Based on this question, how does one analytically compute the asymptotes for the equation defined by:
$$(x+y+2)(x+y)^2 = x+9y-2 ?$$
I can easily plot the curve...
and can manipulate equation for large $x$, but is there a more direct function that provides the asymptotes?
I think this is a good use case for AsymptoticSolve:
eq = (x + y + 2) (x + y)^2 == x + 9 y - 2;
AsymptoticSolve[eq, y, x -> Infinity, Reals]
(*{{y -> -(2/(1/x)^(1/3)) - x}}*)
AsymptoticSolve[eq, y, {x, ∞, 5}, Reals]
$\endgroup$
Linear asymptotes are determined by the top degree terms, which can be obtained by homogenizing with a new variable and then setting it to zero.
poly = (x + y + 2)*(x + y)^2 - (x + 9 y \[Minus] 2);
homogpoly = ResourceFunction["PolynomialHomogenize"][poly, {x, y}, z]
topform = homogpoly /. z -> 0
(* Out[94]= x^3 + 3 x^2 y + 3 x y^2 + y^3 + 2 x^2 z + 4 x y z +
2 y^2 z - x z^2 - 9 y z^2 + 2 z^3
Out[95]= x^3 + 3 x^2 y + 3 x y^2 + y^3 *)
Now solve for one variable in terms of the other.
Union[Solve[topform == 0, x]]
(* Out[96]= {{x -> -y}} *)
eq = (x + y + 2) (x + y)^2 == x + 9 y - 2;AsymptoticSolve[eq, y, x -> Infinity, Reals]? $\endgroup$