AP Calculus AB: Linearization and L’Hôpital’s Rule — Drill 1

About This Drill

AP Calculus AB: Linearization and L’Hôpital’s Rule — Drill 1 is a Math practice drill covering Linearization and L'Hôpital's Rule. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice using local linear approximation to estimate function values and applying L'Hôpital's Rule to evaluate limits in indeterminate forms. These AP Calculus AB skills appear in both multiple-choice and free-response sections of the exam.

Questions in This Drill

1. Let \( f(x) = \sqrt{x} \). Which of the following is the linearization of \( f \) at \( x = 9 \)?
2. What is \( \displaystyle\lim_{x \to 0} \dfrac{\sin 3x}{x} \)?
3. Let \( g \) be a twice-differentiable function with \( g(2) = 5 \), \( g'(2) = -1 \), and \( g''(x) > 0 \) for all \( x \). Which of the following best describes the linearization \( L(x) \) of \( g \) at \( x = 2 \) used to approximate \( g(2.1) \)?
4. What is \( \displaystyle\lim_{x \to \infty} \dfrac{3x^2 + 1}{5x^2 - 2x} \)?
5. A student evaluates \( \displaystyle\lim_{x \to 0} \dfrac{x + 2}{x} \) using L'Hôpital's Rule, differentiating numerator and denominator to obtain a limit of 1. Which of the following best identifies the error?