AP®MathsCollege BoardCalculus ABExam QuestionsUnit 4: Contextual Applications of DifferentiationL'Hospital's Rule

L'Hospital's Rule (College Board AP® Calculus AB)

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L'Hospital's Rule

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L'Hospital's Rule

1
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Find the value of limit as x rightwards arrow 0 of open parentheses fraction numerator sin x over denominator x squared plus 2 x end fraction close parentheses, or show that it does not exist. Justify your answer.

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2
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Use L'Hospital's rule to find the value of limit as x rightwards arrow infinity of open parentheses fraction numerator 7 x minus 3 x squared over denominator 12 x squared plus 5 x end fraction close parentheses.

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3
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A graph of f', the derivative of the function f, consisting of a line segment connecting points (0, 5) and (2, 5), another line segment connecting points (2, 5) and (5, 0), and a semicircle of radius 2 below the x-axis connecting points (5, 0) and (9, 0)

The function f is defined on the closed interval open square brackets 0 comma space 9 close square brackets and satisfies f open parentheses 3 close parentheses equals 2. The graph of f to the power of apostrophe, the derivative of f, consists of two line segments and a semicircle, as shown in the figure above.

Find the value of limit as x rightwards arrow 3 of open parentheses fraction numerator 2 x minus 3 f open parentheses x close parentheses over denominator x squared minus x minus 6 end fraction close parentheses, or show that it does not exist. Justify your answer.

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4
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A graph of the function f consisting of three straight line segments, connecting the points (-2, -2) and (0, -4), (0, -4) and (4, 4), and (4, 4) and (8, 0)

Let f be a continuous function defined on the closed interval negative 2 less or equal than x less or equal than 8. The graph of f, consisting of three line segments, is shown in the figure above. Let G be the function defined by G open parentheses x close parentheses equals integral subscript 0 superscript x f open parentheses t close parentheses space d t.

Find limit as x rightwards arrow 4 of open parentheses fraction numerator x squared minus 3 x minus 4 over denominator G open parentheses x close parentheses end fraction close parentheses.

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5
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Functions f and g are differentiable functions with g open parentheses 3 close parentheses equals 1.

The function g satisfies g open parentheses x close parentheses equals fraction numerator 9 minus x squared over denominator open parentheses f open parentheses x close parentheses close parentheses cubed minus 8 end fraction for x not equal to 3. It is known that limit as x rightwards arrow 3 of g open parentheses x close parentheses can be evaluated using L'Hospital's rule. Use limit as x rightwards arrow 3 of g open parentheses x close parentheses to find f open parentheses 3 close parentheses and f to the power of apostrophe open parentheses 3 close parentheses. Show the work that leads to your answers.

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