1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Karolina [17]
3 years ago
14

Let P and Q be polynomials with positive coefficients. Consider the limit below. lim x→[infinity] P(x) Q(x) (a) Find the limit i

f the degree of P is less than the degree of Q. 0 (b) Find the limit if the degree of P is greater than the degree of Q.
Mathematics
1 answer:
jenyasd209 [6]3 years ago
3 0

Answer:

If the limit that you want to find is \lim_{x\to \infty}\dfrac{P(x)}{Q(x)} then you can use the following proof.

Step-by-step explanation:

Let P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0} and Q(x)=b_{m}x^{m}+b_{m-1}x^{n-1}+\cdots+b_{1}x+b_{0} be the given polinomials. Then

\dfrac{P(x)}{Q(x)}=\dfrac{x^{n}(a_{n}+a_{n-1}x^{-1}+a_{n-2}x^{-2}+\cdots +a_{2}x^{-(n-2)}+a_{1}x^{-(n-1)}+a_{0}x^{-n})}{x^{m}(b_{m}+b_{m-1}x^{-1}+b_{n-2}x^{-2}+\cdots+b_{2}x^{-(m-2)}+b_{1}x^{-(m-1)}+b_{0}x^{-m})}=x^{n-m}\dfrac{a_{n}+a_{n-1}x^{-1}+a_{n-2}x^{-2}+\cdots +a_{2}x^{-(n-2)}+a_{1}x^{-(n-1)})+a_{0}x^{-n}}{b_{m}+b_{m-1}x^{-1}+b_{n-2}x^{-2}+\cdots+b_{2}x^{-(m-2)}+b_{1}x^{-(m-1)}+b_{0}x^{-m}}

Observe that

\lim_{x\to \infty}\dfrac{a_{n}+a_{n-1}x^{-1}+a_{n-2}x^{-2}+\cdots +a_{2}x^{-(n-2)}+a_{1}x^{-(n-1)})+a_{0}x^{-n}}{b_{m}+b_{m-1}x^{-1}+b_{n-2}x^{-2}+\cdots+b_{2}x^{-(m-2)}+b_{1}x^{-(m-1)}+b_{0}x^{-m}}=\dfrac{a_{n}}{b_{m}}

and

\lim_{x\to \infty} x^{n-m}=\begin{cases}0& \text{if}\,\, nm\end{cases}

Then

\lim_{x\to \infty}=\lim_{x\to \infty}x^{n-m}\dfrac{a_{n}+a_{n-1}x^{-1}+a_{n-2}x^{-2}+\cdots +a_{2}x^{-(n-2)}+a_{1}x^{-(n-1)}+a_{0}x^{-n}}{b_{m}+b_{m-1}x^{-1}+b_{n-2}x^{-2}+\cdots+b_{2}x^{-(m-2)}+b_{1}x^{-(m-1)}+b_{0}x^{-m}}=\begin{cases}0 & \text{if}\,\, nm \end{cases}

You might be interested in
Jackson had 1, 3/8 kg of fertilizer he used some to fertilize a flower bdd and he only had 2/3kg left how much fertilizer was us
ddd [48]

Answer:

Jackson had 1, ⅜ kg of fertilizer he used some to fertilize a flower bed and he only had ⅔ kg left how much fertilizer was used in the flower bed

Step-by-step explanation:

The calculator performs basic and advanced operations with fractions, expressions with fractions combined with integers, decimals, and mixed numbers. It also shows detailed step-by-step information about the fraction calculation procedure. Solve problems with two, three, or more fractions and numbers in one expression.

1/3 * (1/2 - 3 3/8) = -23/

24

≅ -0.9583333

Spelled result in words is minus twenty-three twenty-fourths.

How do you solve fractions step by step?

