Asymptote.

Asymptote. Asymptote, a straight line associated with a curve, having the property that as a point moves along the curve to infinity, the distance from the point to the straight line tends toward zero. Some definitions of an asymptote require that after a certain distance along the line, the curve and the line do not intersect. In figure 1, the lines x = 0 and y = x are vertical and slant asymptotes, respectively, to the graph of y = x + 1/ x. The x and y axes are horizontal and vertical asymptotes to the graph of the hyperbola y = 1/x. Horizontal asymptotes can sometimes appear in population growth graphs when the growth of the population is inhibited by some factor, such as a limited amount of food. This type of growth is often modeled by the logistic growth function where P(t) is the size of the population at time t. As the graph of P(t) in figure 2 shows, the population continues to grow closer and closer to the maximum number L without ever reaching it. The line y = L is a horizontal asymptote to the graph of P(t). Asymptotes can be defined formally using the idea of limits in calculus. The line y = L is said to be a horizontal asymptote to the graph of the function y = f(x) if the limit of f( x) as x tends toward infinity or toward minus infinity is equal to L. The line x = L is said to be a vertical asymptote if the limit of f(x) as x approaches L (from either the right or the left of L) is equal to either infinity or minus infinity. Contributed By: William James Ralph Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.