Common Mistakes
Frequent Errors
1. Treating every shared factor as a hole without multiplicity compare. 2. Calling holes vertical asymptotes. 3. Finding $x=c$ but not the $y$-value $L$. 4. Saying $r(c)=L$ (the function is still undefined at $c$). 5. Canceling incorrectly and evaluating the original unsimplified form at $c$ (0/0). 6. Assuming a hole means $L=0$. 7. Mixing hole location with horizontal asymptotes. 8. Leaving a VA case labeled as a hole when den mult is larger.
Why These Happen
“Cancel and you’re done” skips the coordinate. 0/0 on a calculator feels like “nothing,” so students don’t hunt for $L$. VA and hole rules are easy to flip.
How to Fix Them
- Table: shared $c$ | num mult | den mult | hole or VA?
- For holes: simplify → plug $c$ → write $(c,L)$.
- Say: “undefined at $c$, but limit is $L$.”
Avoidance Scripts
- “Num mult ≥ den mult ⇒ hole; den mult greater ⇒ VA.”
- “Hole needs a point, not only an $x$.”
- “Cancel, then evaluate the simplified rule.”
Quick Self-Check
- Did I compare multiplicities?
- Do I have $(c,L)$?
- Did I avoid claiming $r(c)$ exists?