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Suppose that $$Q(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0} $$and $$P(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}.$$ How do I find $$\lim_{x\rightarrow\infty}\frac{Q(x)}{P(x)}$$ and what does the sequence $$\frac{Q(k)}{P(k)}$$ converge to?

For example, how would I find what the sequence $$\frac{8k^2+2k-100}{3k^2+2k+1}$$ converges to? Or what is $$\lim_{x\rightarrow\infty}\frac{3x+5}{-2x+9}?$$

This is being asked in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions.

and here: List of abstract duplicates.

Hans Lundmark
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Eric Naslund
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    Bad form to use $n$ for both the degree of $Q$ and the integer argument of $Q$ in the same problem (but I think we all know what you mean). – Gerry Myerson Apr 20 '11 at 02:07
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    @Gerry Myerson: Good point. Fixed now! – Eric Naslund Apr 20 '11 at 02:09
  • Probably also good to leave a note that the techniques here are useful for finding asymptotes of rational functions. – J. M. ain't a mathematician Apr 20 '11 at 02:10
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    Eric: Such questions and answers should be CW in my opinion. Further, why not simply modify one of the many prior answers on such topics, so to reduce the noise? – Bill Dubuque Apr 20 '11 at 02:14
  • @Bill: Pretty sure Eric did the right thing to flag for wiki-ing beforehand... – J. M. ain't a mathematician Apr 20 '11 at 02:15
  • No moderator seems to be online, however, so... – J. M. ain't a mathematician Apr 20 '11 at 02:26
  • @J.M. It apears that Eric is not marking these CW, e.g. see this similar question where he supplied two non-CW answers to his question. This is probably an issue that deserves to be discussed on meta. – Bill Dubuque Apr 20 '11 at 02:29
  • @Eric: When I originally proposed the "abstract duplicate" questions, the model I had in mind is that such questions and answers should be approved either by mods or 10k+ users, in order to ensure proper organization, quality control, etc. Further, the questions and answers should be CW so that as many experts as possible can contribute to composing optimal such questions and answers. My apologies for not fleshing out my proposal further - time has been very scarce recently. – Bill Dubuque Apr 20 '11 at 02:45
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    @Bill: To answer your questions: I prefer it when a new question is created and devoted solely to the abstract duplicate, rather than just hijacking one that was already asked. I think it makes things nicer and cleaner, however that is a matter of personal taste. As for the community wiki, I think that is a good idea, and have flagged the post for moderator attention. At first glance, I just went with what the other answers on the abstract duplicate page had done. (Upon a quick look again, only one is community wiki, but then again I think most were modifications of other questions) – Eric Naslund Apr 20 '11 at 03:37

2 Answers2

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Short Answer:

The sequence $\displaystyle\frac{Q(k)}{P(k)}$ will converge to the same limit as the function $\displaystyle\frac{Q(x)}{P(x)}.$ There are three cases:

$(i)$ If $n>m$ then it diverges to either $\infty$ or $-\infty$ depending on the sign of $\frac{a_{n}}{b_{m}}$.

$(ii)$ If $n<m$ then it converges to $0$.

$(iii)$ If $n=m$ then it converges to $\frac{a_{n}}{b_{n}}$.

Hans Lundmark
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Eric Naslund
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    And perhaps the easiest way to get to this answer is to divide top and bottom by $x^d$, where $d$ is the smaller of $m$ and $n$. – Gerry Myerson Apr 20 '11 at 02:06
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    Lang had an interesting approach: factor $x^{n}$ out of the numerator and $x^{m}$ from the denominator to get $x^{n-m}$ times a rational function with constant limit $a_{n}/b_{m}$ as $x \rightarrow \infty$. The result Eric quoted now follows by examining the limit of $x^{n-m}$. – Chris Leary Apr 20 '11 at 15:27
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More generally: if a sequence $a_n$ is given by the values of function $f(x)$ that is defined on an interval of the form $(b,\infty)$, $$a_n = f(n),$$ and the limit of $f(x)$ as $x\to\infty$ exists or is equal to $\infty$ or $-\infty$, $$\lim_{x\to\infty}f(x) = L,\qquad L\in\mathbb{R}\cup\{\infty,-\infty\},$$ then the limit of the sequence is the same as the limit of the function: $$\lim_{n\to\infty}a_n = \lim_{n\to\infty}f(n) = \lim_{x\to\infty}f(x).$$

This applies to the case where $\displaystyle f(x)= \frac{P(x)}{Q(x)}$ with $P$ and $Q$ polynomials; also to sequences like $$a_n = \frac{\sin(n)}{n},$$ given by $\displaystyle f(x) = \frac{\sin(x)}{x}$; and even some functions which are more complicated. E.g., $$a_n = \frac{(-1)^n}{n}$$ can be seen as given by the function $$f(x) = \frac{\cos(\pi x)}{x}.$$

Note, however, that it is possible for the limit of $a_n$ to exist, but that of $f(x)$ not to exist. For instance, the sequence $a_n = \sin(n\pi)$ has limit $0$ (because every $a_n$ is equal to $0$), but the limit of $f(x)=\sin(\pi x)$ as $x\to\infty$ does not exist.

Arturo Magidin
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