The method of integration can be traced back to the Egyptians, in the Moscow Mathematical Papyrus circa 1800 BC, which gives the formula for finding the volume of a pyramidal frustrum.[1] Greek geometers are credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create. Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See Archimedes' Quadrature of the Parabola and Archimedes on Spheres & Cylinders.) It was not until the time of Newton that these methods were made obsolete. It should not be thought that infinitesimals were put on rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek Mathematicians accept a proposition as true. (See Archimedes' use of infinitesimals.) The method of exhaustion was rediscovered in China by Liu Hui in the 3rd century AD, who used it to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.[1]

Infinity Limits

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Author: Surachai U.
Created Date: 2007-03-03 03:03:03
Last Update: 2007-06-13 11:01:00

Evaluating limits at infinity for rational functions There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x): -If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients; -If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q; -If the degree of p is less than the degree of q, the limit is 0. If the limit at infinity exists, it represents a horizontal asymptote at x = L. Polynomials do not have horizontal asymptotes; they may occur with rational functions.