AP Calculus AB Question of the Day

1. Yes; $\lim_{x\to0}b(x)=0$ and $b(0)=0$. (correct answer)
2. No; $\lim_{x\to0^-}b(x)=0$ and $\lim_{x\to0^+}b(x)=0$, so the limit does not exist.
3. No; $b(0)$ is undefined.
4. No; $\lim_{x\to0}b(x)=1$ but $b(0)=0$.
5. Yes; continuity requires only that $b(0)$ be defined.

Explanation: To be continuous at x = a, f(a) must exist, lim_{x→a} f(x) must exist, and equal f(a). For b(x) at x = 0, b(0) = 0, left limit x² → 0, right -x² → 0, so limit = 0 and matches, continuous. Often, students forget to evaluate one-sided limits separately in piecewise functions. Despite the sign change, both approach 0. This demonstrates symmetry in limits for continuity. To check continuity at any point a, use this checklist: ensure f(a) is defined, compute lim_{x→a} f(x) and confirm it exists, then check if lim_{x→a} f(x) = f(a).