Study Guide

Learning Objectives

• Determine vertical asymptotes of rational functions.
• Apply: den zero not in num ⇒ VA; den multiplicity > num multiplicity ⇒ VA.
• Describe unbounded behavior near a VA with one-sided limits.
• Distinguish VA from zeros (1.8), HA (1.7), and holes (1.10 preview).

Suggested Practice Sequence

1. Simple VA: den zero only. 2. Multiple den zeros; identify which are VAs. 3. Shared factors with multiplicity compare. 4. One-sided limit descriptions. 5. Contrast items: zero vs VA vs HA. 6. Mixed FRQs with domain + VA list.

Classroom / Check-In Moves

• Require a multiplicity table for shared factors.
• Cold-call one-sided limits at a VA.
• Sort cards: VA / hole / zero / HA.

Misconception Watch-List

• Every den zero is a VA.
• Shared factor ⇒ never a VA.
• VA means $r(a)=\infty$.

Vocabulary to Own

• vertical asymptote
• one-sided limit ($x\to a^+$, $x\to a^-$)
• multiplicity (num vs den)
• unbounded increase/decrease
• domain restriction

Materials Map

• Practice: locate VAs + limits
• Word: applied blow-up / barrier stories
• Mixed / multi-*: contrast with zeros/HA
• Review & practice test: timed VA fluency
• Interactive: quick checks