Limits of rational functions

For a rational function $R(x)=\dfrac{p(x)}{q(x)}$, the technique of Section 9.2.5 (divide by the highest power of $x$ in the denominator) can be packaged into a single rule of thumb that avoids repeating the division every time, based purely on comparing the degrees of $p$ and $q$:

• If $\deg p(x)>\deg q(x)$: the numerator eventually dominates, and $\dfrac{p(x)}{q(x)}\to+\infty$ or $-\infty$ as $x\to\infty$ (the sign depending on the leading coefficients) — in particular the limit does not exist as a finite number.
• If $\deg p(x)<\deg q(x)$: the denominator eventually dominates, and $\displaystyle\lim_{x\to\infty}\frac{p(x)}{q(x)}=0$.
• If $\deg p(x)=\deg q(x)$: the two degrees cancel out under division, and $\displaystyle\lim_{x\to\infty}\frac{p(x)}{q(x)}=\frac{\text{coefficient of the highest power of }x\text{ in }p(x)}{\text{coefficient of the highest power of }x\text{ in }q(x)}$ — simply the ratio of the two leading coefficients. (This matches Section 9.2.5's Illustration 9.5: both degree $2$, leading coefficients $2$ and $1$, ratio $2$.) …