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Find the limit as x approaches 0 of (1 - cos x)/x

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Answer
0\boxed{0}

Checking the result with a small value of xx (like x=0.01x = 0.01), 1−cos⁡(0.01)0.01≈0.005\frac{1 - \cos(0.01)}{0.01} \approx 0.005, which approaches 00.

Problem: Find the limit lim⁡x→01−cos⁡xx\lim_{x \to 0} \frac{1 - \cos x}{x}.

Steps

  1. Direct substitution
    1−cos⁡(0)0=1−10=00\frac{1 - \cos(0)}{0} = \frac{1 - 1}{0} = \frac{0}{0}
  2. Apply L'Hôpital's Rule
    lim⁡x→0ddx(1−cos⁡x)ddx(x)=lim⁡x→0sin⁡x1\lim_{x \to 0} \frac{\frac{d}{dx}(1 - \cos x)}{\frac{d}{dx}(x)} = \lim_{x \to 0} \frac{\sin x}{1}
  3. Evaluate the limit by direct substitution
    sin⁡(0)1=01=0\frac{\sin(0)}{1} = \frac{0}{1} = 0

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