  A generalization of the definition of a one-dimensional improper integral
with a finite limit is presented. The new definition extends the range of valid
integrals to include integrals which were previously considered to not be
integrable. This definition is shown to be equivalent to the infinite limit
definition presented in "Generalized improper integral definition for infinite
limit" (arXiv:0805.3559) via a particular change of variable of integration.
The definition preserves linearity and uniqueness. Integrals which are valid
under the conventional definition have the same value under the new definition.
Criteria for interchanging the order of integration and differentiation, and
for interchanging the order with a second integration, are obtained. Examples
are provided.
