Horizontal asymptotes can be confusing for students who are learning about rational functions.

They show us what happens to a function as ( x ) gets really big or really small. But figuring them out can be a bit tricky.

Common Problems:

• It can be hard to tell the comparison between the degrees of the numerator (the top part) and the denominator (the bottom part).
• Some students think horizontal asymptotes mean a function can never go past that line, which isn't true.
• People may use limits incorrectly when trying to find the asymptotes.

Simple Solutions:

• First, get to know the degrees of the polynomials:
  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is ( y = 0 ).
  • If the degree of the numerator is the same as the degree of the denominator, the horizontal asymptote is ( y = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}} ).
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Practicing different examples can really help make these ideas clearer!