Equivalent Replacement Rules of Local Infinitesimal in Limit Mathematical Operation

Equivalent replacement of infinitesimal is one of the efficient ways to simplify limit mathematical operation,which generally applies to calculate the infinitesimal ratio.Through the comparison and analysis of the original and new formula which is from the equivalent substitution of an infinitesimal in the infinitesimal formula,we can conclude necessary and sufficient conditions that the new formula is equivalent to the original one.Based on this,a further conclusion can be made that as long as the two infinitesimal items in the formula are not the same order infinitesimal with the ratio of-1,the new formula must be equivalent to the old one.Thus,it provides a judgment evidence for weather the individual infinitesimal can be replaced equivalently in the Numerator or denominator formula to simplify the calculation in limit mathematical operation.