The Definition of Limiting Equality

Abstract

IN teaching the calculus to students of applied mathematics and physics I have found that the definitions of limiting value and of limiting equality given in practically all our text-books are unsatisfactory, and in my opinion inadequate. According to these books the test of limiting equality of two magnitudes is that their difference shall become less than any assignable quantity, however small. But this condition is satisfied by any two quantities whatever if they vanish simultaneously, and it affords no justification for the use of statements such as dy=f(x)dx; on the other hand, if the quantities remain finite in the limit the test appears; scarcely to be necessary or j useful in teaching elementary classes.