Please answer all question
(1)
(a) Use the fact that [tex]\sqrt{x^2} = |x|[/tex] for all [tex]x[/tex]. Since [tex]x\to+\infty[/tex], we have [tex]x>0[/tex] and [tex]|x| = x[/tex].
[tex]\displaystyle \lim_{x\to\infty} \frac{\sqrt{9x^2 - 2}}{x + 4} = \lim_{x\to\infty} \frac{\sqrt{x^2} \sqrt{9 - \frac2{x^2}}}{x + 4} \\\\= \lim_{x\to\infty} \frac{ x \sqrt{9 - \frac2{x^2}}}{x+4} \\\\= \lim_{x\to\infty} \frac{\sqrt{9 - \frac2{x^2}}}{1 + \frac4x} \\\\= \frac{\sqrt9}1 = \boxed{3}[/tex]
(b) This time [tex]x\to-\infty[/tex], so [tex]x < 0[/tex] and [tex]|x| = -x[/tex].
[tex]\displaystyle \lim_{x\to\infty} \frac{\sqrt{9x^2 - 2}}{x + 4} = \lim_{x\to\infty} \frac{\sqrt{x^2} \sqrt{9 - \frac2{x^2}}}{x + 4} \\\\= \lim_{x\to\infty} \frac{ \boxed{-x} \sqrt{9 - \frac2{x^2}}}{x+4} \\\\= (-1) \times \lim_{x\to\infty} \frac{\sqrt{9 - \frac2{x^2}}}{1 + \frac4x} \\\\= -\frac{\sqrt9}1 = \boxed{-3}[/tex]
(c) We immediately have
[tex]\displaystyle \lim_{x\to\infty} (x - \sqrt x) = \boxed{\infty}[/tex]
since [tex]x > \sqrt x[/tex] for all [tex]x > 1[/tex].
(d) Introduce a difference of squares by factoring in the limand's conjugate. The rest mirrors what we did in (a)/(b).
[tex]\displaystyle \lim_{x\to\infty} \left(\sqrt{x^2 + 12x} - x\right) = \lim_{x\to\infty} \frac{\left(\sqrt{x^2+12x} - x\right) \left(\sqrt{x^2+12x} + x\right)}{\sqrt{x^2 + 12x} + x} \\\\ = \lim_{x\to\infty} \frac{\left(\sqrt{x^2+12x}\right)^2 - x^2}{\sqrt{x^2+12x}+x} \\\\ = \lim_{x\to\infty} \frac{12x}{\sqrt{x^2+12x}+x} \\\\ = \lim_{x\to\infty} \frac{12}{\sqrt{1 + \frac{12}x} + 1} = \frac{12}{\sqrt1 + 1} = \boxed{6}[/tex]
(e) Divide through by the highest-degree exponential term.
[tex]\displaystyle \lim_{x\to\infty} \frac{12e^{2x} - 3e^{3x}}{2e^{2x} + 4e^{3x}} = \lim_{x\to\infty} \frac{12e^{-x} - 3}{2e^{-x} + 4} = \frac{0 - 3}{0 + 4} = \boxed{-\frac34}[/tex]
(2) By definition of the derivative, we have
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{f(x+h) - f(x)}h[/tex]
For [tex]f(x) = \sqrt{x^2+1}[/tex], the limit becomes
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\sqrt{(x+h)^2+1} - \sqrt{x^2+1}}h[/tex]
Factor in the conjugate.
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\left(\sqrt{(x+h)^2+1} - \sqrt{x^2+1}\right) \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\left(\sqrt{(x+h)^2+1}\right)^2 - \left(\sqrt{x^2+1}\right)^2}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\bigg((x+h)^2+1\bigg) - (x^2+1)}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{2xh + h^2}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]
[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{2x + h}{\sqrt{(x+h)^2+1} + \sqrt{x^2+1}}[/tex]
[tex]\implies \boxed{f'(x) = \displaystyle \lim_{h\to0} \frac{x}{\sqrt{x^2+1}}}[/tex]
(3) The tangent line to
[tex]y = \frac1{x^2+1}[/tex]
at the point (2, 1/5) has slope equal to the derivative [tex]\frac{dy}{dx}[/tex] when [tex]x = 2[/tex]. Compute the derivative; since [tex]y = \frac1{f(x)^2}[/tex] where [tex]f(x)[/tex] is the function from the previous problem, using the chain rule gives
[tex]y = \dfrac1{f(x)^2} \implies \dfrac{dy}{dx} = -\dfrac{2f'(x)}{f(x)^3} = -\dfrac{2 \times \frac{x}{\sqrt{x^2+1}}}{\left(\sqrt{x^2+1}\right)^3} \\\\ \implies \dfrac{dy}{dx} = -\dfrac{2x}{(x^2+1)^2}[/tex]
The tangent line at (2, 1/5) then has slope
[tex]\dfrac{dy}{dx}\bigg|_{x=2} = -\dfrac{2\times2}{(2^2+1)^2} = -\dfrac4{25}[/tex]
Using the point-slope formula, the equation of the tangent line is
[tex]y - \dfrac15 = -\dfrac4{25} (x - 2) \implies \boxed{y = -\dfrac{4x - 13}{25}}[/tex]
How would you describe the plants growth? Please help!!!!
