Rational functions and vertical asymptotes — AP Precalculus
1. Classifying Discontinuities: Removable vs Non-Removable ★★☆☆☆ ⏱ 4 min
For any rational function $f(x) = \frac{N(x)}{D(x)}$, all points where $D(x) = 0$ are discontinuities, since the function is undefined there. To classify a discontinuity at $x=a$, first factor both numerator and denominator completely.
- If $(x-a)$ is a common factor of both $N(x)$ and $D(x)$, the discontinuity is **removable (a hole)**: the two-sided limit exists and is finite, but the function is undefined at $x=a$.
- If $(x-a)$ is only a factor of $D(x)$ and does not cancel after removing all common factors, the discontinuity is **non-removable**, corresponding to a vertical asymptote at $x=a$.
Exam tip: Always factor out all common factors completely before classifying discontinuities; partial factoring can lead to misidentifying holes as vertical asymptotes.
2. The Limit Rule for Locating Vertical Asymptotes ★★★☆☆ ⏱ 4 min
The formal definition of a vertical asymptote relies on one-sided limits: the line $x=a$ is a vertical asymptote if at least one of the following holds:
\lim_{x \to a^-} f(x) = \pm\infty \quad \text{or} \quad \lim_{x \to a^+} f(x) = \pm\infty
For rational functions, this definition simplifies to a straightforward rule that works for all cases: if $f(x) = \frac{N(x)}{D(x)}$ is written in lowest terms (no common factors between numerator and denominator), then every real zero of $D(x)$ is a vertical asymptote. The lowest terms condition is non-negotiable.
Exam tip: On AP Precalculus free-response questions, you must write vertical asymptotes as full equations of lines (e.g., $x=2$, not just $2$) to earn full credit.
3. Analyzing Function Behavior Near Vertical Asymptotes ★★★☆☆ ⏱ 4 min
Once you have identified a vertical asymptote at $x=a$, you will often need to determine whether the function approaches $+\infty$, $-\infty$, or opposite infinities on either side of $a$ for graphing or limit questions. The fastest way to do this is to test the sign of the reduced rational function on each side of $x=a$.
- Repeated root (even exponent on $(x-a)$ in denominator): the term is always positive for $x \neq a$, so the function approaches the same infinity on both sides.
- Distinct root (odd exponent on $(x-a)$): the sign changes across the asymptote, so the function approaches opposite infinities on either side.
Exam tip: When calculating the sign of the function near an asymptote, any even-powered factor can be ignored entirely because it is always positive, cutting down on calculation time.
4. AP-Style Practice Worked Examples ★★★★☆ ⏱ 3 min
Common Pitfalls
Why: Students rush after finding a root of the denominator and forget to check for removable discontinuities.
Why: Confuses vertical asymptotes (vertical lines, constant x-value) with horizontal asymptotes (horizontal lines, constant y-value).
Why: Students mix up properties of horizontal and vertical asymptotes.
Why: Students assume all roots change sign, forgetting that even powers are always positive.
Why: Students do not cancel all common factors after factoring.