Common Mistakes

Frequent Errors

1. Labeling every denominator zero as a VA. 2. Ignoring multiplicity when num and den share a factor. 3. Confusing VA ($x=a$) with HA ($y=b$). 4. Calling a zero of $r$ a vertical asymptote. 5. Writing two-sided limits only when sides disagree. 6. Saying the function “equals infinity” at $x=a$ (it’s undefined there). 7. Using end behavior (large $|x|$) to answer local VA questions. 8. Forgetting that den mult > num mult still yields a VA.

Why These Happen

“Set denominator = 0” is over-applied. Holes and asymptotes compete for attention. Limit notation from 1.6/1.7 gets reused without one-sided care.

How to Fix Them

• Table: den zero | also in num? | mult den vs num | VA?
• Habit: undefined at $a$, not equal to infinity.
• Separate columns: local VA vs far-away HA.

Avoidance Scripts

• “Den zero proposes; numerator/multiplicity disposes.”
• “VA is a vertical line; HA is horizontal.”
• “Near $a$, use left and right limits.”

Quick Self-Check

• Did I check the numerator for each den zero?
• Did I compare multiplicities when shared?
• Are my limits one-sided when needed?