Finding horizontal asymptotes in a limit function

Question

Find the horizontal asymptote(s) of

g(x)= x¹²−2x⁴+2 ÷ x⁹−x⁶−x

If there are none, use limit notation to describe the end behavior.

Answer 1

Let n be the degree of the polynomial in the numerator and let m be the degree of the polynomial in the denominator. You can follow these simple rules to find if there is a horizontal asymptote and determine what it is:

1. If n < m, then the x-axis is the horizontal asymptote
2. If n = m, then the line g(x) = a/b is the horizontal asymptote where a and b are the coefficients of the highest order terms in the numerator and denominator, respectively
3. If n > m, then there is no horizontal asymptote

Describing the end behavior of the function using limit notation entails simply finding the limit as x approaches both infinity and negative infinity. In this case, the easiest way to figure this out is to consider the degrees of the numerator and denominator and see which one dominates as x approaches positive and negative infinity (this is because both numerator and denominator approach infinity individually). If the dominating term is in the numerator, then the limit is equal to infinity (positive or negative depending on your limit). And if the dominating term is in the denominator, the the limit is equal to zero. If the degrees are the same, then the limit is equal to the coefficients of the leading terms (ie: a/b). These last two facts are essentially repeating the rules for horizontal asymptotes. If the horizontal asymptote is the x-axis (n < m), then the function approaches zero from both directions. If the horizontal asymptote is g(x) = a/b, (n = m), then the function approaches that value from both directions.