  It is our aim to establish a general analytic theory of asymptotic expansions
of type f(x)=a_1 phi_1(x)+dots+ a_n phi_n(x)+o(phi_n(x)), x tends to x_0 (*),
where the given ordered n-tuple of real-valued functions phi_1 dots,phi_n forms
an asymptotic scale at x_0. By analytic theory, as opposed to the set of
algebraic rules for manipulating finite asymptotic expansions, we mean
sufficient and/or necessary conditions of general practical usefulness in order
that (*) hold true. Our theory is concerned with functions which are
differentiable (n-1) or n times and the presented conditions involve
integro-differential operators acting on f, phi_1, dots, phi_n. We essentially
use two approaches; one of them is based on canonical factorizations of
nth-order disconjugate differential operators and gives conditions expressed as
convergence of certain improper integrals, very useful for applications. The
other approach, valid for (n-1)-time differentiable functions starts from
simple geometric considerations (as old as Newton's concept of limit tangent)
and gives conditions expressed as the existence of finite limits, as x tends to
x_0, of certain Wronskian determinants constructed with f, phi_1, dots, phi_n.
There is a link between the two approaches and it turns out that the integral
conditions found via the factorizational approach have striking geometric
meanings. Our theory extends to general expansions the theory of polynomial
asymptotic expansions thoroughly investigated in a previous paper. In the first
part of our work we study the case of two comparison functions phi_1, phi_2.
The theoretical background for the two-term theory is much simpler than that
for n >=3 and, in addition, it is unavoidable to separate the treatments as the
two-term formulas must be explicitly written lest they become unreadable.
