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Find the number of all possible asymptotes of: $$f(x)=\sqrt{x^2-4x}+\frac{1}{x^2-1}$$

Since we know $\sqrt{ax^2+bx+c}\approxeq \sqrt{a}\big|x+\frac{b}{2a}\big|$ when $(x\rightarrow\pm\infty)$ so, working with $\sqrt{x^2-4x}$ in $f(x)$, makes two following lines : $$x\rightarrow +\infty\Rightarrow y=x-2\\x\rightarrow -\infty\Rightarrow y=-x+2$$ I see the second part of $f(x)$ tends to zero. So, I agree that this function has 2 oblique asymptotes and one horizontal asymptote $y=0$.

For vertical asymptote, I say $x=\pm1$ are that ones. Overall, Am I thinking right about the number of all asymptotes for $f(x)$? Is taking the function apart to two sections allowed in this function? Thanks.

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  • $\begingroup$ @Gigili: Thanks for editting. $\endgroup$
    – Mikasa
    Sep 7, 2012 at 18:52

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Slight trick question. The function is not defined near $x=1$. So definitely no asymptote there.

"Taking apart" is sort of OK, we are examining behaviour at quite different places. But, as we saw in the first sentence, we cannot separate things completely. If $\sqrt{x^2-4x}$ was well-behaved near $x=\pm 1$, there would be no problem.

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  • $\begingroup$ So, we have 2 oblique asymptotes, one horizontal asymptote and just $x=-1$ for vertical. Right? $\endgroup$
    – Mikasa
    Sep 7, 2012 at 18:55
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    $\begingroup$ @BabakSorouh: Yes, there are three asymptotes, one vertical, two oblique. $\endgroup$
    – André Nicolas
    Sep 7, 2012 at 19:01

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