Use l'Hospital's Rule to evaluate: (a) [8] limx→0 ex−1−x/x2 (b) [8] limx→[infinity] 3x2/ex.
Answers
L'Hopital's rule can be used to evaluate the limits of 0/0 and infinity/infinity. It can be used to evaluate the limits of 0/0 and infinity/infinity. It can be used to evaluate the limits of 0/0 and infinity/infinity.
(a) Let's evaluate the following limit using L'Hopital's rule:\($$\lim_{x \to 0} \frac{e^{x}-1-x}{x^{2}}$$\)
We have an indeterminate form of 0/0, so we can use L'Hopital's rule:
\($$\lim_{x \to 0} \frac{e^{x}-1-x}{x^{2}}\)
\(=\lim_{x \to 0} \frac{e^{x}-1}{2x}$$$$\)
\(=\lim_{x \to 0} \frac{e^{x}}{2}\)
\(=\frac{1}{2}$$\)
Therefore\($$\lim_{x \to 0} \frac{e^{x}-1-x}{x^{2}}\)
\(=\frac{1}{2}$$\)
(b) Now let's evaluate the following limit using L'Hopital's rule:
\($$\lim_{x \to \infty} \frac{3x^{2}}{e^{x}}$$\)
We have an indeterminate form of infinity/infinity, so we can use L'Hopital's rule:
\($$\lim_{x \to \infty} \frac{3x^{2}}{e^{x}}\)
\(=\lim_{x \to \infty} \frac{6x}{e^{x}}$$$$\)
\(=\lim_{x \to \infty} \frac{6}{e^{x}}=0$$\)
Therefore,\($$\lim_{x \to \infty} \frac{3x^{2}}{e^{x}}=0$$\)
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Related Questions
Interest Rates: Different Types and What They Mean to Borrowers
Answers
There are different types of interest rates that borrowers should be aware of: 1. Fixed Interest Rates, 2. Variable/Adjustable Interest Rates, 3. Prime Interest Rates, 4. Annual Percentage Rate (APR)
Interest rates refer to the percentage charged by lenders on borrowed funds, which borrowers must pay in addition to the principal amount. They represent the cost of borrowing money and play a significant role in determining the affordability and overall cost of loans. Higher interest rates imply higher borrowing costs for borrowers, while lower interest rates can make borrowing more affordable.
Interest rates can vary depending on several factors, including the type of loan, the borrower's creditworthiness, prevailing market conditions, and central bank policies. There are different types of interest rates that borrowers should be aware of:
1. Fixed Interest Rates: These rates remain constant throughout the loan term, providing borrowers with predictable monthly payments. Fixed rates are commonly used for mortgages and long-term loans, offering stability and protection against potential rate increases.
2. Variable/Adjustable Interest Rates: Also known as adjustable rates, these rates can change over time based on an underlying benchmark rate, such as the prime rate or the London Interbank Offered Rate (LIBOR). Variable rates are often lower initially but can fluctuate, leading to changes in monthly payments.
3. Prime Interest Rates: The prime rate is the interest rate offered to a bank's most creditworthy customers. It serves as a benchmark for many other interest rates, such as variable rate loans and credit cards. Borrowers with strong credit histories may qualify for loans with rates below the prime rate.
4. Annual Percentage Rate (APR): The APR represents the true cost of borrowing by factoring in both the interest rate and associated fees. It provides a comprehensive measure of the total cost of a loan and helps borrowers compare different loan offers.
Understanding the different types of interest rates and their implications is crucial for borrowers when considering loans. It is essential to evaluate the overall cost of borrowing, including interest rates, fees, and repayment terms, to make informed financial decisions. Borrowers should shop around, compare offers from different lenders, and consider their financial situation and long-term affordability before committing to a loan.
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Predict the number of times you roll an odd number or a two when you roll a six-sided number cube 300 times.
Answers
Answer:
The probability of rolling an odd number or a two on a six-sided die is 1/2 + 1/6 = 2/3. This means that if you roll a six-sided die 300 times, you can expect to roll an odd number or a two approximately 200 times
Step-by-step explanation:
Answer: 400 TIMES
Step-by-step explanation:
1/6 +1/2
4/6
2/3
how many yards are in the perimeter of a square that measures 99 inchs on each side?
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AB is a straight line.
Work out the size of angle x.
Not drawn accurately
(132
Х
A
B
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x:48
A straight line will be 180 degrees so 180－132＝x
132＋48=180
The value of the angle x will be 48° hence correct option will be 48°.
What is an angle?An angle is a geometry in plane geometry that is created by 2 rays or lines that have an identical terminus.
