Home / AP Calculus AB 1.11 Defining Continuity at a Point – MCQs

AP Calculus AB 1.11 Defining Continuity at a Point - MCQs - Exam Style Questions

AP Calculus AB-MCQs and Answers - All Topics

No-Calc Question

\(f(x)=\begin{cases}6+cx,& x<1\\[2pt]9+2\ln x,& x\ge 1\end{cases}\). If \(f\) is continuous at \(x=1\), what is the value of \(c\)?

(A) \(2\)   

(B) \(3\)   

(C) \(5\)   

(D) \(9\)

▶️ Answer/Explanation

Continuity at \(x=1\): \(6+c(1)=9+2\ln 1\Rightarrow 6+c=9\Rightarrow c=3\).
✅ Answer: (B) \(3\)

Calc-Ok Question

If the function \(f\) is continuous at \(x=3\), which of the following must be true?

(A) \(f(3)<\displaystyle\lim_{x\to 3}f(x)\)
(B) \(\displaystyle\lim_{x\to 3^-} f(x)\ne \lim_{x\to 3^+} f(x)\)
(C) \(f(3)=\displaystyle\lim_{x\to 3^-} f(x)=\lim_{x\to 3^+} f(x)\)
(D) The derivative of \(f\) at \(x=3\) exists.
(E) The derivative of \(f\) is positive for \(x<3\) and negative for \(x>3\).

▶️ Answer/Explanation

Continuity at \(x=3\) requires the two-sided limit to exist and equal the function value.
Therefore \(f(3)=\lim_{x\to 3} f(x)=\lim_{x\to 3^-} f(x)=\lim_{x\to 3^+} f(x)\).
Differentiability (D) is not guaranteed by continuity alone.
✅ Answer: (C)

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