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To see this, let us consider the function sin(3x) again, the limit and the
asymptotic sequence The first term in such an expansion, if it exists, will be a
constant (corresponding to n = 0); but in this limit, so the constant is
zero. Perhaps the first term is proportional to for some n > 0; thus we examine

If we are to have (for some n and some constant c), then this limit is
to be L = 1. However, this limit does not exist—it is infinite—for every n > 0. Hence
we are unable to represent sin(3x), as with the asymptotic sequence proposed
(which many readers will find self-evident, essentially because sin(3x) ~ 3x as
If every in the asymptotic expansion is either zero or is undefined, then the
expansion does not exist.
Let us take this one step further; if we have a function, a limit and an appropriate
asymptotic sequence, then the coefficients, are unique. This is readily demonstrated.
From the definition of an asymptotic expansion, we have

consider

and take the limit to give

which determines each
Finally, the terms should not be regarded or treated as a series in
any conventional way. This notation is simply a shorthand for a sequence of
‘varies as’ calculations (as in (1.33), for example); at no stage in our discussion have
we written that these are the familiar objects called series—and certainly not convergent
series. Indeed, many asymptotic expansions, if treated conventionally i.e. select a value
and compute the terms in the series, turn out to be divergent (although,
exceptionally, some are convergent). Of course, numerical estimates are sometimes
relevant, either to gain an insight into the nature of the solution or, more often, to
provide a starting point for an iterative solution of the problem. Because these issues
may be of some interest, we will (in §1.4) deviate from our main development and
offer a few comments and observations. We must emphasise, however, that the thrust