The identical endpoint of the two rays—known as the vertex—is referenced as an angle's sides.
In another word, the angle is the measurement of the angular distance for example for linear motion we have a meter inch but for angular rotation, we don't have the measurement so the angle is useful to measure the angular rotation.
Given that a line with an angle of 132° since in a single line has an angle of 180° so
Angle x + 132° = 180°
x = 180° - 132° = 48° hence it will be the correct answer.
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You work at a grocery store. Potatoes are sold for half a dollar each, including tax. How many cents should a customer expect in change from a $5 bill if 9 potatoes are purchased?
25
50
75
100
125
explain plz
Answers
Answer:
50
Step-by-step explanation:
so half a dollar is $0.50. We multiply by 9, which gives you $4.50. But the customer gave $5, so they would get $0.50 back.
Therefore if they paid with 5 dollars, they’d get 50 cents change
Q7) I CAN FIND THE PROBABILITY OF AN EVENT WHEN THE OUTCOMES ARE EQUALLY LIKELY A bag contains 5 grapes, 2 bananas, and 6 avocados. One fruit is chosen at random. What is the probability an avocado is chosen? I a) 1/5 b) 6/7 c) 6/13 d) 2/13
Answers
Answer:
c
Step-by-step explanation:
Avacado is 6 out of 5 + 2 + 6 fruits
6 out of 13 = 6/ 13
5×23 times to which properties is it
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Answer:
5 × (20+3) it ez math bro 5 th grade
percent of 1hr is 15miute
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Step-by-step explanation:
BRAINLIEST ME PLEASE PLEASE PLEASE PLEASE
what does a linear graph look like?
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Answer:
A linear equation looks like any other equation
Step-by-step explanation:
What is the range of the function shown below?
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Answer:
y ≤ 0
Step-by-step explanation:
The correct option is option C. The range of the given function is less than or equal to zero i.e., y ≤ 0.
What are domain and range of a function?
Range is the set of all values that the given function can output, whereas domain is the input or set of values for which the function is defined.
The graph of a function is shown. We know that the range of function is the output of the input of values we give to the function. Another thing to keep in mind is that we often plot functions with the outputs on the vertical axis and the inputs on the horizontal axis.
From graph it is clear that the function is going down from axis to negative y - axis.
So , it is clear that range will be less than or equal to zero.
i.e.,
Range is :
y ≤ 0
Therefore , the correct option is option C. The range of the given function is less than or equal to zero i.e., y ≤ 0.
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Analyze the table showing the number of days and cost of the trip.
What is the cost per day?
Eternal City Bike Tours
Length of Trip (days) Cost (euros)
3
75
4
100
5
125
6
150
?
euros per day
DONE
Answers
The equation showing the relationship between the number of days and cost of the trip is y = 25x.
The formula of a linear function is:
y = mx + b;
where y,x are variables, m is the slope of the line and b is the y intercept (value of y when x is 0)
Let x represent the length of trip and y represent the amount of money.
From the table, taking two pair of points: (3, 75) and (6, 150), the equation is:
\(y-y_1=\frac{y_2-y_1}{x_2-x_1} (x-x_1)\\\\y-75=\frac{150-75}{6-3}(x-3) \\\\y-75=25(x-3)\\\\y=25x\)
Hence the equation showing the relationship between the number of days and cost of the trip is y = 25x.
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1. Use logarithms to evaluate 0.00196/5.73 What is the logarithm of the answer?
a. 8.0188 – 10
b. 7.5342 – 10
c. 7.0188 – 10
d. 6.5341 – 10
Answers
The logarithm of the answer is approximately -3.46592. None of the given options match this value exactly.
To evaluate 0.00196/5.73 using logarithms, we can take the logarithm of both the numerator and the denominator and then subtract them.
Let's denote log10 as log base 10:
log10(0.00196/5.73) = log10(0.00196) - log10(5.73)
Using a calculator or logarithm tables, we can find the logarithms of 0.00196 and 5.73:
log10(0.00196) ≈ -2.70805
log10(5.73) ≈ 0.75787
Substituting these values into the equation, we have:
log10(0.00196/5.73) ≈ -2.70805 - 0.75787
log10(0.00196/5.73) ≈ -3.46592
The logarithm of the answer is approximately -3.46592.
None of the given options match this value exactly.
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proportional relationships, please help!
Answers
Answer:
y = 5/2 x
Step-by-step explanation:
y varies directly with x meaning this equation : y=kx
you plug in the information given
10 = k(4) and then solve ... k = 10/4 which simplifies to 5/2
If two lines meet at a 90 angle, they are perpendicular.
If two lines are perpendicular, they have opposite reciprocal slopes.