Answer:
increasing
Step-by-step explanation:
it's increasing because throughout the days you can see that the plants growth is slowly progressing in this case going up on the grpah.
Need the answer for a
Answer: 6 10
Step-by-step explanation:
Jaden runs in a circle at constant speed
Radius is 30m. Time is 90s. What is the speed?
Answer:
hope i helped...if anythng not understood please write in the comments
If the temperature is 30, what is the temperature in Fahrenheit?
Answer:
The answer is 86° Fahrenheit.
Step-by-step explanation:
The equation for converting between Fahrenheit and Celsius is °F = 1.8 x °C + 32. So you would just plug 30 in for the °C.
°F = 1.8 x 30° + 32.
°F = 54 + 32
°F = 86°
Answer:
30ºC = 86ºF
Step-by-step explanation:
The temperature T in degrees Fahrenheit (ºF) is equal to 30 degrees Celsius (ºC) times 9/5 plus 32:
T(ºF) = 30ºC × 9/5 + 32 = 86ºF
What are the steps to solving this following equation? Put answer and steps.
(5) * (-9) * (-2)
Answer:
90
Step-by-step explanation:
you can multiply anything here in any order.
you could first do 5 times -9
5 * -9 = -45
and then multiply the -2 to the -45
-2 * -45 = 90
What is the difference between Independent and dependent events?
Choose the correct answer below.
O A Two events are independent when the occurrence of one event does not affect the probability of the occurrence of the other event. Two events are dependent when the occurrence of one event affects the probability
of the occurrence of the other event.
OB. Two events are independent if only one of the two events can occur. Two events are dependent if they can occur at the same time.
OC. Two events are independent when the occurrence of one event affects the probability of the occurrence of the other event. Two events are dependent when the occurrence of one event does not affect the probability
of the occurrence of the other event
OD. Two events are independent if they can occur at the same time. Two events are dependent if only one of the two events can occur.
Answer: A
Step-by-step explanation:
It is the definition of independent and dependent events
what is -3 increased by -9
Answer:
-12
Step-by-step explanation:
-3 + -9 = 12
hope this helps
Answer:
-12
Step-by-step explanation:
-3 + -9
turns into
-3 - 9
Which means on the number line, we start at -3 and move to the left 9 units.
When we move to left on the number line by 9, we are at -12.
Which means -3 - 9 (-3 + -9) = -12
2. Change the following into indirect speech.
a. Anupam says, "Where do you live?”
Answer:
Anupam ask me where l live
hope it's help you (^0^)
test.mapnwea.org
Use the information to answer the question.
A spinner had 5 equal sections, labeled A through E. After 100 spins, the spinner had landed on section A a total of 45 times.
Which two reasons would best explain why the expected probability and observed probability did not match?
A. The surface of the spinner was not even.
B. The speed of each spin was not consistent.
c. The size of each spinner section was equal.
D. The number of spins did not match the number of sections.
O A. The surface of the spinner was not even
UB. The speed of each spin was not consistent.
c. The size of each spinner section was equal.
UD. The number of spins did not match the number of sections.
Sorry hindi ko alam yan i
A go-karts top speed is 607,200 feet per hour. What is the speed in miles per hour?
I need help pls I really need a answer
Answer:-16
Step-by-step explanation:
add all the temperature together =-112
then you divide by the number of temperatures given which is 7 which gives you -16
Answer:
-16 degrees C.
Step-by-step explanation:
When they say mean, they mean the average. You add the numbers together and divide them on how many numbers there are.
The probability of drawing three spades in a from from a standard deck of cards when the drawn card is returned to the deck each time.
