt \large {22} \: \times\:7.303 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \large \rm \pink{ \: Area \: of \: circle }\:\large\blue\implies \tt \large \: 160.666 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [/tex]
Hence , the area of circle is 160.66 m²
Given :-
Radius of circle = 7.15mTo Find :-
Area of circleSolution :-
Area of Circle = πr²
putting the value of π and rArea = 3.14×(7.15)² m²
= 160.52465 , which rounded off to 2DP would be 160.52 m²
PLEASE HELP ASAP
(Picture shown)
Answer:
-2x+3
Step-by-step explanation:
Gradient is change in y over change in x
[tex] \frac{2}{1} [/tex]
=2
and then
+3 because that's where it cuts the Y axis
Brianna bought a total of 8 notebooks and got 16 free pens
Answer: 1:2
Step-by-step explanation:
Well, if you're asking for the ratio, it is 1:2
For every 1 notebook bought, there would be 2 free pens.
Since she bought 8 notebooks, just multiply 2 by 8 which will result in 16.
Sahil chooses a number, divides it by 8 , adds 8 to the answer. Then multiples the answer with 8 . He obtains the result as 952 . The number he chooses in the beginning was
USER IS DEAD
TAKE POINTS
Answer:
888
Step-by-step explanation:
Sahil chooses a number, [We'll call it x]
divides it by 8 , [x/8]
adds 8 to the answer. [(x/8) + 8]
Then multiples the answer with 8 . [((x/8) + 8)*8]
He obtains the result as 952 . [((x/8) + 8)*8 = 952]
The number he chooses in the beginning was
((x/8) + 8)*8 = 952
x + 64 = 952
x = 888
CHECK:
Does (888/8 + 8)*8 = 952?
(119)*8 = 952? YES
The number is 888
Help... will give you 20 points if you answer
Answer:
(-3,5) (-3,3) (-8,3)
Step-by-step explanation:
Answer:
(-3,5) , (-3,3) , (-8,3)
Step-by-step explanation:
bye, bye!!
write the explicit rule for the nth term of the arithmetic sequence. then find the 43rd term (3,5,7….)
Write an equation for a parabola in which the set of all points in the plane are equidistant from the focus and line F(-6,0); x=6
Step-by-step explanation:
Since we have a vertical directrix, the equation of the parabola is
[tex](y - k) {}^{2} = 4p(x - h)[/tex]
Where p is the distance from the vertex to the directrix.
or the distance from the vertex to the focus.
Since we have a sideways parabola, let use the point for the directrix is (-6,0). So let find the midpoint of (-6,0) and (6,0). That would be our vertex.
[tex] \frac{ - 6 + 6)}{2} = 0[/tex]
[tex] \frac{0 + 0}{2} = 0[/tex]
So our vertex is (0,0).
So our equation become
[tex] {y}^{2} = 4px[/tex]
The distance from the focus and directrix is 6.
So p=6.
[tex] {y}^{2} = 24x[/tex]
So p is 6.
Since p is 6,
[tex] \frac{ {y}^{2} }{24} = x[/tex]
Select the correct answer.
What is the output of the function f(p) = 3p – 2 when the input is 2?
A.
0
B.
2
C.
4
D.
6
E.
8
f(p) = 3p - 2
f(2) = 3(2) - 2
f(2) = 6 - 2
f(2) = 4
Therefore, the answer is 4, so option C :)
Kevin and Randy Muise have a jar containing 65 coins, all of which are either quarters or nickels. The total value of the coins in the jar is $8.65 How many of each type of coin do they have?
The number of quarters or nickels Kevin and Randy Muise have in the jar is 27 and 38 respectively.
Simultaneous equationA nickel = 5 centsA quarter = 25 centsn + q = 65
0.05n + 0.25q = 8.65
From equation (1)
n = 65 - q
substitute n = 65 - q into (2)
0.05n + 0.25q = 8.65
0.05(65 - q) + 0.25q = 8.65
3.25 - 0.05q + 0.25q = 8.65
- 0.05q + 0.25q = 8.65 - 3.25
0.20q = 5.40
q = 5.40/0.20
q = 27
Substitute into (1)
n + q = 65
n + 27 = 65
n = 65 - 27
n = 38
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Suppose the mean height for men is 70 inches with a standard deviation of 2 inches. What percentage of men are more than 72 inches tall?
The percentage of men is more than 72 inches tall will be 0.15866.
What is the z-score?The z-score is a statistical evaluation of a value's correlation to the mean of a collection of values, expressed in terms of standard deviation.
The z-score is given as
z = (x – μ) / σ
Where μ is the mean, σ is the standard deviation, and x is the sample.
Suppose the mean height for men is 70 inches, with a standard deviation of 2 inches.
Then the percentage of men are more than 72 inches tall will be
z = (72 – 70) / 2
z = 1
The percentage of men is more than 72 inches tall will be
P(x > 72) = P(z > 1)
P(x > 72) = 1 – P(x < 72)
P(x > 72) = 1 – 0.84134
P(x > 72) = 0.15866
Thun, the percentage of men are more than 72 inches tall will be 0.15866.
