SOLUTION: Please explain in words how to determine the equation of the horizontal asymptote for the graph of a rational function.

Question 437467: Please explain in words how to determine the equation of the horizontal asymptote for the graph of a rational function.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

First you have to decide whether a horizontal asymptote exists.

If the degree of the numerator polynomial is greater than the degree of the denominator polynomial, then there is no horizontal asymptote. (If the degree of the numerator is exactly one greater than the degree of the denominator, then there is a slant or oblique asymptote with equation y = mx + b where mx + b is the quotient excluding the remainder resulting from performing polynomial long division of the denominator into the numerator).

If the degree of the numerator polynomial is less than the degree of the denominator then the horizontal asymptote is the line x = 0, which is to say the x-axis. (Think about it -- if the denominator has a greater degree, the denominator gets bigger much faster than the numerator, and the whole fraction tends toward zero)

If the degree of the numerator polynomial is equal to the degree of the denominator polynomial then the horizontal asymptote is the line

where

is the lead coefficient of the numerator polynomial and

is the lead coefficient of the denominator polynomial. (Think about it -- with the numerator and denominator the same degree, as the value of the independent variable gets very large, the high order term is the only part of the polynomial that makes much difference, and since the high order terms differ only in their coefficients, it is this ratio of coefficients to which the value of the fraction tends)

John

My calculator said it, I believe it, that settles it

The Out Campaign: Scarlet Letter of Atheism

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