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AP Calculus AB 1.3 Estimating Limit Values from Graphs - MCQs - Exam Style Questions

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Calc-Ok Question


The graph of the function \(f\) is shown. Which of the following statements must be false?

(A) \(\displaystyle \lim_{x\to 2^{-}} f(x)=3\)
(B) \(\displaystyle \lim_{x\to 2^{+}} f(x)=\infty\)
(C) \(\displaystyle \lim_{x\to 2} f(x)=f(2)\)
(D) \(\displaystyle \lim_{x\to\infty} f(x)=0\)

▶️ Answer/Explanation

At \(x=2\) there’s a vertical asymptote: left-hand limit is finite (\(=3\)), right-hand limit is \(+\infty\). Hence the two-sided limit at \(2\) does not exist and cannot equal \(f(2)\).
✅ Answer: (C)

Question 

The graph of the function \( f \) is shown above. Which of the following statements is false?

(A) \( \lim_{x \to 2} f(x) \) exists.
(B) \( \lim_{x \to 3} f(x) \) exists.
(C) \( \lim_{x \to 4} f(x) \) exists.
(D) \( \lim_{x \to 5} f(x) \) exists.
(E) The function \( f \) is continuous at \( x = 3 \).

▶️ Answer/Explanation

Solution

Correct Answer: C

At \( x = 4 \), the left-hand limit and right-hand limit are not equal (there’s a jump discontinuity).
Therefore, \( \lim_{x \to 4} f(x) \) does not exist, making statement C false.
All other statements are true based on the graph: limits exist at 2, 3, and 5, and the function is continuous at 3.

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