CalcRef

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.

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

Find lim(x→0) (1 - cos x)/(x sin x).
  1. Rewrite as [(1 - cos x)/x] · [1/sin x] · x/x
  2. = [(1-cos x)/x] · [x/sin x]
  3. lim(x→0)(1-cos x)/x = 0 ... but this gives 0/0 form
  4. Instead multiply top and bottom by (1+cos x): (1-cos²x)/(x sin x (1+cos x)) = sin²x/(x sin x(1+cos x)) = sin x/(x(1+cos x))
  5. lim = 1/(1·2)

Answer: 1/2

Related formulas

Limit of sin(x)/x

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