# Folium of Descartes

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?

## Answers and Replies

LCKurtz
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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##.