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retains the first two (in size), plus the first exponentially small term and then the first
which involves both the sequences and these two specifications define
N and M here.
To proceed, we write in terms of X and expand:

where we have retained terms O(1) and (which constitute M here). Correspond-
ingly, we write (1.66), in terms of x and expand:

where we have retained terms O(1), and which is N here.
Finally, we write (1.68), say, in terms of X:

which is identical to (1.67): the two expansions match (to this order).

This example makes clear that we need not restrict the matching to the first N and
first M terms (in size)—but we must accurately identify N and M and then retain
precisely these terms when the expansions are further expanded. Of course, as we
have seen, there is also no requirement to work to the same number of terms in each
original asymptotic expansion. However, we should offer one word of warning. In the
above example, we included the term which arises from the two otherwise
disjoint asymptotic sequences and To retain such terms, we must ensure
that all terms that might contribute are also included; here, these are and We
may not elect to keep this term alone, and ignore those in and To take this one
step further, if we decided, in this example, to include the term we must also
include the terms of order 1, (And for this same reason,
it is obvious that we must retain all terms of orders 1, when we wish to go as
far as A number of examples of expanding and matching can be found in Q1.19,
1.20.