Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

Let $ P $ and $ Q $ be polynomials. Find $$ \lim_{x \to \infty} \frac{P(x)}{Q(x)} $$ if the degree of $ P $ is (a) less than the degree of $ Q $ and (b) greater than the degree of $ Q $.

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Daniel Jaimes

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Aidana B.

August 31, 2021

Show that f(x) has a jump discontinuity at x=5 by calculating the limits from the left and right at x=5.

Missouri State University

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

01:05

$$ \begin{array}{c}{\text …

02:30

Let P and Q be polynomials…

01:03

If $P$ and $Q$ are polynom…

03:58

Suppose $p$ and $q$ are po…

00:49

01:28

Let $p(x)$ be a polynomial…

03:15

02:29

In each part, find example…

01:46

this is problem number fifty five of this tour Calculus eighth edition, Section two point six Let Pete and Q B polynomial. Lt's kind of the limit is X approaches Infinity Lovely quotient, Pete of x o ver que mix. If the degree of PS lessen the degree of Q and being if the agree of peace is greater than the degree of Q so to illustrate. And to prove this on the outcomes for these we Karl P. Polynomial that is representative represented here by coefficients of eh seven. Yeah, and an amount of coefficients multiplied by on X term Such a exit, an exit and mass one and so on. So this would be a polynomial of order and for peace, the degree of the polynomial is equal. And for cue, the provisions are crew a group of B coefficients B seven and the terms were occupied by that. Our exit DMX they minus one and so on in this sort of have a degree of him. No, when we evaluate this limit and as we have seen in the past past problems that have been worked out in this chapter, we know that on ly the leading coefficient will be significant. And only the leaving coefficients for two ponytails being divided by each other are going to have an effect on the limit itself. So that in mind, we're going to write the two coefficient polynomial for pain and part of milk you and then we're gonna discuss the effects of if the degree of Pete is less Thank you. And if it is created them the degree of cute. So party, what if and is less than ember. Okay, if we rearrange our limits, what scene? For now, I'm okay. And that in our matter will have peace of M s O s. A bend about a piece of him X um m minus. And if we combined, the two exploring ex idiom is in the bottom. But if we bring decks to the get down as a negative exponents, that's it. The way the term would look like if and his less than them. That would be this famous scene and minus end. It's created that zero. So this experience, it's positive. As X approaches infinity, this value purchase infinity. So we're gonna have a very large valued that a nominator and no matter what the coefficients air in this case, a very large positive number or even negative, in this case, the call. Okay, Constant about away an infinity of some sort always approaches zero. So element in this case will be zero as experts is infinity. And this is true for the general case where the polynomial on the numerator has a degree less than the polynomial in the denominator for a party. We look at the opposite case, Ace Um d degree of the polynomial Pete is greater than the degree of the polynomial for cute. So we will rearrange the terms in the limit differently. We're still doing this ratio of coefficients a seven or a piece of him, but instead we're going to have X and minus M. So we're going to take Nixon and then Dorothy ex idiom to the top, making of negative exponents. Little combined with this next to the end, and I'll be extra the unrest M. And the reason we do this is because here we rearrange and raise him and we see that this differences greater than zero again because the degree of P it's created some degree of Q in a dead isso in this terrible urges positive infinity in this case. And we see that over on this wall. That bridge depending on what the coefficients are the coefficients Mabel to be positive or both negative. In that case, the limit ever just departed infinity if they are opposite of each other. One is that anyone is positive. A little beverage to negative infinity. But regardless we can say conclusively that if the degree of the polynomial p it's great, thank you overall the limit old average either to positive infinity or negative infinity.

View More Answers From This Book

Find Another Textbook

01:10

'geometry geometry geometry'

01:34

'Writethe largest 5-digit number ending with zero and write its prime f…

01:17

'write down the squares of 3x+1/3x1 (v) 3x+ 3 3x'

01:32

'tick the correct answer'

00:57

"Evaluate :- 795 (-25) + (795) x 75 By using distributive property of m…

03:03

'Subtract the sum of -8/7 and 5/4 from the sum of 3/2 and -31/28 class …

02:36

'Please help me with questionJust tell me the steps and it is ok.The on…

01:58

"answer my question and it is correct question. don't post irrelev…

00:09

'1. Make a Robot a. Make Robot with different coloured sheets. b. Calcu…