Common Mistakes

Frequent Errors

1. Thinking every rational function has HA $y=0$. 2. Using constant terms instead of leading coefficients for equal-degree HA. 3. Ignoring the slant case when deg(num)=deg(den)+1. 4. Confusing vertical asymptotes with horizontal end behavior. 5. Writing HA when the numerator degree is much larger (no HA). 6. Forgetting limit notation meaning of HA $y=b$. 7. Comparing wrong degrees after a messy expansion. 8. Saying outputs “equal $b$” instead of approach $b$ for large $|x|$.

Why These Happen

Students import “HA at $y=0$” from simple examples like $1/x$. Equal-degree ratios feel optional. Slant asymptotes are easy to skip. Vertical asymptote lessons bleed into end-behavior questions.

How to Fix Them

• Always write deg(num) ? deg(den) first.
• Equal → ratio of leading coefficients.
• Den larger → $y=0$.
• Num larger by 1 → slant; by more → polynomial-like ends, no HA.
• HA sentence: approach $b$, not equal $b$ for all large $x$.

Avoidance Scripts

• “Compare degrees, then leading terms.”
• “Tie goes to the ratio; bottom wins goes to zero.”
• “Approach, don’t land on, the asymptote.”

Quick Self-Check

• Did I compare degrees?
• Is my HA the leading-coefficient ratio when degrees match?
• Did I consider slant?