AP Calculus AB : 1.4 Estimating Limit Values from Tables- Exam Style questions with Answer- MCQ

Question

The table above gives values of the function f at selected values of x. Which of the following statements must be true?

A) lim_x→2 f(x) = 3
B) lim_x→2 f(x) = 8
C) lim_x→2 f(x) does not exist
D) lim_x→2 f(x) cannot be definitively determined from the data in the table

▶️ Answer/Explanation

Solution

Correct Answer: D

Analysis of each option:

Option A: lim_x→2 f(x) = 3

✕ Not necessarily true. While f(2) = 3, the limit depends on approaching values.

Option B: lim_x→2 f(x) = 8

✕ Not necessarily true. The values approach ≈8 from both sides, but f(2) = 3 creates a discontinuity.

Option C: lim_x→2 f(x) does not exist

✕ Not necessarily true. The left and right limits both approach ≈8, but we can’t be certain without more precise data.

Option D: lim_x→2 f(x) cannot be definitively determined

✓ Correct. The table shows:

– Values approaching ≈8 from both sides

– But f(2) = 3 is different

– Without knowing if this is a removable discontinuity or error, we can’t determine the limit definitively

Key Observation:

The table suggests a possible removable discontinuity at x=2, but we cannot confirm this without either:

1) More precise data points closer to x=2

2) The actual function definition

Question

The table above gives values of a function \( f \) at selected values of \( x \). Which of the following conclusions is supported by the data in the table?

A) \( \lim_{x \to 5} f(x) = 8 \)
B) \( \lim_{x \to 5^{-1}} f(x) = 8 \)
C) \( \lim_{x \to 5^{+1}} f(x) = 8 \)
D) \( \lim_{x \to 8^{+1}} f(x) = 8 \)

▶️ Answer/Explanation

Solution

Correct Answer: C

As \( x \to 5^+ \), \( f(x) \) approaches 8.
Table values: 7.999 (5.0001), 7.99 (5.001), 7.9 (5.01).

Why others are not supported:
A) \( \lim_{x \to 5} f(x) \) does not exist
B) \( \lim_{x \to 5^-} f(x) \to -726 \)
D) No data near \( x = 8 \)

Question

The table above gives selected values for a continuous function \( f \). Based on the data in the table, which of the following is the best approximation for \( \lim_{x \to 4} f(x) \)?

A) 0
B) 4
C) 7
D) There is no best approximation, because the limit does not exist.

▶️ Answer/Explanation

Solution

Correct Answer: C

As \( x \to 4^- \), \( f(x) \): 7.018, 7.007, 7.002
As \( x \to 4^+ \), \( f(x) \): 6.998, 5.982, 5.887
Values near \( x = 4 \) approach 7, and since \( f \) is continuous, \( \lim_{x \to 4} f(x) \approx 7 \).

Why others are wrong:
A) 0: Limit is not 0
B) 4: Limit is not 4
D) Limit exists due to continuity

Question

Find the limit: \(\lim_{t\rightarrow -3}\frac{t+3}{t^{2}+9}\)

A) \(-\frac{1}{3}\)
B) 0
C) \(\frac{1}{3}\)
D) 1

▶️ Answer/Explanation

Solution

Correct Answer: B

Step-by-Step Solution:

Direct Substitution:

\(\frac{(-3)+3}{(-3)^{2}+9} = \frac{0}{9+9} = \frac{0}{18} = 0\)

AP Calculus AB : 1.4 Estimating Limit Values from Tables- Exam Style questions with Answer- MCQ

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