Asymptote
An asymptote is a line that a curve approaches as it extends toward infinity or a specific value. It can be classified as linear or curvilinear. A linear asymptote can be horizontal, vertical, or slant, depending on its orientation. For instance, a horizontal asymptote occurs when the curve approaches a constant line as it heads toward positive or negative infinity. Conversely, a vertical asymptote is present when the curve gets arbitrarily close to a vertical line at a particular x-value, often caused by a denominator approaching zero in rational functions. Curvilinear asymptotes arise when a curve approaches a more complex function rather than a straight line. Rational functions, which are ratios of polynomials, can exhibit various types of asymptotes based on the degrees of their numerator and denominator. Notably, the behavior of curves near asymptotes can sometimes lead to intersections, although there is no consensus among mathematicians regarding the significance of such intersections.
Subject Terms
Asymptote
If a real-valued curve f(x) approaches a line y = mx + b arbitrarily closely as x approaches +∞ or −∞, or approaches the vertical line x = k arbitrarily closely as x approaches a real constant k, that curve f(x) is said to have a linear asymptote and the line is called the linear asymptote.
If a real-valued curve f(x) approaches another functiong(x) arbitrarily closely as x approaches +∞ or −∞, where g(x) is not a line, the curve f(x) is said to have a curvilinear asymptote and g(x) is called the curvilinear asymptote.
There is usually no ambiguity about whether an asymptote is linear or curvilinear, so the terms "linear" or "curvilinear" are frequently not stated. A linear asymptote is horizontal if it is the horizontal line y = k. It is vertical if it is the vertical line x = k. If it is neither horizontal nor vertical, then it is a slant, or oblique, asymptote.
Overview
For example, the real-valued function logex approaches the y-axis as x approaches 0 from the positive side. Hence the y-axis is a vertical asymptote for logex.
The hyperbolay = 1/x approaches the x-axis from above as x goes to +∞ and from below as x approaches −∞. Therefore, y = 1/x has the x-axis as a horizontal asymptote. In addition, y = 1/x approaches +∞ as x approaches zero from the left and approaches −∞ as x approaches zero from the right, thus y = 1/x has the y-axis as a vertical asymptote.
The function f(x) = x + 1/x approaches the line y = x as x approaches +∞ and as x approaches −∞, hence y = x is a slant (linear) asymptote. f(x) also has the y-axis as a vertical asymptote.
The hyperbola
,
which has both foci on the x-axis, has two slant asymptotes, namely
.
Asymptote for a Rational Function
No polynomial has an asymptote. However, rational functions (which are the quotients of two polynomials) can have asymptotes. If the degree of the numerator is less than the degree of the denominator, the x-axis is the horizontal asymptote. If the denominator goes to zero then there is a vertical asymptote x = k where k is the value at which the denominator becomes zero.
If the degree of the numerator is equal to the degree of the denominator, there is a horizontal asymptote y = r where r is the ratio of the leading coefficients of the numerator and denominator.
If the degree of the numerator is one greater than the denominator, there is a slant linear asymptote y = the quotient in the long division of the fraction.
If the degree of the numerator is at least two greater than the degree of the denominator, there is a curvilinear asymptote y = the quotient in the long division of the fraction with degree equal to the difference in the degrees of the denominator and numerator.
Intersections with the Asymptote
There is no established convention among mathematicians as to whether a curve can be an asymptote if the first function intersects it. For example, the curve y = e-x sin x (a damped oscillation, a sine wave decaying in amplitude) approaches the x-axis, which it crosses an infinite number of times whenever x is an integral multiple of π.
Bibliography
Aufmann, Richard , Vernon Barker, and Richard Nation. College Algebra. 7th ed. Belmont CA: Cengage Learning, 2010.
Blitzer, Robert. College Algebra. 6th ed. Upper Saddle River, NJ: Pearson, 2012.
Swokowski, Earl W., Jeffery A. Cole. Algebra and Trigonometry with Analytic Geometry. Belmont CA: Cengage Learning, 2011.