L'Hôpital's Rule Practice

Sample questions

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1. Sample 1difficulty 2/5

   Evaluate $\displaystyle\lim_{x\to 0}\dfrac{\sin(2x)}{\tan(3x)}$.

   • A

     $0$

   • B

     $\tfrac{2}{3}$

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   • C

     $\tfrac{3}{2}$

   • D

     $1$

   Why

   L'Hôpital: $2\cos(2x)/(3\sec^2(3x)) \to 2/3$.

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2. Sample 2difficulty 2/5

   Evaluate $\displaystyle\lim_{x\to 0}\dfrac{e^x - 1 - x}{x^2}$.

   • A

     $1$

   • B

     $\tfrac{1}{2}$

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   • C

     $0$

   • D

     $2$

   Why

   L'Hôpital twice: $(e^x - 1)/(2x) \to e^x/2 \to 1/2$.

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3. Sample 3difficulty 2/5

   Evaluate $\displaystyle\lim_{x\to 1}\dfrac{x^3 - 1}{x^2 - 1}$.

   • A

     $1$

   • B

     $3$

   • C

     $\tfrac{3}{2}$

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   • D

     $2$

   Why

   L'Hôpital: $3x^2/(2x) = 3x/2 \to 3/2$ at $x = 1$.

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4. Sample 4difficulty 2/5

   Evaluate $\displaystyle\lim_{x\to 0}\dfrac{1 - \cos x}{x^2}$.

   • A

     $0$

   • B

     $1$

   • C

     $-1$

   • D

     $\tfrac{1}{2}$

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   Why

   $0/0$. L'Hôpital twice: $\sin x/(2x) \to 1/2 \cdot \cos x \to 1/2$.

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5. Sample 5difficulty 2/5

   Evaluate $\displaystyle\lim_{x\to 0}\dfrac{\sin x}{x}$ using L'Hôpital.

   • A

     $0$

   • B

     $1$

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   • C

     $\tfrac{1}{2}$

   • D

     Does not exist

   Why

   $0/0$ form. L'Hôpital: $\cos x/1 \to 1$ at $x = 0$.

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