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For an assignment, I was asked to find the vertical asymptote of the function $$g(x)= \frac{\frac{1}{2}x^3-4x^2+6x}{7x^2-56x+84}.$$ According to my text, a reliable method of finding the asymptote is to factor the numerator and denominator, and what left in the denominator that was not cancelled out is the asymptote.

I factored the numerator to $\frac{1}{2}(x^2-6x)(x-2)$, and the denominator factored to $7(x-2)(x-6)$, therefore $(x-2)$ cancelled out, leaving $(x-6)$ in the denominator.

However, 6 was not accepted as the answer, and I would like to know why.

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Jason, can you add some parentheses to your function $g(x)$? It is difficult to tell what function $g$ is precisely. –  JavaMan Feb 13 '11 at 2:49
(1) Probably you have an extra $1/$ in the beginning. (2) $2x^3-4x^2+6x = 2x(x^2-2x+3)$ does not factor further (the discriminant is $2^2-4\cdot 3 = -8$). –  Yuval Filmus Feb 13 '11 at 3:00
@JDC, @Yuval: It seems likely that it was meant to be as I have edited. –  Arturo Magidin Feb 13 '11 at 4:06
@Jason: Please don't use in-line fractions with /; they are extremely hard to parse. –  Arturo Magidin Feb 13 '11 at 4:06
@Jason: Note that formulas don't have vertical asymptotes. Functions and graphs do. –  Arturo Magidin Feb 13 '11 at 4:21
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3 Answers

up vote 4 down vote accepted

You did not finish factoring the numerator. That's where the problem is. You started out all right, but then you had to keep factoring that $x^2-6x$: it is not degree $1$, and it is not irreducible quadratic, so it can still be factored. In fact, it has a factor of $x$. So you really have: $$\frac{1}{2}x^3 - 4x^2+6x = \frac{1}{2}\left(x^3 - 8x^2 + 12x\right) = \frac{1}{2}x\left(x^2-8x+12\right) = \frac{1}{2}x(x-6)(x-2).$$ In fact, both $x-2$ and $x-6$ cancel out in $g(x)$.

As a function, $g(x)$ is equal to $$g(x) = \left\{\begin{array}{ll} \frac{1}{14}x &\mbox{if $x\neq 2$ and $x\neq 6$,}\\ \mbox{undefined} &\mbox{if $x=2$ or $x=6$.} \end{array}\right.$$ So it does not have any vertical asymptotes.

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Thanks for the help! I didn't recognize that the numerator could be factored any more, and thatks for pointing that out. The tough part now is figuring out whose answer to accept! –  Jason Feb 13 '11 at 15:58
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No factorization is needed. First, separate the rational function $\rm\:f/g\:$ into its integral and fractional parts by using the division algorithm to divide the numerator by the denominator, namely

$$\rm f\ =\ q\ g + r\ \ \Rightarrow\ \ \frac{f}g\ =\ q + \frac{r}g,\quad\quad deg\ r\ <\ deg\ g$$

Since the integral part $\rm\:q\:$ is a polynomial it has no vertical asymptotes. Rather, such asymptotes arise only from the "proper" fractional part $\rm\:r/g\:,\: $ namely at roots of the denominator $\rm\:g\:,\: $ which are not also roots of the numerator $\rm\:r\:.\: $ But in your example, performing the simple division shows that remainder $\rm\:r = 0\:,\: $ so there are no such asymptotes. Generally this will be a much simpler approach than immediate factorization since generally division (and gcd) algorithms are much simpler than factorization algorithms.

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@downvoter: If something is not clear please feel free to ask questions and I am always glad to elaborate. –  Bill Dubuque Feb 13 '11 at 18:15
+1: This is the approach I would prefer in general. –  Jonas Meyer Feb 13 '11 at 18:17
@Jonas: Thanks. Perhaps the downvoter thought that was no need for injecting any generality. But even in rote problems I think it's worthwhile to attempt to encourage students to think abstractly. If one is going to invest any nontrivial amount of time into a problem then it is usually worthwhile to spend a little extra time introspecting - to abstract out essential insights discovered in the solution in the hope that may be reused in future problems. Such introspectional learning processes go a long way towards building up a strong arsenal of general methods for attacking diverse problems. –  Bill Dubuque Feb 13 '11 at 18:38
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Note that $\displaystyle f(x) = \frac{\frac{1}{2}x^3-4x^2+6x}{7x^2-56x+84} = \frac{1}{2}\frac{x^3-8x^2+12x}{7x^2-56x+84} = \frac{1}{2}\frac{1}{7}\frac{x^3-8x^2+12x}{x^2-8x+12} = \frac{x}{14}$.

So the function is "almost" a straight line passing through origin with a slope $\frac{1}{14}$ except at $x=2$ and $x=6$.

The function is not defined at $x=2$ and $x=6$. But $\displaystyle \lim_{x \rightarrow 2} = \frac{1}{7}$, $\displaystyle \lim_{x \rightarrow 6} = \frac{3}{7}$.

There are no asymptotes for the problem.

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oops... essentially the same answer as Arturo. I didn't see Arturo's answer since I left for a while in the middle of typing my answer. –  user17762 Feb 13 '11 at 4:24
Not a problem. I did add a \displaystyle to yours, though, as it was very hard to read... –  Arturo Magidin Feb 13 '11 at 4:25
Thanks for the help! –  Jason Feb 13 '11 at 15:59
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