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Squeeze Theorem

AP Calculus AB· difficulty 2/5

Evaluate lim⁡x→0tan⁡(3x)x\displaystyle\lim_{x\to 0}\dfrac{\tan(3x)}{x}.

  • A

    11

  • B

    33

    check_circle
  • C

    00

  • D

    13\tfrac{1}{3}

Explanation

tan⁡(3x)/x=(sin⁡3x)/(xcos⁡3x)=3⋅(sin⁡3x)/(3x)⋅1/cos⁡3x→3⋅1⋅1=3\tan(3x)/x = (\sin 3x)/(x\cos 3x) = 3 \cdot (\sin 3x)/(3x) \cdot 1/\cos 3x \to 3 \cdot 1 \cdot 1 = 3.

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