By V. I. Smirnov and A. J. Lohwater (Auth.)
Read or Download A Course of Higher Mathematics. Volume I PDF
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Additional info for A Course of Higher Mathematics. Volume I
Concerning this, it is useful in the theory of limits to add a rider to the definition of a bounded magnitude, viz, there is no need to demand t h a t | y \ < M for all values of y ; it is sufficient to take the more general definition: a magnitude y is said to be bounded, if there exists a positive number M and a value of y, such that \ y j < M for all subsequent values. The proof of the second property of infinitesimals remains unchanged with this definition of a bounded magnitude. For an enumerated variable, the first definition of a bounded magnitude follows from the second, so t h a t the second is not less 27] THE LIMIT OF A VARIABLE 49 general.
62 FUNCTIONAL RELATIONSHIPS AND THE THEORY OP LIMITS [28 Furthermore, there is no need to consider all the values of a variable x when defining its limit; we need only take values subsequent to some arbitrarily given value. Another point: if a variable x tends to a limit a, it will differ from a by as little as is desired, after a certain initial moment of its variation, and hence it is all the more a bounded variable. An ordered variable does not always have a limit, as already mentioned. 111, .
From (6), the point Mx (xv yx) also lies on the parabola. The proof is similar for the remaining points. e. the abscissae of the points of intersection are solutions of the equation : x \* V* 1 h- b s 0 / fi( ) = M ) · 1* <*,-'υ / TB 1 -r x ι 'y ώ-Qâ} \-2 \r *i 1 f J 2 -/;y ^ ια This fact can readily be used · solve a qudaratic equation approximately. Having constructed as accurately as possible the graph of the parabola y = x2 (6J on a piece of millimetre graph paper we can now find the roots of the quadratic equation x2 = ρχ + q (7) as the abscissae of the points of intersection of the parabola (6X) and the straight line y = px + q.
A Course of Higher Mathematics. Volume I by V. I. Smirnov and A. J. Lohwater (Auth.)