Free response without a calculator (Part B)

Limits, asymptotes and the squeeze theorem

Units 1 and 2
9 points
About 15 minutes

A multi-part problem you solve by hand, often from a graph, a table, an equation or a differential equation. You show your work, use exact values, and justify answers with calculus reasons such as a sign change in a derivative or the conditions of a theorem. On the exam: 4 questions in Part B of the free-response section (60 minutes, no calculator). The whole free-response section is 6 questions in 90 minutes and counts for 50% of the score. AB and BC use the same format; on BC, Part B usually includes a series question and a question on BC-only topics.

The question

Answer each part without a calculator. Show the work that leads to your answers.

Suggested time: 15 minutes

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Part (a)

2 points

Let f be the function defined by f(x) = (√(x + 5) − 3)/(x − 4) for x ≠ 4, and f(4) = k, where k is a constant. Find the value of k for which f is continuous at x = 4. Justify your answer.

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Part (b)

3 points

Let g be the function defined by g(x) = (2x² − 8)/(x² − x − 2). The graph of g has a removable discontinuity at x = 2. Find lim (x→2) g(x). Then find the equations of all vertical asymptotes and all horizontal asymptotes of the graph of g. Use limits to support your answers.

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Part (c)

2 points

The function p satisfies 4x − 1 ≤ p(x) ≤ x² + 3 for all real numbers x. Find lim (x→2) p(x). Justify your answer.

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Part (d)

2 points

Find lim (h→0) [tan(π/4 + h) − 1]/h. Show the work that leads to your answer.

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