Curvilinear Asymptotes in GeoGebra

GeoGebra has a very useful function called Asymptote: if you have something like $f(x)=\frac{3x^2+4x+3}{x-1}$, typing Asymptote(f) in the input bar gives a list of the linear asymptotes: $\{y=3x+7;x=1\}$. Very nice, very useful.

But something like $f(x)=\frac{2x^4+3x^3+2x+4}{x^2+3x+2}$ is more tricky: GeoGebra only returns the two linear asymptotes, $x=-1$ and $x=-2$. However, there’s also a curvilinear asymptote that GeoGebra doesn’t return. Can we get GeoGebra to find it?

Of course we can. It’s a tiny bit tricky, but it’s not as bad as I first thought.

The first step is to split the function into a numerator and a denominator, a top and a bottom:

• $N(x)=2x^4+3x^3+2x+4$
• $D(x)=x^2+3x+2$

The key is then to use the Division command:

• $L=Division(N,D)$

This gives a list containing the quotient and remainder. Here, we need the quotient:

• $q(x)=Element(L,1)$ ¹.

And that’s it! $q(x)$ is the curvilinear asymptote to $f(x)$.

Footnotes:

1. Note that GeoGebra lists start counting from 1