Is there a systematic way of determining the "correct" asymptotic approximation? Consider these two quadratic equations,
$$\text{i)} \quad x^2+4x-5-\epsilon$$
$$\text{ii)} \quad \quad x^2+(4+\epsilon)x+4-\epsilon = 0$$
If we attempt to find an asymptotic approximation of the form 
$$x = x_0 + \epsilon x_1+...$$
for i) this works out fine, in  ii) we get to an equation $-3 = 0$, which is rubbish.
From using the quadratic formula we find
$$\text{i)} x = -2 \pm \sqrt{9+\epsilon}$$
$$\text{ii)} \quad x = \frac{-4-\epsilon \pm \sqrt{\epsilon}\sqrt{\epsilon+12}}{2}$$
The radical epsilon factor leads me to beleive that we should try an approximation of the form 
$$x = x_0 + \sqrt{\epsilon} x_1+ \epsilon x_2 + ...$$
This works out fine. From this I would like to know if 

Was this the correct way to deduce the new form of the asymptotic approximation? Is is there a way to "spot" that a standard asymptotic approximation is going to fail? For instance, above I did not have any idea that it was going to fail until it failed!


Compute the differential of the equation.


*

*$(2x+4)dx-d\epsilon=0$. This shows that $\dfrac{dx}{d\epsilon}$ is finite a the roots.

*$(2x+4+\epsilon)dx+(x-1)d\epsilon=0$. At the double root $x=-2$, $\dfrac{dx}{d\epsilon}$ is infinite so that an "ordinary" approximation (entire function) cannot work.
