A Course of Higher Mathematics. Volume I by V. I. Smirnov and A. J. Lohwater (Auth.)

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

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Este reconocido libro aborda el precálculo desde una perspectiva novedosa y reformada que integra l. a. tecnologa de graficación como una herramienta esencial para el descubrimiento matemático y para l. a. solución efectiva de problemas. A lo largo del texto se explican las ecuaciones paramétricas, las funciones definidas por partes y l. a. notación de límite.

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We recall the definition of infinitesimal: for any given positive ε, there exists a value of the variable x, such t h a t for all subsequent values, | # | < ε . I t follows immediately, t h a t in proving t h a t a given variable x tends to zero, we can confine ourselves to considering only those values of x t h a t succeed a certain definite value of x, where this definite value can be chosen arbitrarily. 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.

We suppose t h a t x, y and z tend respectively to limits a, b and c. We show t h a t the sum tends to the limit a — b + c. We have by hypothesis [27]: x = a + a, y = b + ß, z = c + y , where a, ß, y are infinitesimals. We can write for the sum: x - V + z = (a + a) - (6 + ß) + (c + y) = = (a-b + c) + (a- β + γ). The first bracket on the right-hand side of this equation is a constant, and the second is an infinitesimal [26]. Hence: lim (x — y -\- z) = a — 6-)-c = lim x — lim y + lim z. 2. The limit of the product of a finite number of variables is equal to the product of their limits.

I n fact, both the equations y — f(x) and x = q>(y) give the same functional relationship between x and y. Suppose an arbitrary x is given in the direct function. If we mark off an interval from the origin 0 32 FUNCTIONAL BELATIONSHIPS AND THE THEORY OF LIMITS [20 along the axis OX, corresponding to the number x, then erect a perpendicular to OX from the end of this interval as far as its intersection with the graph, we obtain the value of y corresponding to the chosen x as the length of this perpendicular, with the corresponding sign.

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