AP Calculus AB 1.13 Removing Discontinuities - MCQs - Exam Style Questions
Calc-Ok Question
If \(f\) is a function that has a removable discontinuity at \(x=3\), which of the following could be the graph of \(f\) ?
▶️ Answer/Explanation
A removable discontinuity is a “hole”: the limit exists but \(f(3)\) may differ or be undefined.
Among the given sketches, option (C) shows a line with an open circle at \(x=3\) (and a filled point elsewhere).
✅ Answer: (C)
Among the given sketches, option (C) shows a line with an open circle at \(x=3\) (and a filled point elsewhere).
✅ Answer: (C)
▶️ Answer/Explanation
Solution
Correct Answer: A
1. For continuity at x=3, left and right limits must be equal:
\[k^3 + 3 = \frac{16}{k^2 – 3}\]
2. Solving for k > 0 gives k ≈ 2.081
3. With this k, \(\lim_{x\to3}f(x)\) exists and equals f(3)
4. Therefore, f is continuous at x=3 when k=2.081