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Given a function $a(x)$ is there a general method to writing a function $f(x)$ so that is has the asymptote $a(x)$?

As far as I can tell from here you can sort of do this as long as $a(x)$ is a polynomial and you don't mind vertical asymptotes. Then $f(x)=a(x) + 1/x$ does the trick. But what about doing it without vertical asymptotes and for more general $a(x)$?

Hudson Hochstedler
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  • Can you simply choose $f(x)=a(x)$? If not, what properties do you require $f(x)$ to have? Also, is $a(x)$ always that specific function or can it be more general?—because the limits of $f'$ won't be compatible with general $a(x)$. – Greg Martin Jul 25 '23 at 15:08
  • @Greg Martin: I guess you could choose $f=a$ but its not very interesting. I've removed my specific case to be more concise. I'm more interested in general $a(x)$. – Hudson Hochstedler Jul 25 '23 at 21:01

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