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The side of an equilateral triangle is given as 8cm, correct to the nearest centimeter. What is the possible least lenght of its perimeter?
Answer:
An equilateral triangle is a triangle with all 3 sides the same in size.
If one side is 8cm, then 8x3=24 cm
The perimeter is possibly 24 cm.
Max's Market sells 6 eggs for £2.04.
Groceries Galore sells 8 eggs for £2.64.
Fancy Foodstuffs sells 12 eggs for £4.08.
Which shop's price per egg is cheaper?
Answer:
Groceries Galore
Step-by-step explanation:
1) Find the value of 1 egg for each shop.
Max's Market:
2.04 ÷ 6
= £0.34
Groceries Galore:
2.64 ÷ 8
= £0.33
Fancy Foodstuffs:
4.08 ÷ 12
= £0.34
Therefore, Groceries Galore's price per egg is cheaper.
Answer:
Groceries Galore
Step-by-step explanation:
we have to calculate the price of 2 eggs
in each shop apart ,and then we can infer
which shop offers the cheaper price.
…………………………………………………………
• At Max's market :
2 egg costs 2,04÷3 = £0,68
• At Groceries Galore :
2 egg costs 2,64÷4 = £0,66
• At Fancy Foodstuffs :
4,08÷6 = £0,68
………………………………
Conclusion :
Groceries Galore Offers the Lower price.
Note : we can just determine
the price of one egg in each store ,and then compare.
Parallelogram ABCD has vertices: A(-3, 1), B(3, 3), C(4, 0), and D(-2, -2). In two or more complete sentences, explain how you can use the coordinates of the vertices to prove that parallelogram ABCD is a rectangle.
please help
Step-by-step explanation:
via the distance formula (based on Pythagoras) I can prove that ABC is a right-angled triangle, and that ADC is fully congruent to ABC.
so, together these 2 triangles (joined at their Hypotenuses, the diagonal AC) build a rectangle.
3. a 12 inch apple pie is sliced into 9 slices. what is the area of one slice of pie?
Answer:
4 pi ( = 12.566 in^2 )
Step-by-step explanation:
TOTAL area = pi r^2 = pi (6)^2 = 36 pi in^2
divide into 9 pieces 36 pi / 9 = 4 pi in ^2
A 1/25 scale model sports car is 5.5 in long. What was the length of the actual car to the nearest in?
Answer: 137.5 in
Step-by-step explanation: All we have to do is multiply 25 to 5.5 to get the original result. Since the scaling is 1/25, it means it is 25 times smaller. Thus, just multiply 25 to 5.5 and you get 137.5 in, the original result.
On a coordinate plane, a right triangle has points A (negative 5, negative 6), B (negative 5, negative 2), C (negative 2, negative 2).
Examine the right triangle ABC. Which rise and run would create a similar right triangle on the same line?
a rise of 6 and a run of 8
a rise of 8 and a run of 6
a rise of 6 and a run of 5
a rise of 5 and a run of 6
A rise of 8 and a run of 6. Then the correct option is B.
What is a coordinate geometry?Coordinate geometry is the study of geometry using the points in space.
On a coordinate plane, a right triangle has points A (-5, -6), B (-5, -2), and C (-2, -2).
The distance between AB will be
AB² = (-5 + 5)² + (-2 + 4)²
AB² = 4²
AB = 4
The distance between BC will be
BC² = (-2 + 5)² + (-2 + 2)²
BC² = 3²
BC = 3
Then the rise and run will be
rise / run = 4 / 3
Then this can be written as
rise / run = 8 / 6
Then the correct option is B.
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I dive to 17 metres for 23 minutes. after a 30 minute surface interval, i plan to dive to 16 metres. what is the maximum allowable time for the second dive
Answer:
i don't know
Step-by-step explanation:
i don' know
Carmela opened her piggy bank and found she had $15.30. If she had only nickels, dimes, quarters, and half-dollars and an equal number of coins of each kind, how many coins did she have in all
Answer:
68 coins
Step-by-step explanation:
We can use the given information to write two equations involving the numbers of coins. One will express the total number of coins; the other will express their value.
__
setupSince there are equal numbers of coins of each type, we can let n represent the number of each kind of coin. The total number coins will be ...
n +n +n +n + N . . . the sum of numbers of nickels, dimes, quarters, halves
The value of the coins is ...
0.05n +0.10n +0.25n +0.50n = 15.30
solutionThe second equation lets us find the value of n:
0.90n = 15.30
n = 17 . . . . . . . . divide by 0.90
Then the number of coins in all is ...
4n = N
4(17) = N = 68
Carmela had 68 coins in all.
Use the drawing tool(s) to form the correct answer on the number line.
Graph the solution to this inequality on the number line.
-4x+7>x-13
Answer:
Let's solve your inequality step-by-step.
