Common Mistakes
Frequent Errors
1. Treating the leftmost written term as leading when the polynomial is not standard-ordered. 2. Stating only $\lim_{x\to\infty}$ and skipping $x\to-\infty$. 3. Confusing even-degree “same ends” with odd-degree “opposite ends.” 4. Letting zeros or turning points change the end-behavior call. 5. Writing $\lim_{x\to\infty}p(x)=0$ for a nonconstant polynomial. 6. Mixing the sign of $a_n$ with the sign of a constant term. 7. Saying “increases” without specifying which end / which limit. 8. Forgetting that dominance justifies ignoring lower-degree terms.
Why These Happen
End behavior feels graphical, so students eyeball the middle of a sketch. Limit notation is new, so they under-specify. Messy algebra hides the true leading term.
How to Fix Them
- Always rewrite or highlight $a_n x^n$ before concluding.
- Template: two limit lines, then one dominance sentence.
- Memory: even = same; odd = opposite; then apply the sign of $a_n$.
Avoidance Scripts
- “Leading term first, limits second, story last.”
- “Both ends, both limits.”
- “Bumps don’t change infinity.”
Quick Self-Check
- Did I use the actual highest power?
- Do my two limits match even/odd + sign?
- Did I mention dominance?