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I am teaching high school precalculus and have a textbook that gives the following preamble to its rules for finding horizontal and slant asymptotes of rational functions:

Suppose $f(x)=\frac{a(x)}{b(x)}$ is a rational function, where $a(x)$ and $b(x)$ are polynomials, $b(x)\neq 0$, and $a(x)$ and $b(x)$ have no common factors other than $\pm 1$.

The text then lists the standard rules for finding horizontal and oblique asymptotes. (Asymptote at $y=0$ if $n < m$, etc.)

My question: Is it necessary for the polynomials to be fully simplified (no common factors other than $\pm 1$) to apply these rules? It seems to me that this does not affect the limit behavior. Am I missing something?

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    $\begingroup$ We can't answer this question without the full statement of the standard rules, unfortunately. $\endgroup$
    – Opal E
    Jul 29, 2022 at 5:02
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    $\begingroup$ For vertical asymptotes, do you look for zeros of the denominator? So would the rule say that the rational function $x/x$ has vertical asymptote $x=0$? $\endgroup$ Jul 29, 2022 at 10:15
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    $\begingroup$ @GeraldEdgar That seems to violate the rule that the numerator and denominator be relatively prime. $\endgroup$
    – Xander Henderson
    Jul 29, 2022 at 11:48
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    $\begingroup$ @XanderHenderson ... and that was exactly the question: "is it necessary that $a(x)/b(x)$ be fully simplified?" I hoped Chad could answer his question by thinking of my example. $\endgroup$ Jul 29, 2022 at 11:52
  • $\begingroup$ A side note: I think it's really good that you ask this question, but I find it irritating (though, sadly, not too surprising) that this is not discussed in the textbook itself. Having a list of "rules" about asympotes of rational functions seems to be completely pointless unless it is discussed in detail why these rules are true; and discussing why they are true should certainly involve a discussion of the relevant assumptions. $\endgroup$ Jul 31, 2022 at 9:46

2 Answers 2

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Summarizing from comments on the OP:

For horizontal and slant asymptotes there is no need to stipulate that the numerator and denominator share no common factors.

For vertical asymptotes you do need this stipulation so that you can distinguish between removable discontinuities and vertical asymptotes.

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One of your standard rules will be that if the degrees of your polynomials match, then there is a horizontal asymptote found by a simple calculation.

However, consider the function $f(x) = \frac{1}{1}$. Would you say the graph of this function has a "horizontal asymptote at $y=1$?" I would not. Maybe you would?

It is possible that the author is trying to avoid such issues entirely by stipulating that there are no common factors? It's hard to say for sure.

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    $\begingroup$ Since I prefer inclusive definitions, I would just define a $y=b$ as a horizontal asymptote if $\lim_{x \to \infty} f(x) = b$. So $f(x) = \frac{x}{x}$ would have a horizontal asymptote at $y=1$. So would $f(x) = \frac{x + \sin(x)}{x}$. $\endgroup$ Jul 31, 2022 at 0:32
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    $\begingroup$ Hmm, could you clarify which rule might lead to the conclusion that $f(x) = x/x$ has a horizontal asymptote at $y = 0$? I don't think I can see your point right now. $\endgroup$ Jul 31, 2022 at 8:54
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    $\begingroup$ I was also confused but not about the $y=0$ part. It seems like you have an issue with the function value at $x=0$ but that has nothing to do with horizontal asymptotes. $\endgroup$
    – Thierry
    Jul 31, 2022 at 20:48
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    $\begingroup$ Yeah, there is no issue with this example, either. Would you also say that straight lines don't have tangents? Such definitions excluding the "trivial" cases just lead to unnecessary casework. $\endgroup$ Aug 1, 2022 at 6:16
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    $\begingroup$ Ah, now I see your point. I still don't see an issue but I have heard of people defining asymptotes as, roughly, "lines that come close to a graph but don't touch it". That seems like a first attempt at a definition that will then be refined to Steve Gubkin's definition above. $\endgroup$
    – Thierry
    Aug 1, 2022 at 12:44

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