−4x+7>x−13
Step 1: Subtract x from both sides.
−4x+7−x>x−13−x
−5x+7>−13
Step 2: Subtract 7 from both sides.
−5x+7−7>−13−7
−5x>−20
Step 3: Divide both sides by -5.
−5x/−5>−20/−5
x<4
Answer:
x<4
IMAGINE AN ARROW ABOVE THE NUMBERS
⇔
−7−6−5−4−3−2−1 0 1 2 3 4 5 6 7
4 is where the line is circled and the arrow is facing left
←Ф
like that
Answer:
x<4
Step-by-step explanation:
So you will put an open circle on positive 4 and a ray going to the left of the graph.
On a coordinate plane, the x-axis is labeled minutes and the y-axis is labeled Blocks. Line A goes through points (10, 4), (15, 12), and (20, 20). Line B goes through points (5, 8), (10, 16), (15, 24). Line C goes through points (5, 24), (15, 16), (20, 12).
The table shows Jose’s rate of bicycle riding.
A 2-column table with 3 rows. Column 1 is labeled Minutes with entries 5, 10, 15. Column 2 is labeled Blocks with entries 8, 16, 24.
Which line on the graph shows the proportional relationship in the table?
Line
on the graph shows the proportional relationship.
The proportional relationship given in the table is y = 1.6x, and line B shows this relationship.
What is a proportional relationship?A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:
y = kx
In which k is the constant of proportionality.
The constant of proportionality, considering the points on the table, is given as follows:
[tex]k = \frac{24}{15} = \frac{16}{10} = \frac{8}{5} = 1.6[/tex].
Hence the equation is:
y = 1.6x.
Line B gives the same points as the table, hence it shows the proportional relationship.
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I ne subject of the formula below iS y.
y = 4x + 3
Rearrange the formula to make x the subject.
Answer:
x=(y-3)/4
Step-by-step explanation:
y=4x+3
y-3=4x
x=(y-3)/4
When a patient with hypertension takes a particular type of blood pressure medication, the effects on the systolic pressure, S(t), can be measured by the following piecewise defined function: S of t is equal to the piecewise function of 140 minus 6 times t if 0 is less than or equal to t is less than or equal to 4 and 116 if 4 is less than t is less than 8 and 96 plus 8 times t if 8 is less than or equal to t is less than or equal to 12 where t is the time, in hours, since taking the medication. Based on the graph of the piecewise function, if the patient takes the blood pressure medication at 8 a.m., in which interval will their systolic pressure be lowest?
Based on the graph, the interval will their systolic pressure be lowest is 4pm to 5pm.
What is the interval in blood pressure ?The lowest systolic pressure can be seen from 5 and 8 hours after taken the medication.
If we look at the time details then this will fall inbetween the interval of 2pm to 5pm.
According to question, the time range is 4pm to 5pm.
Hence, the interval will their systolic pressure be lowest is 4pm to 5pm.
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Answer:
The answer is B - 12 P.M to 2 P.M
Step-by-step explanation:
Solve for x. Round to the nearest tenth, if necessary
The value of x in the triangle to the nearest tenth is calculated as; 1.1
How to use trigonometric ratios?
We have different trigonometric ratios such as;
sin θ = opp/hyp
cos θ = adj/hyp
tan θ = opp/adja
Thus, from the given triangle we can use the formula for cos θ to get the value of x. Thus;
cos 41° = 0.8/x
x = 0.8/cos 41
x = 1.06
To the nearest tenth, we have x = 1.1
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16t^2 + 48t + 160 =0
Answer:
Simplifying -16t2 + 48t + 160 = 0
Reorder the terms: 160 + 48t + -16t2 = 0
Solving 160 + 48t + -16t2 = 0 Solving for variable 't'.
Factor out the Greatest Common Factor (GCF), '16'. 16(10 + 3t + -1t2) = 0
Factor a trinomial. 16((5 + -1t)(2 + t)) = 0 Ignore the factor 16.
hope it helps.
Answer all the questions. 1 The first three terms in a sequence of numbers, T1, T2, T3,... are given below: (a) T₁=1+3=4 T₂=4+5=9 T3=9+7=16 Find T4. Answer T4=
Answer:
16 + 9 = 25
Step-by-step explanation:
For each term in the sequence, we can see that the second number being added increases by 2 each time (3, 5, 7)
And that the first number of each term increases by the value of the second term
(1 increases by 3; 4 increases by 5)
we can follow out these two observations to find T4
the first term (9) will increase by the value of the second term (7)
{the sum of the previous term is the first number}
so the first number of T4 will be 16.
Because we left off on +7, we are now on +9
so, T4 is
16 + 9 {and then do the math accordingly}
16 + 9 = 25
So, T4 is
16 + 9 = 25
hope this helps!! have a lovely day!! :)
On Friday, a local hamburger shop sold a combined total 312 of hamburgers and cheeseburgers. The number of cheeseburgers sold was two times the number of hamburgers sold. How many hamburgers were sold on Friday?
