The terms infinity and infinite have a variety of related meanings in mathematics. The adjective finite means “having an end,” so infinity may be used to refer to something having no end. In order to give a precise definition, the mathematical domain of discoursemust be specified.
Set theory provides a simple and basic example of an infinite collection—the class of natural numbers, or positive integers. A fundamental property of positive integers is that after each integer there follows a next one, so that there is no last integer. Now it is necessary in mathematics to treat the collection of all positive integers as an entity, and this entity is the simplest infinity, or infinite collection.
The term infinity appears in mathematics in a different sense in connection with limits of functions. For example, consider the function defined by y = 1/x. When x tends to 0, yapproaches infinity, and the expression may be written as shown below. 
Precisely, this means that for an arbitrary number a > 0, there exists a number b > 0 such that when 0 < x < b, then y > a, and when −b < x <>y < −a. This example indicates that it is sometimes useful to distinguish +∞ and −∞. The points +∞ and −∞ are pictured at the two ends of the y axis, a line which has no ends in the proper sense of euclidean geometry.
In geometry of two or more dimensions, it is sometimes said that two parallel lines meet at infinity. This leads to the conception of just one point at infinity on each set of parallel lines and of a line at infinity on each set of parallel planes. With such agreements, parts of euclidean geometry can be discussed in the terms of projective geometry. For example, one may speak of the asymptotes of a hyperbola as being tangent to the hyperbola at infinity.
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