  A nice example illustrating this distinction is the following: Consider the quadratic equation \(x^2 + x - 9\varepsilon = 0\). Think of \(\varepsilon\) as a small expansion or perturbation parameter. The equation has two roots for any value of \(\varepsilon\) as \(\varepsilon \rightarrow 0\). In a well-defined sense, the solutions to this quadratic equation as \(\varepsilon \rightarrow 0\) smoothly approach solutions to the “unperturbed” \((\varepsilon = 0)\) equation \(x^2 + x = 0\); namely, \(x = 0, -1\). On the other hand, the equation \(x^2\varepsilon + x - 9 = 0\) has two roots for any value of \(\varepsilon \gt 0\) but has for its “unperturbed” solution only one root; namely, \(x = 9\). The equation suffers a reduction in order when \(\varepsilon = 0\). Thus, the character of the behavior in the limit \(\varepsilon = 0\) differs fundamentally from the character of its limiting behavior. Not all singular limits result from reductions in order of the equations. Nevertheless, these latter singular cases are much more prevalent than the former.