Rational functions and end behavior

Take r(x) = (5x³ − x) / (2x³ + 8). For large |x|, the numerator is essentially 5x³ and the denominator essentially 2x³, so r(x) is essentially 5x³ / 2x³ = 5/2. The lesser terms fade for exactly the reason they fade in polynomials: higher powers outgrow lower ones without bound.

That is the master move for every rational end-behavior question: replace top and bottom by their leading terms, then simplify the single power of x that remains. Constants, middle terms, and the location of any vertical asymptotes contribute nothing to the ends.

Let n be the numerator's degree and m the denominator's. The leading-term ratio leaves x to the power n − m, and three cases cover everything:

1. n < m (bottom-heavy): the ratio shrinks to 0, so the graph has horizontal asymptote y = 0.
2. n = m (balanced): the powers cancel, leaving the ratio of leading coefficients: horizontal asymptote y = a/b.
3. n > m (top-heavy): the ratio still grows, so there is no horizontal asymptote. Divide to find the quotient: if n − m = 1 the ends follow a slant line (the quotient), and if n − m ≥ 2 they follow that higher-degree polynomial quotient.

The rules are consequences, not incantations. "Ratio of leading coefficients" is only true when the degrees match, because only then do the powers of x cancel; reciting it elsewhere is how the classic wrong answers get generated.

A horizontal asymptote is a claim about the ends: as x → ±∞, the outputs approach the line. It is not a fence. In the middle of the graph the function may cross its horizontal asymptote, touch it, or wander back and forth over it; r(x) = x / (x² + 1) has horizontal asymptote y = 0 and crosses it exactly at the origin.

Contrast with vertical asymptotes, which the graph genuinely cannot touch, since the function is undefined there. The horizontal line constrains long-run behavior only. If a graph equals its asymptote value at some x, the line is still the asymptote, because the ends still approach it.

Horizontal asymptote y = L translates to a pair of statements: as x → ∞, r(x) → L, and as x → −∞, r(x) → L. For a rational function the leading-term ratio governs both directions at once, so a horizontal asymptote, when it exists, is the same line on the left and on the right.

In the top-heavy case, the honest statements are about unbounded growth: for r(x) = (x³ + 1)/(x − 2), dividing gives quotient x² + 2x + 4, so as x → ±∞ the graph hugs that parabola and r(x) → ∞ on both ends. Saying "no horizontal asymptote" is the start of the answer; naming the quotient the ends follow is the finish.