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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)

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L'Hospital's Rule

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L'Hospital's Rule

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1
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Which of the following limits could you not evaluate using L'Hospital's rule?

  • limit as x rightwards arrow 0 of open parentheses x over x close parentheses

  • limit as x rightwards arrow 1 of open parentheses fraction numerator x squared plus 1 over denominator 1 minus x end fraction close parentheses

  • limit as x rightwards arrow 2 of open parentheses fraction numerator x squared plus 4 x minus 12 over denominator x minus 2 end fraction close parentheses

  • limit as x rightwards arrow 3 of open parentheses fraction numerator x squared minus 9 over denominator sin open parentheses fraction numerator pi x over denominator 3 end fraction close parentheses end fraction close parentheses

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2
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limit as x rightwards arrow 4 of fraction numerator x cubed minus 64 over denominator x minus 4 end fraction equals

  • 0

  • infinity

  • 48

  • 12

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3
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limit as x rightwards arrow 1 of fraction numerator x squared plus 4 x minus 5 over denominator x squared plus x minus 2 end fraction equals

  • 0

  • 1

  • 2

  • infinity

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4
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limit as x rightwards arrow 0 of fraction numerator sin x over denominator x end fraction is

  • 0

  • 1

  • 2

  • nonexistent

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5
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limit as x rightwards arrow 0 of fraction numerator 1 minus cos x over denominator x end fraction is

  • 0

  • 1

  • 2

  • nonexistent

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1
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Which of the following limits could you not attempt to evaluate using L'Hospital's rule?

  • limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos x over denominator sin x end fraction close parentheses

  • limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos x over denominator x sin x end fraction close parentheses

  • limit as x rightwards arrow pi over 2 of open parentheses fraction numerator tan x over denominator sin 2 x end fraction close parentheses

  • limit as x rightwards arrow pi of open parentheses fraction numerator sin x over denominator sin 7 x end fraction close parentheses

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2
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If f to the power of apostrophe open parentheses x close parentheses equals sin x minus 1 and g to the power of apostrophe open parentheses x close parentheses equals 2 for all x, and if f open parentheses 0 close parentheses equals g open parentheses 0 close parentheses equals 0, then limit as x rightwards arrow 0 of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction is

  • nonexistent

  • negative 1 half

  • 0

  • 1 half

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3
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limit as x rightwards arrow fraction numerator 3 pi over denominator 2 end fraction space of fraction numerator sin space x plus 1 over denominator cos space x end fraction is

  • 0

  • infinity

  • 1

  • nonexistent

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4
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limit as x rightwards arrow 0 space of fraction numerator tan 3 x over denominator sin 5 x end fraction equals

  • negative 3 over 5

  • negative 9 over 25

  • 9 over 25

  • 3 over 5

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5
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limit as x rightwards arrow 0 space of fraction numerator 1 minus cos 3 x over denominator 1 minus cos 7 x end fraction equals

  • negative 3 over 7

  • negative 9 over 49

  • 9 over 49

  • 3 over 7

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1
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If a not equal to 0, then limit as x rightwards arrow a of fraction numerator x squared minus a squared over denominator x to the power of 5 minus a to the power of 5 end fraction is

  • nonexistent

  • 0

  • fraction numerator 2 over denominator 5 a cubed end fraction

  • fraction numerator 3 over denominator 7 a cubed end fraction

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2
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limit as x rightwards arrow 0 of fraction numerator 1 minus cos squared open parentheses 3 x close parentheses over denominator open parentheses 3 x close parentheses squared end fraction equals

  • 0

  • 1 third

  • 1 over 9

  • 1

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3
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limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos x over denominator x sin 2 x end fraction close parentheses is

  • 0

  • 1 fourth

  • 1 half

  • nonexistent

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4
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limit as x rightwards arrow 0 of x cot 3 x equals

  • 0

  • 1 third

  • 1

  • 3

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5
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limit as x rightwards arrow 1 of fraction numerator sin open parentheses sin pi x close parentheses over denominator x squared minus 1 end fraction is

  • negative pi over 2

  • 0

  • pi over 2

  • nonexistent

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