Question

Find the asymptotes of the curve 𝑥2𝑦 − 𝑥𝑦2 + 𝑥𝑦 + 𝑦2 + 𝑥 − 𝑦 = 0.

Answer 1

$x^2y-xy^2+xy+y^2+x-y=0\\Horyzontal\,\,asymptotes:\\x=\frac{y^2-y-1\pm \sqrt{\left( y^2-y-1 \right) ^2-4y\left( y^2-y \right)}}{y}\\y\rightarrow C,x\rightarrow \infty \\C=0\\\underset{y\rightarrow 0}{\lim}\frac{y^2-y-1-\sqrt{\left( y^2-y-1 \right) ^2-4y\left( y^2-y \right)}}{y}=\left[ \frac{-2}{0} \right] =\infty \\y=0-horyzontal\,\,asymptote\\Vertical\,\,asymptotes:\\\left( 1-x \right) y^2+\left( x^2+x-1 \right) y+x=0\\y=\frac{-x^2-x+1\pm \sqrt{\left( x^2+x-1 \right) ^2-4x\left( 1-x \right)}}{1-x}\\x\rightarrow C,y\rightarrow \infty \\C=1\\\underset{x\rightarrow 1}{\lim}\frac{-x^2-x+1+\sqrt{\left( x^2+x-1 \right) ^2-4x\left( 1-x \right)}}{1-x}=\left[ \frac{2}{0} \right] =\infty \\x=1-vertical\,\,asymptote$