Rational Functions and End Behavior

Quotient dominance

A rational function $r(x)=\dfrac{p(x)}{q(x)}$ is a tug-of-war between two polynomials. For large $|x|$ only the leading terms matter, so the end behavior of $r$ is the end behavior of the ratio of leading terms. Compare degrees $n=\deg p$ and $m=\deg q$:

• $n
• $n=m$: neither side dominates; the limit is the ratio of leading coefficients $\frac{a_n}{b_m}$ — horizontal asymptote $y=\frac{a_n}{b_m}$.
• $n=m+1$ (top-heavy by one): the quotient of the division is linear — a slant asymptote $y=mx+b$, and $r(x)\to\pm\infty$.
• $n>m+1$: the quotient itself is a polynomial of degree $\ge2$; the graph hugs that polynomial at the ends.

AP Precalculus insists on limit notation. "Horizontal asymptote $y=3$" earns the point only when backed by a statement such as $\lim_{x\to\infty}r(x)=3$ and $\lim_{x\to-\infty}r(x)=3$.