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AP Calculus AB Unit 2 Practice Test: Limits and Continuity
The AP Calculus AB exam is a rigorous test of your understanding of fundamental calculus concepts, and Unit 2, Limits and Continuity, forms the absolute bedrock of the entire course. Without a solid grasp of limits, mastering derivatives in later units becomes an exercise in memorization rather than true comprehension. On the flip side, this comprehensive practice test is designed to assess your understanding of the key ideas in Unit 2, from evaluating limits algebraically and graphically to understanding the precise definitions of continuity and the Intermediate Value Theorem. Work through these problems carefully, paying close attention to the detailed explanations provided for each answer Small thing, real impact..
Part 1: Multiple Choice Questions (No Calculator)
Instructions: Choose the best answer for each question. Questions 1-4 refer to the graph of ( f(x) ) shown below.
The graph of ( f(x) ) has a vertical asymptote at ( x = 2 ), a removable discontinuity (a hole) at ( x = -1 ), and a jump discontinuity at ( x = 3 ) Small thing, real impact..
1. Which of the following statements about ( f(x) ) is true? A) ( \lim_{x \to 2} f(x) = \infty ) B) ( \lim_{x \to -1} f(x) ) does not exist. C) ( f(x) ) is continuous at ( x = 3 ). D) ( \lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) )
2. What is the value of ( \lim_{x \to -1} f(x) )? A) 0 B) 2 C) The value where the hole is located. D) The limit does not exist.
3. At which of the following values of ( x ) is ( f(x) ) continuous? A) ( x = -1 ) B) ( x = 2 ) C) ( x = 3 ) D) ( x = 4 ) (assuming the graph is continuous at this point)
4. Which of the following best describes the behavior of ( f(x) ) as ( x ) approaches 2? A) The function approaches a finite value. B) The function oscillates without approaching a single value. C) The function increases or decreases without bound. D) The function is defined at ( x = 2 ) Took long enough..
5. Evaluate ( \lim_{x \to 3} \frac{x^2 - 9}{x - 3} ) A) 0 B) 3 C) 6 D) Does not exist
6. What is ( \lim_{x \to 0} \frac{\sin(x)}{x} )? A) 0 B) 1 C) ( \infty ) D) Does not exist
7. The expression ( \frac{\sqrt{x+4} - 3}{x - 5} ) is indeterminate at ( x = 5 ). After rationalizing the numerator, the limit ( \lim_{x \to 5} \frac{\sqrt{x+4} - 3}{x - 5} ) is equal to: A) ( \frac{1}{6} ) B) ( \frac{1}{3} ) C) 3 D) 6
8. If ( \lim_{x \to a} f(x) = L ) and ( \lim_{x \to a} g(x) = M ), which of the following limit laws is NOT always true? A) ( \lim_{x \to a} [f(x) + g(x)] = L + M ) B) ( \lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M ) C) ( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M} ) (provided ( M \neq 0 )) D) ( \lim_{x \to a} f(g(x)) = f(M) ) (provided ( f ) is continuous at ( M ))
9. For what value of ( c ) is the function ( f(x) = \begin{cases} x^2 + c & \text{if } x < 2 \ 3x - 2 & \text{if } x \geq 2 \end{cases} ) continuous at ( x = 2 )? A) -2 B) 0 C) 2 D) 4
10. The Intermediate Value Theorem (IVT) guarantees that the function ( f(x) = x^3 - 2x - 5 ) has a root in which of the following intervals? A) [0, 1] B) [1, 2] C) [2, 3] D) [3, 4]
Part 2: Free Response Questions (Calculator Permitted)
Instructions: Show all work clearly. You may use a calculator for calculations, but you must demonstrate the mathematical steps.
11. (4 points) Consider the function ( g(x) = \frac{x^3 - 8}{x^2 - 4} ). a) Find ( \lim_{x \to 2} g(x) ). b) Is ( g(x) ) continuous at ( x = 2 )? Why or why not?
12. (6 points) The graph of a function ( h(x) ) is given below. (Imagine a graph with a vertical asymptote at x=1, a hole at x=-2, and smooth curves elsewhere). a) Find all values of ( x ) where ( h(x) ) is discontinuous. b) For each discontinuity identified in part (a), classify its type (removable, jump, or infinite). c) Find the value of ( \lim_{x \to -2} h(x) ) if it exists Small thing, real impact..
13. (4 points) Use the precise definition of a limit (( \epsilon-\delta )) to prove that ( \lim_{x \to 4} (2x - 3) = 5 ). (Note: A full proof is not required for this practice test, but outline the key steps you would take).
14. (6 points) Find the values of ( a ) and ( b ) that make the following function continuous everywhere: [ f(x) = \begin{cases} a x + b & \text{if } x < 1 \ 4 & \text{if } x = 1 \ b x^2 - a & \text{if } x > 1 \end{cases} ]
Detailed Solutions and Explanations
Part 1: Multiple Choice
1. Correct Answer: A The description states there is a vertical asymptote at ( x = 2 ). By definition, as ( x
