Asymptote to Folium of Descartes

Theorem
Consider the folium of Descartes $F$, given in parametric form as:
 * $\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$

The straight line whose equation is given by:
 * $x + y + a = 0$

is an asymptote to $F$.

Proof
First we note that from Behaviour of Parametric Equations for Folium of Descartes according to Parameter:


 * when $t = 0$ we have that $x = y = 0$


 * when $t \to \pm \infty$ we have that $x \to 0$ and $y \to 0$


 * when $t \to -1^+$ we have that $1 + t^3 \to 0+$, and so:
 * $x \to -\infty$
 * $y \to +\infty$


 * when $t \to -1^-$ we have that $1 + t^3 \to 0-$, and so:
 * $x \to +\infty$
 * $y \to -\infty$

We have that:

So setting $t = -1$:

The result follows.