How do I know what scales to look for for asymptotic expansion? E.g. cubic case? How do I know what scales to look for for asymptotic expansion? E.g. cubic case?
How do I check for their validity?
Such as in the example here:
http://www.macs.hw.ac.uk/~simonm/ae.pdf
where they find that
$x^3-x^2-(1+\epsilon)x+1=0$
has double root at $x=1$ and it suggests them to try an expansion in powers of $\epsilon^{1/2}$.
But one could want to solve e.g.
$x^3-5x^2+4x+\epsilon=0$
Given here:
http://sycon.rutgers.edu/~speer/528s17/Perturbation-DG.pdf
which has roots $x=0,x=1,x=4$.

In the first equation you have
$$
(x+1)(x-1)^2=ϵx\iff (x-1)^2=\frac{ϵx}{1+x}
$$
Close to $x=1$ this gives approximations for the roots there of
$$
x=1\pm\sqrt{\frac{ϵx}{1+x}}\approx 1\pm\sqrt{\fracϵ2}
$$
For the root close to $-1$ you get similarly
$$
x=-1+\frac{ϵx}{(1-x)^2}\approx -1+\fracϵ4
$$

For the second equation goes the same,
$$
x(x-1)(x-4)=-ϵ
$$
gives approximations close to the roots of the undisturbed equation of
$$
-\fracϵ4,~~1+\fracϵ3,~~ 4-\fracϵ{12}
$$
