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
A jar has 32 marbles. Some are blue and the rest are green. There are three times as many green marbles as blue. How many green
ella [17]

Answer: 24

Step-by-step explanation:

Number of marbles in jar = 32

Let blue marbles be represented by b.

Since there are three times as many green marbles as blue. This means that green will be = 3 × b = 3b

We then add them together. This will be:

3b + b = 32

4b = 32

b = 32/4

b = 8

There are 8 blue marbles

Since green is 3× blue. Therefore,

Green = 8 × 3 = 24 marbles

7 0
2 years ago
A square has vertices (a,0) , (0,a) , and (a, 2a) find the fourth vertex
I am Lyosha [343]

Notice the first two vertices are reverse from each other (a,0) to (0,a)

the fourth vertex would be the reverse of the third one.

The fourth vertex would be (2a, a)

3 0
3 years ago
Read 2 more answers
What is the surface area for this?
DanielleElmas [232]

It is 6 sq. units. I hope you do well. Have a wonderful day.

6 0
3 years ago
A cell phone company charges a monthly fee of $50 for 300 minutes. Each addition minute costs $0.05. Determine if the function i
Ymorist [56]

Answer:

linear

Step-by-step explanation:

The graph would be linear since it's .05 every minute, which would mean that every minute it would cost .05 --> adding up minutes. The flat fee at the beginning is trying to trick you up, don't get fooled ;)

3 0
2 years ago
Read 2 more answers
Can someone please help! Thank you so much!
torisob [31]

Answer:

a. 3x (150 degrees)

b. 9x (180 degrees)

c. 3x - 20 (220 degrees)

3 0
3 years ago
Other questions:
  • 17.2=h+4 what does h equal
    10·2 answers
  • On a Saturday, a library checked out 52 books. If 24 of the books were fiction, what ratio of npn-fiction books to fiction books
    13·1 answer
  • Find the area of the circle. Round to the nearest tenth.
    9·1 answer
  • What is the absolute value of -0.575
    12·1 answer
  • This graph displays a linear function. What is the rate of change?
    15·1 answer
  • You invest an initial $100 in an account that has an annual interest rate of 3%, compounded quarterly. How much money will you h
    13·2 answers
  • Find the area of the regular polygon...<br> Uh how?
    8·2 answers
  • Students
    15·2 answers
  • Helpppppppppppppppppppppppppppp
    8·2 answers
  • I can bake 3/4 of a cake in 5 minutes. How much of the cake canlbake in 1 minute?​
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!