Problem 55 Hard Difficulty

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 $.

Answer

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Video Transcript

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.