Often, among several infinitesimals that take part in the same process of variation, one of them, let us say y, is assumed to be the principal, and the rest are compared with it. If y is infinitesimal here, then it is said that z is an infinitesimal of a higher order than y. The latter fact is often written in the form z = o(y), which reads “ z is o small with respect to y”. The variable z is called infinitesimal with respect to y if z/y is infinitesimal.

In considering several variables involved in the same process of variation, the variables y and z are called equivalent if lim (z/y) = 1 if in this case y is infinitesimal, the y and z are called equivalent infinitesimals. The theory of the infinitesimal is one of the methods of constructing a theory of limits. Thus, the concept of an infinitesimal can serve as the basis of a general definition of the limit of a variable. If the limit of the variable y is finite and equal to a, then lim (y – a) = 0, and conversely. For example, the quantity y = 1/ x is an infinitesimal for an argument x that approaches infinity, but for an x that approaches zero it proves to be infinitely large. In order that the concept of an infinitesimal may have an exact meaning, it is necessary to indicate the process of variation in which the given quantity becomes an infinitesimal. In mathematics, a variable quantity that approaches a limit equal to zero.

Infinitesimals hsmr