By V. I. Smirnov and A. J. Lohwater (Auth.)

**Read Online or Download A Course of Higher Mathematics. Volume I PDF**

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**Additional resources for A Course of Higher Mathematics. Volume I**

**Sample text**

Hence: The denominator of the fraction on the right of this equation is the product of two factors, and tends to b2. e. the fraction is bounded. The term (aß — ba) is an infinitesimal. Hence [26], the difference a/b — x/y is an infinitesimal, and ,. x a lima; lim — = -=-- = -y. y b limy The theorems proved are of fundamental importance in the theory of limits. The proofs have been given for the general case, and not for the case of enumerated variables, as when proving the properties of infinitesimals.

In the general case of non-enumerated variables we can look on x, y, z as functions of some ordered variable t: x = x(t)r y = y(t), z = z(t). Variables x, y, z are themselves ordered, so that if t = V precedes t = £", then x(t') precedes x{t"), etc. The sum w(t) = x(t) + y(t) + z(t) , obtained by adding the x, y, and z corresponding to the same value of t, is also ordered. The proof is as above, for enumerated variables. In this latter case, t has the role of subscript; or the subscript can be looked on as an increasing, integral t.

The proofs have been given for the general case, and not for the case of enumerated variables, as when proving the properties of infinitesimals. But the remark we made when proving the first property of infinitesimals should be borne in mind. Take the case of a product. We take x and y as functions of some ordered variable t : x = = x(t)\ y = y(t). Then x and y are themselves ordered variables. The same can be said of their product: w(t) = x(t) · y(t). The subscript plays the part of t in enumerated variables, increasing through integral values.