Home / AP Calculus AB : 1.8 Determining Limits Using the Squeeze Theorem- Exam Style questions with Answer- MCQ

AP Calculus AB : 1.8 Determining Limits Using the Squeeze Theorem- Exam Style questions with Answer- MCQ

AP Calculus AB-Questions and Answers - All Topics

Question

Let f be a function of x. The value of \(\lim_{x \to a}{f(x)}\) can be found using the squeeze theorem with the functions g and h. Which of the following could be graphs of f, g, and h?

A)Graph option A

B)Graph option B

C)Graph option C

D)Graph option D

▶️ Answer/Explanation

Solution

Answer: D

The squeeze theorem requires: 1. \(g(x) \leq f(x) \leq h(x)\) 2. \(\lim_{x\to a}g(x) = \lim_{x\to a}h(x)\) Only option D shows f completely bounded between g and h with equal limits.

Question
The function g is given by \(g(x)=\frac{1}{x^{2}-4x+5}\). The function h is given by \(h(x)=\frac{2x^{2}-8x+10}{x^{2}-4x+6}\). If f is a function that satisfies \(g(x) \leq f(x) \leq h(x)\) for \(0 < x < 5\), what is \(\lim_{x \to 2}{f(x)}\)?
A) 0
B) 1
C) 2
D) The limit cannot be determined from the information given.
▶️ Answer/Explanation
Solution
Correct Answer: B
Evaluate limits of g(x) and h(x) at x=2:
\(\lim_{x \to 2} g(x) = \frac{1}{2^2-4(2)+5} = \frac{1}{4-8+5} = 1\)
\(\lim_{x \to 2} h(x) = \frac{2(2)^2-8(2)+10}{2^2-4(2)+6} = \frac{8-16+10}{4-8+6} = \frac{2}{2} = 1\)
Apply Squeeze Theorem:
Since \(g(x) \leq f(x) \leq h(x)\) and both g(x) and h(x) approach 1 as x→2,
by the Squeeze Theorem, \(\lim_{x \to 2} f(x) = 1\)
Question
The function f is defined for all x in the interval 3<x<6. Which of the following statements, if true, implies that \(\lim_{x \to 5}{f(x)}=12\)?
A) There exists a function g with f(x)≤g(x) for 3<x<6, and lim g(x)=12

B) There exists a function g with g(x)≤f(x) for 3<x<6, and lim g(x)=12

C) There exist functions g and h with g(x)≤f(x)≤h(x) for 3<x<6, and lim g(x)=10 and lim h(x)=14

D) There exist functions g and h with g(x)≤f(x)≤h(x) for 3<x<6, and \(\lim_{x \to 5}{g(x)}=\lim_{x \to 5}{h(x)}=12\)
▶️ Answer/Explanation
Solution
Correct Answer: D
Explanation:
Option D satisfies all conditions of the Squeeze Theorem:
1. \(g(x) \leq f(x) \leq h(x)\) for all x in (3,6)
2. \(\lim_{x \to 5}{g(x)} = \lim_{x \to 5}{h(x)} = 12\)
Therefore, by the Squeeze Theorem, \(\lim_{x \to 5}{f(x)} = 12\)
lim f(x)=12
Question
Find the limit: \(\lim_{x\rightarrow -\infty }\frac{\sin \Theta }{\Theta }\)
(A) 0
(B) −1
(C) −∞
(D) The limit does not exist
▶️ Answer/Explanation
Solution
Correct Answer: A
Explanation:
1. The sine function is bounded: \(-1 \leq \sin \Theta \leq 1\) for all real Θ
2. As Θ approaches -∞, the denominator grows without bound: \(|\Theta| \to \infty\)
3. Therefore, the fraction \(\frac{\sin \Theta}{\Theta}\) is a bounded quantity divided by an infinitely large quantity
By the Squeeze Theorem: \(\lim_{\Theta \to -\infty} \frac{\sin \Theta}{\Theta} = 0\)
Note: The same result holds for Θ → +∞
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