<s>[INST] Finding the asymptotes of $(y-a)^2(x^2-a^2)=x^4+a^4$. Problem: Finding the  asymptotes of $(y-a)^2(x^2-a^2)=x^4+a^4$.  
My efforts:  $(y-a)^2(x^2-a^2)=x^4+a^4\implies (y-a)^2=\dfrac{x^4+a^4}{x^2-a^2}\implies x=\pm a$ are vertical asymptotes. What are the remaining asymptotes?
 [/INST] This doesn't satisfy 
the usual definition 
of asymptote but ...
If
$(y-a)^2(x^2-a^2)=x^4+a^4
$
then
$\begin{array}\\
(y-a)^2
&=\dfrac{x^4+a^4}{x^2-a^2}\\
&=\dfrac{x^4-a^2x^2+a^2x^2+a^4}{x^2-a^2}\\
&=\dfrac{x^2(x^2-a^2)+a^2(x^2+a^2)}{x^2-a^2}\\
&=x^2+\dfrac{a^2(x^2-a^2+2a^2)}{x^2-a^2}\\
&=x^2+a^2+\dfrac{2a^4}{x^2-a^2}\\
\text{so}\\
y-a
&=\sqrt{x^2+a^2+\dfrac{2a^4}{x^2-a^2}}\\
&=x\sqrt{1+a^2/x^2+\dfrac{2a^4}{x^2(x^2-a^2)}}\\
&\approx x(1+a^2/(2x^2)+\dfrac{2a^4}{2x^2(x^2-a^2)})
\qquad\text{for large }x\\
&= x+a^2/(2x)+O(1/x^3)\\
\end{array}
$
so $y = x+a$
is a "asymptote".
</s>