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
Which of the following is an extraneous solution of sqrt 4x + 41 = x+5
stiv31 [10]

Answer:

x=-8

Step-by-step explanation:

\sqrt{4x+41}=x+5

on squaring both sides

4x+41=(x+5)^2\\\\4x+41=x^2+10x+25\\\\x^2+6x-16=0\\\\x^2+8x-2x-16=0\\\\x(x+8)-2(x+8)=0\\\\(x-2)(x+8)=0

either x-2=0  or  x+8=0

either x=2 or x=-8

Putting x=2 in original equation, we get

\sqrt{4\times 2+41}=2+5

7=7 hence, it is not an extraneous solution

Putting x=-8 in original equation

\sqrt{4\times (-8)+41}=-8+5

i.e. 3=-3

Hence, x=-8 is an extraneous solution (since it doesn't satisfy the original equation)

3 0
3 years ago
Read 2 more answers
What are the signs of the coordinates of a point in each of the four quadrants
guajiro [1.7K]
On the Cartesian plane in two dimensions, the quadrants are <span>I (+,+), II (−,+), III (−,−), and IV (+,−). The numbering, starting in the upper right corner, goes counter-clockwise.</span>

6 0
2 years ago
Does anyone know this? Please help! ASAP!!!
Fudgin [204]
The answers:
a=4
b=5
c=6
d=1
8 0
2 years ago
Read 2 more answers
What is the answer and how to do this?​
zhenek [66]

Answer:

<u>a</u>

Step-by-step explanation:

Given :

⇒ P (Sumit) = 1/2

⇒ P (Sujan) = 1/3

⇒ P (Rakesh) = 1/a

⇒ P (total) = 3/4

============================================================

Solving :

⇒ 1/2 × 1/3 × 1/a = 3/4

⇒ 1/6 × 1/a = 3/4

⇒ 2/12 × 1/a = 9/12

⇒ a = <u>9/2</u>

6 0
1 year ago
Just answer number 7 and 8
melamori03 [73]
We are given the equation:
F = 2.25+0.2(m-1)
where:
F is the fare
m is the number of miles.

In question 7, we are given that the fare (F) is equal to $6.05 and we need to get the number of miles. To do so, we will simply substitute with the fare in the given equation and solve for the number of miles (m) as follows:
F = 2.25 + 0.2(m-1)
6.05 = 2.25 + 0.2(m-1)
6.05-2.25 = 0.2(m-1)
3.8 = 0.2(m-1)
3.8/0.2 = m-1
19 = m-1
m = 19+1
m = 20 miles

Number 8 is exactly the same, but we will substitute F=7.65 and again solve for m
4 0
3 years ago
Other questions:
  • Write the name of the period that has the digits 208
    9·1 answer
  • X-yz for x=-2 y=5 solve for z
    13·1 answer
  • HELP!!! URGENT!!!! 40 POINTS!!!
    10·2 answers
  • In the ordered pair -3/2,-7 is called the
    8·1 answer
  • Dillard's checking account charges a $13.25 monthly service fee and a $0.24 per check fee. if Dillard writes 18 check per month,
    9·1 answer
  • HURRY GETTING TIMED<br> What is the value of 73 n=12<br> 1,891<br> 2,610<br> 2,635<br> 2,701
    13·2 answers
  • I need plz and thank you I am time
    14·2 answers
  • Solve for x:
    6·1 answer
  • If population function x has mean M(x)=2 and M(x²)=8, find its standard deviation​
    14·1 answer
  • The two-way table below shows the number of tickets sold for performances to the school play.
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!