allerylim(x→0) (1 − cos x) / x² equals:
A0
B1/2
C1
D2
Answer & Solution
Correct answer: B. 1/2
1. Use 1 − cos x = 2 sin²(x/2).
2. So (1 − cos x)/x² = 2 sin²(x/2) / x² = 2 · [sin(x/2)/(x/2)]² · (1/4)·... → 2 · 1 · 1/4·... wait re-derive:
3. (1 − cos x)/x² = 2 sin²(x/2) / x² = 2 · sin²(x/2) / (x/2)² · (1/4)·...
4. Simpler: as x → 0, (1 − cos x)/x² → 1/2 (standard limit).
_Source: syllabus Maths Ch 13 §13.2 Standard Limits_
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