Folium of Descartes: Proving Asymptote

[mahler1]

I am given the parametrized curve $\alpha:(-1,\infty) \to \mathbb R^2$ as $\alpha(t)=(\dfrac{3at}{1+t^3},\dfrac{3at^2}{1+t^3})$.

I am asked to show that the line $x+y+a=0$ is an asymptote. So, I have to prove that when $t \to \infty$, the curve tends to that line. My doubt is: The limit of $\alpha(t)$ when $t \to \infty$ is $(0,0)$, how is it possible that the curve tends to the origin and at the same time to that line? How could I show that the line is in fact an asymptote of the curve?

 

[LCKurtz]

I am given the parametrized curve $\alpha:(-1,\infty) \to \mathbb R^2$ as $\alpha(t)=(\dfrac{3at}{1+t^3},\dfrac{3at^2}{1+t^3})$.

I am asked to show that the line $x+y+a=0$ is an asymptote. So, I have to prove that when $t \to \infty$, the curve tends to that line. My doubt is: The limit of $\alpha(t)$ when $t \to \infty$ is $(0,0)$, how is it possible that the curve tends to the origin and at the same time to that line? How could I show that the line is in fact an asymptote of the curve?

Your x and y coordinates will get large when $t \rightarrow -1$.