Horizontal and Vertical Asymptotes

Vertical Asymptote A term which results in zero in the denominator causes a vertical asymptote when the function is graphed, providing that the function is in its lowest terms. The vertical asymptote is found at the term which causes the zero in the denominator.

Example 1 Find the vertical asymptote in the following function:

Note that x = 5 results in a zero in the denominator. The fraction is in simplest form. Therefore, there is a vertical asymptote at x = 5

Example 2

would appear to have a vertical asymptote at x = -1. The reason that it does not is that the fraction may be re written as

This simplifies to (x-1), which does not result in a zero in the denominator. This does not result in a vertical asymptote.

Horizontal Asymptotes Horizontal asymptotes, when they exist, are determined by the value approached by the function as x gets either extremely large or extremely small. When graphed, asymptotes are expressed as dashed lines which the values graphed approach but never meet

Three types of rational expressions There are three types of rational expressions, as determined by the relationships of the greatest power in the numerator and denominator. If the greatest power in the denominator is greater than the greatest power in the numerator, there is a horizontal asymptote at y = 0

Example 3 The greatest power in the numerator is 1. The greatest power in the denominator is 2. Therefore, as |x| gets larger and larger, there is a horizontal asymptote at x = 0

If the greatest power in the numerator is equal to the greatest power in the denominator, there is a horizontal asymptote at the ratio of the leading coefficients.

Example 4 The greatest power in the numerator is one. The greatest power in the denominator is one. The ratio of the leading coefficients is 3/1, or 3. There is a horizontal asymptote at 3.

Proof using a table of values XF(x)

Proof using Algebra: Multiply numerator and denominator by 1/x Note that the numerator simplifies to 3 – 4/x, and the denominator simplifies to 1 + 7/x At extremely high and extremely low values of x, the values of -4/x and 7/x approach zero, resulting in a horizontal asymptote at 3/1, or 3

If the greatest power in the numerator is one greater than the greatest power in the denominator, an oblique asymptote generally results.