AP Calculus AB: Squeeze Theorem, IVT, and Mixed Limit Skills — Drill 1

About This Drill

AP Calculus AB: Squeeze Theorem, IVT, and Mixed Limit Skills — Drill 1 is a Math practice drill covering Limits and Continuity. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice applying the Squeeze Theorem to evaluate limits involving bounded functions, use the Intermediate Value Theorem to guarantee the existence of roots and fixed points, and work with piecewise functions and continuity. These AP Calculus AB skills appear on both the multiple-choice and free-response sections.

Questions in This Drill

1. Using the Squeeze Theorem, what is \( \lim_{x \to 0} x^2 \sin\!\left(\dfrac{1}{x}\right) \)?
2. The function \( f \) is continuous on \( [1, 5] \) with \( f(1) = -3 \) and \( f(5) = 8 \). Which of the following is guaranteed by the Intermediate Value Theorem?
3. Let \( f \) be defined by \( f(x) = 2x + 1 \) for \( x \leq 1 \) and \( f(x) = x^2 + bx \) for \( x > 1 \). For what value of \( b \) is \( f \) continuous at \( x = 1 \)?
4. A student wants to use the Intermediate Value Theorem to show that \( f(x) = x^3 - 4x + 1 \) has a root on the interval \( [0, 2] \). Which of the following correctly justifies this conclusion?
5. Suppose \( f \) is continuous on \( [-2, 6] \) with \( f(-2) = 4 \) and \( f(6) = 2 \). Let \( g(x) = f(x) - x \). Which of the following must be true?
   
   (A) There exists \( c \in (-2, 6) \) such that \( f(c) = 3 \).
   (B) There exists \( c \in (-2, 6) \) such that \( f(c) = c \).