Question

Find the asymptotes of y^2(a+x) =x^2(3a-x) parallel to coordinate axes

Answer 1

$Horyzontal\,\,asymptotes:\\\underset{x\rightarrow \infty}{\lim}y\left( x \right) =C\\C^2=\underset{x\rightarrow \infty}{\lim}\frac{x^2\left( 3a-x \right)}{a+x}=\infty \\No\,\,horyzontal\,\,asymptotes\\Vertical\,\,asymptotes:\\\underset{x\rightarrow x_0}{\lim}y\left( x \right) =\infty \Rightarrow \underset{x\rightarrow x_0}{\lim}y^2\left( x \right) =+\infty \Rightarrow \\\Rightarrow \underset{x\rightarrow x_0}{\lim}\frac{x^2\left( 3a-x \right)}{a+x}=+\infty \Rightarrow \left\{ \begin{array}{c} a+x=0\\ 3a-x\ne 0\\\end{array} \right. \\Thus\,\,for\,\,a=0 no\,\,vertical\,\,asymptotes\\For\,\,a\ne 0: x=-a\,\,-vertical\,\,asymptote$