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It is true that If two lines meet at a 90 angle, they are perpendicular. If two lines are perpendicular, they have opposite reciprocal slopes.
What is perpendicular?In geometry, perpendicular lines are two lines that intersect at a 90-degree angle. A right angle is formed where the lines cross each other, so that one line is at a 90-degree angle to the other. This means that the lines are orthogonal to each other, and their slopes are negative reciprocals of each other. For example, consider the Cartesian plane where the x-axis and y-axis are perpendicular lines. The slope of the x-axis is 0, and the slope of the y-axis is undefined, so their negative reciprocals are also undefined and 0, respectively. Perpendicular lines are used extensively in mathematics, physics, and engineering to describe relationships between objects and forces. In construction, perpendicular lines are important for building structures that are stable and level. In computer graphics, perpendicular lines are used to create images and animations with precise angles and proportions.
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Complete question:
If two lines meet at a 90 angle, they are perpendicular. If two lines are perpendicular, they have opposite reciprocal slopes. True or False.
please help i have been struggling
Answers
Answer:
6000
Step-by-step explanation:
What is the measure of SQ?
1. 95
2. 190
3. 265
4. 360
Answers
Answer:
The measure of arc SQ is 95° ⇒ (1)
Step-by-step explanation:
The measure of any circle is 360°The measure of the subtended arc to an inscribed angle is twice the measure of this angleIn the given circle
∵ S lies on the circumference of the circle
∴ ∠QSR is an inscribed angle
∵ ∠QSR is subtended by arc QR
→ By using the 2nd rule above
∴ m arc QR = 2 × m∠QSR
∵ m∠QSR = 95°
∴ m arc QR = 2 × 95
∴ m arc QR = 190°
→ By using the 1st rule above
∵ m of the circle = m arc QR + m arc SQ + m arc SR
∵ m arc SR = 75° and m arc QR = 190°
→ Substitute them in the equation above
∴ 360 = 190 + m arc SQ + 75
→ Add the like term in the right side
∴ 360 = 265 + m arc QS
→ Subtract 265 from both sides
∵ 360 - 265 = 265 - 265 + m arc SQ
∴ 95° = m arc SQ
∴ The measure of arc SQ is 95°
pls pls pls pls pls help me
Answers
Answer:
2 and 8
Step-by-step explanation:
those are the only two numbers that will be divided by 6 and equal a whole number
Please help will give brainliest
Answers
Answer:
16p - 2
Step-by-step explanation:
Answer:
20p - 2
Step-by-step explanation:
To find a perimeter, you take the values of each side and add them together.
Let's start there.\((p-8)+(9p-7)+(p-8)+(9p-7)\)
Bring all values with a p to one side\(p+9p+9p+p-8+7-8+7\)
Simplify, knowing values with an attached variable cannot be added to values without an attached variable.\(20p-8+7-8+7\)
Simplify the other numbers.\(20p-2\)
Jayden multiples -4yx 1/5y and gets an answer of -4/5y. How can you tell without multiplying that his answer is incorrect
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Multiplication is not possible with so similar variables in the denominator, in the given fractions and hence the answer is incorrect.
What are fractions?Any number of equal parts is represented by a fraction, which also represents a portion of a whole. A fraction, such as one-half, eight-fifths, or three-quarters, indicates how many components of a particular size there are when stated in ordinary English. There are three main categories of fractions in mathematics. Proper fractions, incorrect fractions, and mixed fractions are these three types. The expressions with a numerator and a denominator are called fractions.So, -4y × 1/5y:
The answer to this multiplication is -4/5y.Which is a wrong answer.We can easily observe in the second term that the denominator is 5y and in the 1st term there is no denominator, so 1 will be represented as a denominator.And y will not be present.Hence, multiplication is not possible with so similar variables in the denominator, in the given situation.Theefore, multiplication is not possible with so similar variables in the denominator, in the given fractions and hence the answer is incorrect.
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is anyone expert here in data forecasting methods? I need some help in some topics like time series(holts, holts winter), naive method, regression, acf, pacf, arima, stl method and multivariate time series. please reply if you can help me with these topics
Answers
Yes, there are experts here in data forecasting methods who can help you with the topics you've mentioned including time series (holts, holts winter), naive method, regression, acf, pacf, arima, stl method and multivariate time series.
