AP Calculus AB - Limits - 7 Questions 1. What is the limit as x approaches 2 of (3x^2 – 4x + 2)? (a) 10 b) 8 (c) 6 (d) 4 (a) 10 Explanation: By direct substitution, plugging in x = 2 yields (3(2)^2 – 4(2) + 2) = 10. 2. Evaluate the limit as x approaches 3 of (x^3 – 27)/(x – 3). (a) 18 b) 10 ( c) 14 d) 6 Answer: (a) 18 Explanation: This is a factorable form. By factoring the numerator as (x – 3)(x^2 + 3x + 9), the (x – 3) terms cancel out, leaving x^2 + 3x + 9. Plugging in x = 3 yields 3^2 + 3(3) + 9 = 18. 3. Determine the limit as x approaches 0 of (sin(x))/x. (a) 1 b) 0 (c) -1 (d) undefined Answer: (a) 1 Explanation: This is a well-known limit in calculus. As x approaches 0, the function sin(x) approaches 0, and the denominator x also approaches 0. Applying L’Hôpital’s Rule or using trigonometric identities, the limit simplifies to 1. 4. Find the limit as x approaches infinity of (2x^2 – 5x + 3)/(3x^2 + 4). (a) 2/3 (b) 3/2 (c) 1/5 (d) 4/3 Answer: (a) 2/3 Explanation: As x approaches infinity, the highest power terms dominate, so we divide the leading coefficients to obtain 2/3. 5. Evaluate the limit as x approaches -2 from the right of sqrt(x^2 – 4). (a) 4 (b) 2 c) -2 (d) undefined Answer: (d) undefined Explanation: Since the expression inside the square root becomes negative as x approaches -2 from the right, the square root of a negative number is undefined. 6. Determine the limit as x approaches 4 of (x^2 – 16)/(x – 4). a) 8 (b) 4 (c) 2 d) 0 Answer: (a) 8 Explanation: This is a factorable form. By factoring the numerator as (x – 4)(x + 4), the (x – 4) terms cancel out, leaving x + 4. Plugging in x = 4 yields 4 + 4 = 8. 7. Find the limit as x approaches infinity of e^(-x). (a) 0 (b) 1 c) -1 (d) undefined Answer: (a) 0 Explanation: As x approaches infinity, the exponential function e^(-x) approaches 0. Loading … Back to AP Practice Test
