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E1.7 Another example of
Here, we are given

with for x = O(1) we write

and then two applications of the binomial expansion yields

The domain of f is and so we must consider and in either
case the asymptotic expansion (1.49) breaks down. For the breakdown occurs
where (from for the breakdown is where
(from In the former case, we introduce to give

(which, we note, recovers the correct value on X = 0). For the other breakdown, we
introduce and so

Thus the function requires three different asymptotic expansions, valid for different
sizes of x, and two of these have been determined by examining the breakdown. (We
note that these choices are evident from the original function, although this is not
how we deduced the scalings in this example.) Furthermore, expansion (1.50) is valid
as and expansion (1.51) is valid for there are no further breakdowns
(based on the information available in these asymptotic expansions).

Before we continue the discussion of these ideas, and their consequences, we must
adjust the definition of an asymptotic expansion with a parameter; see (1.42). We
have already encountered functions such as these cannot be represented