关于无穷小量代数和的等价代换的注记

A Note on Equivalent Substitution of Algebraic Sum of Infinitesimal Quantities

当两个无穷小量商的广义极限存在且不为-1时,等价无穷小量代换可用于求和的极限.这个结论在两个无穷小量商的极限为-1或广义极限不存在时失效.针对这一情形,给出了无穷小量代数和可以等价代换的两个充分条件,并举例说明应用它们可以更方便地求一些极限问题.

When the generalized limit of the quotient of two infinitesimal quantities exists and is not-1,the equivalent infinitesimal substitution can be used in solving limit with additive factors.This result is invalid when the limit of the two infinitesimal quotient is equal to-1 or the generalized limit dose not exist.Under this case,two sufficient conditions on the equivalent substitution of algebraic sum of infinitesimal quantities are given.Some examples are given to show that they provide a more convenient application in solving some limit problems.