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

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

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L'Hospital's Rule

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1
Sme Calculator4 marks

For each of the following limits, determine whether or not L’Hospital’s rule may be used to evaluate the limit, giving a reason for your answer. In each case, if L’Hospital’s rule may be used, then use the rule to evaluate the limit.

(i)  limit as x rightwards arrow 0 of space fraction numerator sin space x over denominator x squared plus 2 x end fraction

(ii) limit as x rightwards arrow 0 of space fraction numerator cos space x over denominator x squared plus 2 x end fraction

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2
Sme Calculator3 marks

Evaluate limit as x rightwards arrow infinity of space open parentheses fraction numerator 7 minus 3 x over denominator 12 x plus 5 end fraction close parentheses, showing your full reasoning.

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31 mark

Use L'Hospital's Rule to evaluate limit as x rightwards arrow 0 of fraction numerator e to the power of x minus e to the power of negative x end exponent over denominator 2 x end fraction .

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4
Sme Calculator1 mark

Evaluate limit as x rightwards arrow negative 2 of open parentheses fraction numerator x cubed minus x squared minus 5 x plus 2 over denominator 5 x cubed plus 12 x squared minus 3 x minus 14 end fraction close parentheses showing your full reasoning.

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5
Sme Calculator3 marks

Given that space p greater than 0 is a constant, evaluate limit as x rightwards arrow p of open parentheses fraction numerator p minus x over denominator ln space p space minus space ln space x end fraction close parentheses showing your full reasoning.

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1
Sme Calculator2 marks

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
Sme Calculator2 marks

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
Sme Calculator3 marks

Find the value of limit as x rightwards arrow 0 of open parentheses fraction numerator negative sin open parentheses 5 pi x close parentheses over denominator 2 pi x end fraction close parentheses, or show that it does not exist. Justify your answer.

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4
Sme Calculator2 marks

Functions f and g are differentiable functions with g open parentheses 2 close parentheses equals 1.

The function g satisfies g open parentheses x close parentheses equals fraction numerator 4 minus x squared over denominator f open parentheses x close parentheses minus 3 end fraction for x not equal to 2. It is known that limit as x rightwards arrow 2 of g open parentheses x close parentheses can be evaluated using L'Hospital's rule.

Use limit as x rightwards arrow 2 of g open parentheses x close parentheses to find f open parentheses 2 close parentheses, showing the work that leads to your answer.

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5
Sme Calculator3 marks

Find the value of limit as x rightwards arrow 0 of open parentheses fraction numerator 2 e to the power of x open parentheses e to the power of x minus 1 close parentheses over denominator 3 sin x cos squared x end fraction close parentheses or show that it does not exist. Justify your answer.

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1
Sme Calculator3 marks
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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2
Sme Calculator2 marks
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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3
Sme Calculator4 marks

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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4
Sme Calculator4 marks

Let f be the function defined by f open parentheses x close parentheses equals e to the power of x sin open parentheses 2 x close parentheses.

Let g be a differentiable function such that g open parentheses pi close parentheses equals 0. The function g apostrophe, the derivative of g, is a piecewise linear function such that:

  • g apostrophe open parentheses x close parentheses equals x minus 2 space on space open parentheses 0 comma space pi close square brackets

  • g apostrophe open parentheses x close parentheses equals negative x over 2 plus fraction numerator 3 pi over denominator 2 end fraction minus 2 space on space open parentheses pi comma space 2 pi close square brackets

Find the value of limit as x rightwards arrow pi of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction or state that it does not exist. Justify your answer.

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5
Sme Calculator5 marks

Evaluate limit as x rightwards arrow infinity of open square brackets e to the power of x plus 2 x close square brackets to the power of 1 over x end exponent using L'Hospital's Rule.

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