Conversion a mixed number 3 3/

8

to a improper fraction: 3 3/8 = 3 3/

8

= 3 · 8 + 3/

8

= 24 + 3/

8

= 27/

8

To find new numerator:

a) Multiply the whole number 3 by the denominator 8. Whole number 3 equally 3 * 8/

8

= 24/

8

b) Add the answer from previous step 24 to the numerator 3. New numerator is 24 + 3 = 27

c) Write a previous answer (new numerator 27) over the denominator 8.

Three and three eighths is twenty-seven eighths

Subtract: 1/

2

- 27/

8

= 1 · 4/

2 · 4

- 27/

8

= 4/

8

- 27/

8

= 4 - 27/

8

= -23/

8

For adding, subtracting, and comparing fractions, it is suitable to adjust both fractions to a common (equal, identical) denominator. The common denominator you can calculate as the least common multiple of the both denominators - LCM(2, 8) = 8. In practice, it is enough to find the common denominator (not necessarily the lowest) by multiplying the denominators: 2 × 8 = 16. In the next intermediate step the fraction result cannot be further simplified by canceling.

In words - one half minus twenty-seven eighths = minus twenty-three eighths.

Multiple: 1/

3

* the result of step No. 2 = 1/

3

* (-23/

8

) = 1 · (-23)/

3 · 8

= -23/

24

Multiply both numerators and denominators. Result fraction keep to lowest possible denominator GCD(-23, 24) = 1. In the next intermediate step the fraction result cannot be further simplified by canceling.

In words - one third multiplied by minus twenty-three eighths = minus twenty-three twenty-fourths

5 0
2 years ago
Which graph represents the solution set of the system of inequalities?
Lyrx [107]
The answer to this is D
4 0
2 years ago
Read 2 more answers
Angles
coldgirl [10]
<h3>Answer: XWY and STR</h3>

I tend to think of parallel lines as train tracks (the metal rail part anyway). Inside the train tracks is the interior region, while outside the train tracks is the exterior region. Alternate exterior angles are found here. Specifically they are angles that are on opposite or alternate sides of the transversal cut.

Both pairs of alternate exterior angles are shown in the diagram below. They are color coded to help show how they pair up and which are congruent.

A thing to notice: choices B, C, and D all have point W as the vertex of the angles. This means that the angles somehow touch or are adjacent in some way due to this shared vertex point. However, alternate exterior angles never touch because parallel lines never do so either. We can rule out choices B,C,D from this reasoning alone. We cannot have both alternate exterior angles on the same exterior side of the train tracks. Both sides must be accounted for.

4 0
3 years ago
Food A contains 150 calories in 3/4 of a serving. Food B contains 250 calories in 2/3 of a serving. Find each unit rate. Which f
Leviafan [203]
The unit rate is going to be calories per serving....calories / serving

Food A :
150 / (3/4) =
150 * 4/3 =
600/3 =
200 calories per serving

Food B :
250 / (2/3) =
250 * 3/2 =
750/2 =
375 calories per serving

Therefore, Food A has fewer calories per serving <==
5 0
3 years ago
Read 2 more answers
Find x such that -1/3 = 7/x
Setler [38]

Answer:

x= -21

Step-by-step explanation:

-1/3=7/x

-1x=21

-1x/-1=21/-1

x= -21

7 0
2 years ago
Other questions:
  • Simplify 12p-8+6p+14
    5·2 answers
  • What is 0.6÷0.0024 ?
    7·2 answers
  • What is the surface area, in square inches, of a 8-inch cube?
    7·2 answers
  • Solve for x: -3(x + 1)= -3(x + 1) - 5
    5·1 answer
  • Can someone Please help me?
    9·1 answer
  • Helppppp i need help ASAP
    15·1 answer
  • 5/6 x 12 in simplest form?
    15·2 answers
  • Nigel went to a sandwich shop for lunch. The menu to the above shows the types of cheeses and meats available. If Nigel chooses
    15·2 answers
  • This is the mastery test:(
    12·1 answer
  • What is the slope of the line going through A and B?
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!