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lim_x→0 (1 − cos x) / x² = 1/2

The limit of (1 − cos x) / x² as x tends to 0 is 1/2.

Proof

Use 1 − cos x = 2 sin²(x/2).
(1 − cos x)/x² = 2 sin²(x/2)/x² = ½ × [sin(x/2)/(x/2)]².
As x → 0 the bracket tends to 1, so the limit is ½.

Standard limits

Limit	Value
lim_x→0 sin x / x	1
lim_x→0 tan x / x	1
lim_x→0 (1 − cos x) / x²	1/2
lim_x→0 (1 − cos x) / x	0
lim_x→0 (eˣ − 1) / x	1
lim_x→0 (aˣ − 1) / x	log a
lim_x→0 log(1 + x) / x	1
lim_x→0 (1 + x)^(1/x)	e
lim_x→∞ (1 + 1/x)ˣ	e
lim_x→∞ (1 + a/x)ˣ	eᵃ
lim_x→a (xⁿ − aⁿ) / (x − a)	n aⁿ⁻¹
lim_x→0 sin⁻¹x / x	1
lim_x→0 tan⁻¹x / x	1
lim_x→∞ sin x / x	0
lim_x→0⁺ x log x	0
lim_x→0⁺ xˣ	1

Questions students ask

What is the limit of (1 − cos x) / x² as x tends to 0?

1/2.

How do you prove lim x→0 (1 − cos x) / x² = 1/2?

Use 1 − cos x = 2 sin²(x/2). (1 − cos x)/x² = 2 sin²(x/2)/x² = ½ × [sin(x/2)/(x/2)]². As x → 0 the bracket tends to 1, so the limit is ½.

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