When limit problem ($\epsilon =0$) differs in an important way from the limit $\epsilon \rightarrow 0$). For example, a root is lost, or a derivative is lost in a DE.

Problems that are not singular, are called regular

For algebraic equations, often when a root is lost, it's because it goes to $\infty$ as $\epsilon \rightarrow 0$.

It's first term in the expansion may be then $\frac{1}{\epsilon}$, for example.

For the iterative method, different functions $g$ may be needed to find different perturbed roots of an algebraic equation, so that condition $g'(x^*; \epsilon) \rightarrow 0$ as $\epsilon \rightarrow 0$ is satisfied.

Regularization method

Scale variables so that the problem becomes regular.

For instance, if first term in the expansion is $\frac{1}{\epsilon}$, rescale $x =X/\epsilon$.

Indeed, the problem of finding the correct starting point for an expansion, is equivalent to the problem of finding a suitable scaling to regularize the singular problem.

Finding the right scaling

Systematic approach:general rescaling

Let $x=\delta(\epsilon)X$, with $X$ strictly of order $1$ as $\epsilon \rightarrow 0$

Vary $\delta$ from small to large to identify dominant balances in which at least two terms are of the same order of magnitude as $\epsilon \rightarrow 0$, while others are smaller. Scalings that result on dominant balances are called distinguished limits

Alternative approach: pairwise comparison

quicker when there are a small number of terms. Try to create dominant balance between terms pairwise, and see if you can get it, consistently. That way you can find the dominant limits.