Rational Functions and End Behavior

What AP Precalc asks here

For a rational function $r(x) = p(x) / q(x)$, the end behavior is determined by the relationship between the degrees of $p$ and $q$. When the denominator outranks the numerator, $r \to 0$. When degrees match, $r$ approaches the ratio of leading coefficients. When the numerator outranks the denominator by exactly 1, polynomial long division produces a slant asymptote.

Read the degrees of numerator and denominator, then pick the matching case.

Divide $p(x)$ by $q(x)$. The polynomial quotient is the slant asymptote; the remainder/divisor term tends to zero.

Once the asymptote is known, look at the sign of $r(x) - (\text{asymptote})$ for large $x$ to decide whether the function approaches from above or below.