Below are brief explanations of each of these terms:
Time Series: A time series is a sequence of observations of a particular quantity measured over time. Holts Method: The Holt’s method is a forecasting method that forecasts the data by taking into account the trend component along with the level component. Holts Winter Method: Holt's winter model is used to forecast seasonal univariate time series.Naive Method: The naive method is a forecasting method that uses the most recent observation as a forecast for the next time period.Regression: Regression is a statistical method used to estimate the strength and direction of the relationship between two or more variables.ACF & PACF: Autocorrelation function (ACF) and partial autocorrelation function (PACF) are statistical tools used to determine the nature of the correlation between a variable and its lag.ARIMA: ARIMA stands for AutoRegressive Integrated Moving Average. ARIMA is a forecasting technique that uses past data points to predict future values.STL Method: STL is a time series decomposition method that separates a time series into three components: trend, seasonality, and random.Multivariate Time Series: Multivariate time series analysis deals with the analysis of time series data that involves more than one variable.Based on the topics you've mentioned, you may want to ask specific questions regarding these topics to get more detailed answers.To know more about data forecasting, visit
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Danna is buying breakfast cereal for a sleepover. She can buy 12-oz boxes for $3.00 each or 17-oz boxes for $3.57 each. Which is the better value? Explain.
Answers
Answer: 17-oz boxes
Step-by-step explanation:
To determine which breakfast cereal is a better value, we need to compare the cost per ounce of each box.
For the 12-oz boxes, the cost per ounce is:
Cost per ounce = $3.00 / 12 oz = $0.25/oz
For the 17-oz boxes, the cost per ounce is:
Cost per ounce = $3.57 / 17 oz ≈ $0.21/oz
Therefore, the 17-oz boxes are a better value because they have a lower cost per ounce than the 12-oz boxes.
Danna would get more cereal for her money by buying the 17-oz boxes.
the graph of the function f(x)= a^x is shown on the accompanying set of axes. On the same set of axes sketch the reflection of f(x) in the y-axis
Answers
Answer:
To flip or reflect (vertically) about the horizontal x-axis, replace y = f(x) with y = -f(x).
four light bulbs use different amount of electricity as shown below
A) 400 watt in 10 hours
B) 200 watts in 8 hours
C) 180 watts in 3 hours
D) 140 watts in 4 hours
which light bulb uses the least amount of electricity per hour?
Answers
Answer:
The answer is light bulb letter B
Step-by-step explanation:
A = 40
B = 25
C = 60
D = 35
All these are their Unit values.
Explain why this common mistake is wrong. Many people might say that (2, 4) is an intercept of the equation 5x + 2y = 18.
Answers
Answer:
Step-by-step explanation:
(2,4) is point lying on the line represented by 5x + 2y = 18.
They are not intercepts.
x-intercept is point where the line crosses the x-axis and so, y-co ordinate at x-intercept is 0.
y-intercept is point where the line crosses they-axis and so,x-co ordinate at y-intercept is 0.
5x + 2y = 18
To find x-intercept, put y = 0 in the equation
5x = 18
x = 18/5
x = 3.6
To find y-intercept, put x = 0 in the equation
2y = 18
y = 18/2
y = 9
x-intercept : (3.6 , 0)
y-intercept :(0, 9)
could you help me out with a question
Answers
The center of the circle is at the point (-5,11) and has the radius of 3 units.
The equation of the circle will be,
\(\begin{gathered} (x-(-5))^2+(y-11)^2=3^2 \\ (x+5)^2+(y-11)^2=9 \end{gathered}\)The sum of 1/2 and 2/5 is greater than one yes or no
Answers
Find the total surface area of the cylinder shown. Leave the answer in terms of π. A cylinder with radius 4.25 inches and height 6 inches. SA=2πrh+2πr2
Answers
Answer:
\(87.125\pi in^{2} or 87\frac{1}{8}\pi in^{2}\)
Step-by-step explanation:
Given Surface Area of Cylinder = \(2\pi rh+2\pi r^{2}\),
and also given r = 4.25 inches and h = 6 inches.
Let's Substitute r and h into the formula to find the total surface area.
Surface Area of Cylinder = \(2\pi (4.25)(6)+2\pi (4.25)^{2} \\=51\pi +36.125\pi \\=87.125\pi in^{2} or 87\frac{1}{8} \pi in^{2}\)
Is 1+7/x=y a Linear equation
Answers
Answer:
No
Step-by-step explanation:
1+7/x=y cannot be a linear equation because x is the denominator. A variable in the denominator means it has restrictions to what it can or cannot be. For example it can never be 0.
volume of a cuboid whose edges are 4 cm, 5 cm and 6 cm
Answers
Answer:
120
Step-by-step explanation:
Formula for cuboid: L × W × H
Plug in:
L × W × H
4 × 5 × 6
^ ^
20 × 6
^ ^
120
Hope this helps.
Solve equation.
2x-5=-3+2x
Answers
Answer:
x = 2
Step-by-step explanation: