TITLE: Why don't parabolas have asymptotes?
QUESTION [6 upvotes]: As given on the Wikipedia page,

an asymptote is a line which becomes the tangent of the curve as the  $x$ or $y$ cordinates of the curve tends to infinity.

Hyperbola has asymptotes but parabolas ( both being an open curve and a conic section) do not.
I am just curious to know why don't parabolas have an asymptote ? Is there any mathematical proof to show that ?
Thanks in advance:)

REPLY [3 votes]: If a parabola had an asymptote, then we could translate-rotate it so that the asymptote coincides with the $y$ axis.
Now, consider the general conic section (which, of course, includes all rotated parabolas):
$$A x^2 + B x y + C y^2 + D x + E y + F= 0 \tag1$$
Or
$$ y( C y  + B x + E) + A x^2 + D x + F=0 \tag2$$
and let's see under what conditions it could happen its graphic has the $y$ axis as an asymptote, i.e. that as $x\to 0$ $y \to \pm\infty$.
In that limit, we must have $$( C y  + B x + E) \to -\frac{F}{y} \to 0\tag3$$
But this requires $C=0$ and $E=0$
Then it cannot be a parabola (or an ellipse), only an hyperbola.