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
Find the distance between (–6, 5) and (3, 0). round to the nearest tenth if necessary. a 106.1 b –10.3 c 10.3 d 5.8
AleksandrR [38]

\sqrt{81 + 25  }  =  \sqrt{106}
it will be around 10,3
4 0
3 years ago
Which expression is equivalent sqrt10/4sqrt8
N76 [4]

Answer:

The correct option is A

Step-by-step explanation:

The given expression is:

√10/4√8

We have to eliminate the √ from the denominator

4√8 = (2^3)^1/4

Multiply the whole expression by (2)^1/4

=(2)^1/4 * √10/ 2^(1/4)*(2^3)^(1/4)

= 2^(1/4) · 10^(2/4)  / 2^1/4+3/4

=2^(1/4) · 10^(2/4) / 2^1+3/4

=2^(1/4) · 10^(2/4) / 2^4/4

=2^(1/4) · 10^(2/4) /2

=2^(1/4) · 100^(1/4) /2

=200^1/4 /2

= 4√200 /2

Thus the correct option is A....

3 0
4 years ago
M∠a=(48−x)°m∠b=(9x−38)°m∠c=90°
Fudgin [204]

Answer:

Option C:

  • m∠a = 38°
  • m∠b = 52°
  • m∠c = 90°

Step-by-step explanation:

Hello!

The sum of all angles in a triangle is 180°.

Add up all the angles and solve for x by setting the equation to 180°.

<h3>Solve for x</h3>
  • ∠a + ∠b + ∠c = 180°
  • (48 - x)° + (9x - 38)° + 90° = 180°
  • 10° + 8x + 90° = 180°
  • 8x = 180° - 90° - 10°
  • 8x = 80°
  • x = 10°

Now that we solved for x, we can plug it back into each equation to solve for each angle.

<h3>Angle A</h3>
  • 48° - x
  • 48° - 10°
  • 38°
<h3>Angle B</h3>
  • 9x - 38°
  • 9(10°) - 38°
  • 90° - 38°
  • 52°
<h3>Angle C</h3>
  • 90°

The answer is Option C:

  • m∠a = 38°
  • m∠b = 52°
  • m∠c = 90°
4 0
2 years ago
Mr. Allway’s math class surveyed all the seventh-grade students to find out their favorite sport. The following circle graph sho
JulijaS [17]

Answer:

101 is the answer of the question

3 0
3 years ago
Read 2 more answers
Convert NPR 180000 into <br>1. USD <br><br>​
creativ13 [48]

Answer:

Solution here,

1 US dollar=NRS 120.74

Therefore Rs 180000= 180000/120.74

=$1490.99

5 0
3 years ago
Other questions:
  • in the state of manie,the ratio of moose to people is 5 to 2.If a certain county in Maine has a population of 1,120 people, how
    7·1 answer
  • the local theater is showing a matinee and offering a special deal for the community. A ticket for an adult costs $11 and a tick
    14·1 answer
  • Blue cassettes hold 45 minutes of music. Green cassettes hold 55 minutes
    8·1 answer
  • Which best summarizes the purpose of a W-4 form?
    6·2 answers
  • Need help on this slope question the second one
    15·1 answer
  • [Help asap, will mark brainliest] Find the area of the right triangle.
    14·1 answer
  • 1) Find the quotient:<br>(2x4 - 3x2 - 1) = (x - 1)​
    12·1 answer
  • Write and evaluate the expression. Then, complete the statements. Ivan lost twenty dollars. Evaluate when d = $35.
    15·1 answer
  • The number of customers visiting a local business is 61,874 and has been continuously declining at a rate of 3.5% each year. Wha
    12·1 answer
  • Issac drove 100 miles at a speed of 50 mph and then 120 miles at a speed of 30 mph what was his average speed for the entire tri
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!