Asymptote Rules

Asymptote Rules Asymptote Rules In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Som e sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.

In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity.

In coordinate Geometry, an Asymptote of a curve is basically a line which approaches the curve arbitrarily so close that it tends to meet the curve at infinite, but it never intersects the line.

In classical mathematics it was said that an asymptote and the curve never meets but modernly such as in algebraic geometry, an asymptote of a curve is a line which is Tangent to that curve at infinity.

Let us see in the following graph:

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It shows 1/P, which has a Vertical Asymptote at P = 0 and a Horizontal Asymptote at Q = 0.

The line approaches the P-axis (Q=0), infinitely close but it never touches the axis.

A graph to show asymptote, here the x axis is denoted by ‘P’ and y axis is denoted

by Q There are two types of asymptotes: 1. vertical asymptotes 2. Horizontal asymptotes.

Let’s start with some Asymptote rules: As for vertical asymptotes they occur where the denominator equals zero. And for horizontal asymptotes, following rules are applicable: Horizontal asymptotes = Leading Coefficient / Leading Coefficient 1. If the degree of the numerator is greater than the degree of the denominator then there will be no horizontal asymptote. Even if it is greater by exactly 1 there can be a slant asymptote.

Horizontal asymptotes = Denominator Example: f(x) =

11x3 /(x2 + 7)

limx→¥f(x) = ¥

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limx→-¥f(x) = -¥ 2. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote will be the fraction of the leading coefficients.

Example: f(x) = 5x / 3x + 5 3. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote must be the line y = 0.

Example: f(x) = (x3 + 7x) /( x5 +18x4 –x)

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