AP Calculus AB 1.12 Confirming Continuity over an Interval - MCQs - Exam Style Questions
No-Calc Question
The function \(f\) is defined as shown. Which of the following statements is true?
\[ f(x)= \begin{cases} \dfrac{x^{2}+5x+6}{x^{2}-4}, & x\ne -2\text{ and }x\ne 2,\\[6pt] -\dfrac{1}{4}, & x=-2,\\[6pt] \dfrac{1}{4}, & x=2. \end{cases} \]
(A) \(f\) is continuous for all \(x\).
(B) \(f\) is discontinuous at \(x=-2\) only.
(C) \(f\) is discontinuous at \(x=2\) only.
(D) \(f\) is discontinuous at \(x=-2\) and \(x=2\).
▶️ Answer/Explanation
For \(x\ne\pm2\): \(\dfrac{x^{2}+5x+6}{x^{2}-4} =\dfrac{(x+2)(x+3)}{(x+2)(x-2)}=\dfrac{x+3}{x-2}\).
\(\displaystyle\lim_{x\to -2}\dfrac{x+3}{x-2}=\dfrac{1}{-4}=-\dfrac{1}{4}=f(-2)\) ⇒ continuous at \(-2\).
\(\displaystyle\lim_{x\to 2}\dfrac{x+3}{x-2}\) is infinite ⇒ discontinuous at \(2\).
✅ Answer: (C)