When studying the end behavior of rational functions, one often asks: can a rational function have more than one horizontal asymptote? This question touches on the fundamental definition of a rational function, the concept of limits at infinity, and the geometric interpretation of horizontal asymptotes. In the following sections we will explore each of these ideas in depth, provide clear examples, and explain why the answer is a definitive no for a pure rational function.
What Is a Rational Function?
A rational function is any function that can be written as the quotient of two polynomials:
[ f(x)=\frac{P(x)}{Q(x)}, ]
where (P(x)) and (Q(x)) are polynomials and (Q(x)) is not the zero