Answer:
1/64
Step-by-step explanation:
P(pick 3 consecutive spades with replacement of drawn cards) =
(13/52)(13/52)(13/52) =
(1/4)(1/4)(1/4) =
1/64
Please help me with this i will give you brainlesist
Answer: 4
pls give brainliest
PLEASE EXPLAIN AND PLEASE HELP ME AND EXPLAIN I NEED HELP
its right because
6 x 5 is the area of the leftmost rectangle
4 x 5 is the area of the bottom rectangle
3 x 5 is the area of the rightmost rectangle
5 x 5 is the area of the slanted square on the top
the last term is the area of the trapezoid, which is (base1 + base2) x height /2
they split the trapezoid thing into a square and a triangle, which gets you the same answer
they multiplied it by 2 because there are 2 trapezoids
5. The bus fair oe 8 passangers is Rs 640 If the fair decreased by Rs 5 per passanger How many passangers can travel for Rs 900 HL Hill ini mill Your Answer ! lii IR 3 HIE Answer
Answer:
12 passengers can travel in Rs. 900.
Step-by-step explanation:
see the above attachment.
if you face any problem comment me, I am there for help.
hope this helps you.
Which number can each term of the equation be multiplied by to eliminate the fractions before solving? -3/4m-1/2=2+1/4m
Answer:
* 2/3 / formula then multiply the numbers then you get the answer
What is the missing term of the geometric sequence below?
7, ?, 63,...
A: 28
B: 3
С: 35
D: 21
9514 1404 393
Answer:
D: 21
Step-by-step explanation:
The first term is (a1). The third term is (a1)r². The product of these terms is ...
(a1)(a1)r² = ((a1)r)², the square of the second term.
The missing term is √(7·63) = √441 = 21.
_____
Additional comment
In consideration of the above, you can also see that the common ratio can be found from ...
a3/a1 = ((a1)r²)/(a1) = r² = 63/7 = 9
Then the common ratio is r = √9 = 3, and the second term is 7·3 = 21.
Dividing whole numbers
Answer:
137.8
when round of:138
Answer:
137.8
Step-by-step explanation:
Can't really break it down further than this.
What is 1,432 rounded to the nearest 100?
Answer:
1400
Step-by-step explanation:
1432 is closer to 1400 than 1500 because it ends in 32 and is below 50 on the number line
How much money per candy bar is it if theres
4 for 5.50
Alex bought two hots dogs, fries, and a water at the food truck. The hot dogs were
$2.35 each and the fries were $1.25. When he paid with a $10 bill he received $1.45 in
change. How much was the water?
Poems written for entertainment in ancient Sumer were called _______.
a.
Scribes
c.
Epics
b.
Cuneiform
d.
Haiku
Please select the best answer from the choices provided
A
B
C
D
Help pls! T^T I will make you brainylist
Write an equation for a line that passes through the points (1, 2) and (11, 22). The form is your choice.
y=mx+b
-3=3(3)+b
-3=9+b
-12=b
Please help
100 point question
I will mark brainliest
Please answer the questions in the picture
Thanks :)
Answer:
5. yes It still has the same density because it is the same thing.
6. a. float less than one
b. Float less than one
C. Sink greater than one
7. 1652.4 This is because the density equation is mass/volume. We have the density and mass but no volume. do get it we need to re arrange the equation. mass x density = volume.
Step-by-step explanation:
To find simple interest, you multiply the principal (in dollars), the interest rate (as a decimal), and the time in years.
The equation 22.80 = 300 •019•4 shows how to find the simple interest for a certain account after 4 years. What
is the interest rate (as a percent)? How much is the simple interest? What is the principal?
What is the interest rate (as a percent)?
A. 1.9%
B. 22.80%
C. 0.019%
D. 300%
Here we have $400 originally and it is invested at an interest rate of 0.015 for 4 years.
Therefore we can say that the interest rate is 0.015 * 4 * 100 = 6% for the 4 years.
Simple interest is 6% / 4 = 1.5%.
And the principal is $400.
What is the answer to this question number 2
Answer:
B
Step-by-step explanation:
number of adults=a
number of children=1
cost of adult tickets=$9 per adult
cost of children's tickets=$6 per child
Total cost=$78
9(a)+6(1)=78
9a+6=78 B
do have a great day ahead...
Your boss sends you to the hardware store to buy tie down rope. He tells you to buy the best deal. What is the savings per unit between the best deal and worst deal?
Answer:
Explain more
Step-by-step explanation:
You need to explain more like add an image or tell us how much much money is a rope.
What does 11x11-6x6+18=
Answer:
103
Step-by-step explanation:
Hope this Helps :)
Susie is visiting Greece from the U.S. One day she stopped by a market and found a stall selling tomatoes for 1.92 euro per kilogram. If 1 euro was worth 1.09 dollars that day, how much did the tomatoes cost in dollars per pound?
can you help me this 4^x x ^8^x^+^1 = 2⁴