Topic 1.7 Rational Functions and End Behavior: determine the end behavior of a rational function by comparing the degrees of its numerator and denominator, identifying horizontal or slant asymptotes.

What this topic is asking

The College Board (Topic 1.7) wants you to find the end behavior of a rational function $\frac{p(x)}{q(x)}$ by comparing the degree of the numerator with the degree of the denominator. The comparison tells you whether the graph approaches a horizontal asymptote, follows a slant (oblique) asymptote, or grows without bound at the ends.

Comparing degrees

These four cases parallel the degree rule for limits at infinity: divide every term by the highest power of $x$ and let the small terms vanish.

Horizontal asymptotes

A horizontal asymptote is a horizontal line the graph approaches as $x \to \pm\infty$. When the degrees are equal, the leading terms dominate and the function levels off at the ratio of leading coefficients. When the denominator wins, the fraction shrinks toward $0$. In limit notation, a horizontal asymptote $y = L$ means $\lim_{x \to +\infty} f(x) = L$ and usually $\lim_{x \to -\infty} f(x) = L$ as well.

Slant asymptotes

The slant asymptote is the whole linear quotient, including its constant term, not just its slope.

When the ends grow without bound

If the numerator's degree exceeds the denominator's by two or more, the quotient from long division is a polynomial of degree $2$ or higher, so the function behaves like that polynomial at the ends and has no straight-line asymptote. The exam expects you to recognize this case and describe the end behavior using the leading-term reasoning of Topic 1.6 rather than searching for an asymptote that does not exist.

A subtlety worth holding onto is that an asymptote describes only the far-out behavior, and a graph may cross its horizontal asymptote in the middle. The asymptote constrains where the curve settles as $x \to \pm\infty$, not where it goes near the origin. So finding the asymptote answers the end-behavior question completely while saying nothing about the interesting features (zeros, vertical asymptotes, holes) in the middle, which are the subjects of Topics 1.8 to 1.10.

Try this

Q1. Find the horizontal asymptote of $f(x) = \frac{3x + 5}{x^2 + 1}$. [1 point]

• Cue. Denominator degree ($2$) is larger, so the horizontal asymptote is $y = 0$.

Q2. Does $g(x) = \frac{x^3}{x + 1}$ have a horizontal asymptote, a slant asymptote, or neither? [1 point]

• Cue. Numerator degree ($3$) exceeds denominator degree ($1$) by two, so neither; the ends grow without bound.