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:

• $q(x) = Element(L, 1)$ ((Note that GeoGebra lists start counting from 1)).

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