AP Calculus AB: 1.11 Defining Continuity at a Point - Exam Style questions with Answer- MCQ

Question

For any function f, which of the following statements must be true?

I. If f is defined at x = a, then \(\lim_{x \to a}f(x) = f(a)\)
II. If f is continuous at x = a, then \(\lim_{x \to a}f(x) = f(a)\)
III. If f is differentiable at x = a, then \(\lim_{x \to a}f(x) = f(a)\)

A) III only
B) I and II only
C) II and III only
D) I, II, and III

▶️ Answer/Explanation

Solution

Correct Answer: C

Analysis of each statement:

I: False. A function can be defined at a point but not have a limit there (e.g., piecewise functions with jump discontinuities).

II: True. By definition, continuity at x = a requires that \(\lim_{x \to a}f(x) = f(a)\).

III: True. Differentiability implies continuity, which in turn implies the limit equals the function value.

Conclusion:

Only statements II and III must be true for all functions, making option C correct.

Key Concept: Differentiability ⇒ Continuity ⇒ Limit exists, but not conversely.

Question

If the function f is continuous for all real numbers and if \(f(x)=\frac{x^{2}-4}{x+2}\) when \(x \neq -2\), then f(-2) =

A) -4
B) -2
C) -1
D) 0
E) 2

▶️ Answer/Explanation

Solution

Correct Answer: A

Simplify: \(f(x) = \frac{(x-2)(x+2)}{x+2} = x-2\) when \(x \neq -2\)

By continuity: \(f(-2) = \lim_{x \to -2} (x-2) = -4\)

Question

Let f be the function defined by: \[f(x) = \begin{cases} x^2 + 2 & \text{for } x \leq 3 \\ 6x + k & \text{for } x > 3 \end{cases}\] If f is continuous at x=3, what is the value of k?

A) -7
B) 2
C) 3
D) 7
E) There is no such value of k

▶️ Answer/Explanation

Solution

Correct Answer: A

1. Left limit (x→3⁻): \(3^2 + 2 = 11\)

2. Right limit (x→3⁺): \(6(3) + k = 18 + k\)

3. For continuity: \(11 = 18 + k\) ⇒ \(k = -7\)

Question

Let f be the piecewise function defined by: \[ f(x) = \begin{cases} 2x + 3 & \text{for } x < 1 \\ 5 & \text{for } x = 1 \\ -4x + 9 & \text{for } 1 < x < 3 \\ 4 & \text{for } x = 3 \\ x – 6 & \text{for } x > 3 \end{cases}\] Which of the following statements is false?

A) f is continuous at x=1
B) f is continuous at x=2
C) f is continuous at x=3
D) f is continuous at x=4

▶️ Answer/Explanation

Solution

Correct Answer: C

1. At x=3: \(\lim_{x\to3^-}f(x) = -3\) and \(\lim_{x\to3^+}f(x) = -3\)

2. But \(f(3) = 4\)

3. Since \(\lim_{x\to3}f(x) \neq f(3)\), f is not continuous at x=3

AP Calculus AB: 1.11 Defining Continuity at a Point – Exam Style questions with Answer- MCQ

Resources

Members

Company