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Limits & Continuity · Special Limits

Limit of (cos x - 1)/x

lim⁡x→0cos⁡x−1x=0\lim_{x \to 0} \frac{\cos x - 1}{x} = 0

A special limit related to the derivative of cosine at x = 0.

  • 2 worked examples
  • 18 in Limits & Continuity

Worked examples

Find lim(x→0) (cos x - 1)/x².
  1. Use the identity cos x - 1 = -2 sin²(x/2)
  2. -2 sin²(x/2)/x² = -2 · [sin(x/2)]²/x² = -(1/2) · [sin(x/2)/(x/2)]²
  3. As x→0, sin(x/2)/(x/2) → 1

Answer: -1/2

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Find lim(x→0) (1 - cos x)/(x sin x).

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Related formulas

Limit of sin(x)/x

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Common questions

What is the Limit of (cos x - 1)/x?

A special limit related to the derivative of cosine at x = 0. It is one of the limits & continuity formulas in the CalcRef reference.

How do you use the Limit of (cos x - 1)/x?

Worked example. Find lim(x→0) (cos x - 1)/x². Use the identity cos x - 1 = -2 sin²(x/2). -2 sin²(x/2)/x² = -2 · [sin(x/2)]²/x² = -(1/2) · [sin(x/2)/(x/2)]². As x→0, sin(x/2)/(x/2) → 1. Answer: -1/2.

Practice

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