AP PRECALCULUS — UNIT 1B · POLYNOMIAL & RATIONAL FUNCTIONS
1.9 — Rational Functions and Vertical Asymptotes
Notes — Where the Denominator Drives the Graph to Infinity
💡 Learning Objectives By the end of this lesson you will be able to (AP CED 1.9.A):
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1. What Causes a Vertical Asymptote?
When the denominator of a rational function approaches zero while the numerator approaches a nonzero number, the magnitude of the quotient grows without bound. Graphically, the curve shoots up to +∞ or down to −∞ near that input. We call the vertical line x = a a vertical asymptote of r if |r(x)| grows without bound as x approaches a from at least one side.
💡 Definitions Let r(x) = N(x) / D(x) be a rational function, with N and D sharing no common factors. The line x = a is a vertical asymptote (VA) of the graph of r if and only if D(a) = 0.
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💡 Quick Check Predict the vertical asymptotes of r(x) = (x + 2) / ((x − 1)(x − 4)).
Vertical asymptotes: x = 1 and x = 4. |
2. Behavior Near a VA — Left vs. Right
A vertical asymptote has two sides. On each side, r(x) heads off to either +∞ or −∞. To determine which, test a value very close to a on that side and check the sign of the result. The magnitude of the factor (x − a) in the denominator (and whether it's raised to an even or odd power) governs whether the two sides agree or disagree.
Four VA behaviors at x = 1 depending on sign of numerator and parity of the denominator factor. Odd-power factors give opposite signs on the two sides; even-power factors give the same sign.
📘 Example Example 1 — One-sided limits at a VA Find the one-sided limits of r(x) = (x + 3) / (x − 2) at x = 2. As x → 2⁺: numerator → 5 (positive), denominator → 0⁺ (tiny positive). So r(x) → +∞. As x → 2⁻: numerator → 5 (positive), denominator → 0⁻ (tiny negative). So r(x) → −∞. In limit notation: lim r(x) as x → 2⁺ = +∞ and lim r(x) as x → 2⁻ = −∞. |
⚠️ Common Mistake Do NOT assume all vertical asymptotes look like 1/x (opposite signs on either side). For even-power factors like 1/(x − a)², both sides blow up with the SAME sign. Always test a specific value on each side. |
3. A Systematic Procedure
To find the vertical asymptotes of r(x) = N(x) / D(x):
- Factor both N(x) and D(x) completely.
- Cancel any common factors — these become holes (Topic 1.10), not vertical asymptotes.
- Set the simplified denominator equal to zero; each solution is a vertical asymptote.
- For each VA x = a, determine lim r(x) as x → a⁺ and x → a⁻ by testing nearby values.
📘 Example Example 2 — A function with cancellation Find the vertical asymptotes of r(x) = (x² − 1) / (x² − 3x + 2). Factor: N(x) = (x − 1)(x + 1), D(x) = (x − 1)(x − 2). The factor (x − 1) cancels. Simplified: r(x) = (x + 1) / (x − 2) for x ≠ 1. Denominator zero: x = 2 (VA). Hole at x = 1. So there is ONE vertical asymptote, x = 2, and a hole at x = 1. |
📘 Try This Identify vertical asymptotes (if any) for:
Answers: x = 5 ; none (denom never zero) ; x = 3 only (the factor at x = −3 cancels, making a hole). |
🎯 AP Tip An AP free-response often asks you to write a one-sided limit such as lim r(x) as x → 3⁻. To earn the point, write the limit exactly, with the correct one-sided notation (⁺ or ⁻) and the correct value (+∞ or −∞). Stating 'goes to infinity' is not sufficient — specify WHICH infinity. |
4. Summary
- Vertical asymptotes occur where the denominator is zero AND the factor does not cancel with the numerator.
- Each VA has two sides; the sign of r(x) on each side determines +∞ or −∞.
- Express behavior using one-sided limit notation: lim r(x) as x → a⁺ and x → a⁻.
- Even-power factors give matching signs on both sides; odd-power factors give opposite signs.