2.4.3 Horizontal and Oblique Asymptotes

Determining horizontal asymptotes using degree comparison and finding oblique asymptotes through polynomial long division.

定义

Horizontal and oblique asymptotes are lines that a rational function approaches as \(x\) approaches infinity or negative infinity. A horizontal asymptote is a horizontal line \(y = c\) that the graph of a function approaches as \(x \to \infty\) or \(x \to -\infty\). An oblique (or slant) asymptote is a non-horizontal, non-vertical line \(y = mx + b\) (where \(m \neq 0\)) that the function approaches as \(x \to \infty\) or \(x \to -\infty\). For a rational function \(f(x) = \frac{p(x)}{q(x)}\) where \(p(x)\) and \(q(x)\) are polynomials, the existence and form of these asymptotes depend on the degrees of the numerator and denominator polynomials.

核心公式

• \(\text{If } \deg(p) < \deg(q), \text{ then } y = 0 \text{ is a horizontal asymptote}\)
• \(\text{If } \deg(p) = \deg(q), \text{ then } y = \frac{a_n}{b_m} \text{ is a horizontal asymptote, where } a_n \text{ and } b_m \text{ are leading coefficients}\)
• \(\text{If } \deg(p) = \deg(q) + 1, \text{ then an oblique asymptote exists: } y = mx + b \text{ where } mx + b \text{ is the quotient from polynomial long division}\)
• \(\text{If } \deg(p) > \deg(q) + 1, \text{ then no horizontal or oblique asymptote exists}\)
• \(f(x) = \frac{p(x)}{q(x)} = (mx + b) + \frac{r(x)}{q(x)}, \text{ where } \deg(r) < \deg(q) \text{ and } y = mx + b \text{ is the oblique asymptote}\)

易错点

• ⚠️ ["Confusing the conditions for horizontal asymptotes: students often forget that a horizontal asymptote exists only when the degree of the numerator is less than or equal to the degree of the denominator, not when it is greater.", "Incorrectly finding the oblique asymptote by only looking at the leading terms instead of performing complete polynomial long division, which can lead to missing the correct linear equation.", "Assuming that a function cannot cross its asymptote: in reality, rational functions can intersect their horizontal or oblique asymptotes at finite points, even though they approach the asymptote as \(x \to \infty\).", "Forgetting to simplify the rational function before determining asymptotes, which can result in identifying extraneous vertical asymptotes at points where the function has removable discontinuities (holes)."]