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for X = O(1) as Then the value of (1.55), on X = 0, recovers the correct
boundary value, (1.54). Furthermore, the asymptotic expansion (1.55) is not uniformly
valid in as it breaks down where i.e. x = O(l), which
is the variable previously used to generate (1.53).

This example prompts a number of additional and important observations. For the
purposes of determining a relevant scaling, from the breakdown of an expansion, it is
quite sufficient for this to occur only in one direction i.e. expansion A breaks down,
producing a scaling used to obtain expansion B, but B does not necessarily break down
to recover the scaling used in A. This is evident here when we compare (1.53) and
(1.55); expansion (1.55) breaks down, but (1.53) does not. Indeed, as we have seen,
there is no clue in (1.53) that we have a problem—this is only evident when we return
to the original function, (1.52), or we already have available the expansion (1.55). It
is possible to extend the asymptotic expansion given in (1.53) and thereby make plain
the nature of the breakdown; this will prove to be a useful adjunct in some of our later
work.
The breakdown that must exist in the expansion of (1.52), for x = O(l) as
arises from the exponential term. However, to include this term in the asymptotic
expansion would mean, apparently, the inclusion of all terms based on the sequence
because is smaller than for any n (see Q1.5). Of course, there is
no need to write them down explicitly; we could indicate their presence by the use of
an ellipsis (i.e. ...) or, which is the usual practice, simply to state which terms we will
retain. So we might expand (1.52), for x = O(1) and retaining O(1),
and terms only, to give

In a sense, the omitted terms are understood, but not explicitly included and, more
significantly, any further manipulation of (1.55) that we employ will use only the terms
written down. It is clear that, with or without the use of ellipsis, the expansion (1.55)
breaks down as for, eventually, the exponential term becomes O(1)—the
same size as the first term. (The fact that there is an infinity of breakdowns, where x
satisfies for each n, is immaterial; we have a well-defined breakdown the
other way—from (1.55)—which is sufficient. Further, an intimate relation between
different expansions of the same function, which we discuss later, shows that this
infinity of breakdowns plays no rôle.)
The inclusion of the exponentially small term in (1.55) may seem superfluous,
and it is in a strictly numerical sense, but it contains important information about the
nature of the underlying function (and it helps us better to understand the breakdown).
Because we are interested in the behaviour of functions (as and not simply
numerical estimates, we shall retain such terms when they provide useful and relevant
information.