Limit of $\frac{1 - \cos\left(x\right)}{x^{2}}$ as $x \to 0$

Q: What is the limit of (1 - cos(x)) / x^2 as x approaches 0?

The limit of 1 - ≤ft(x)x^2 as x → 0 is 12.

The limit of $\frac{1 - \cos\left(x\right)}{x^{2}}$ as $x \to 0$ is $\frac12$ .

\lim_{x \to 0} \frac{1 - \cos\left(x\right)}{x^{2}} = \frac12

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Graph of (1 - cos(x)) / x^2 near x = 0. The open circle marks the point x = 0, where the function may be undefined; the curve still heads to 1/2.

How to work it out

1
We want what (1 - cos(x)) / $x^{2}$ approaches as x → 0.
$\lim_{x \to 0} \frac{1 - \cos\left(x\right)}{x^{2}}$
2
Direct substitution gives an indeterminate form: 0/0. It does not tell us the limit.
3
L'Hôpital's rule: when the form is 0/0, the limit equals the limit of the ratio of the derivatives.
$\lim \frac{1 - \cos\left(x\right)}{x^{2}} = \lim \frac{\sin\left(x\right)}{2 x}$
4
Still 0/0 (both vanish again), so apply it once more.
$\lim \frac{\sin\left(x\right)}{2 x} = \lim \frac{\cos\left(x\right)}{2}$
5
Now substitute x = 0: the denominator is not 0 any more.
$\frac{1}{2} = 0.5$
6
Method: multiply by 1 + cos x, or use the series cos x = 1 − x²/2 + ….
7
Result:
$\lim_{x \to 0} \frac{1 - \cos\left(x\right)}{x^{2}} = \frac12$

LaTeX source of the formulas on this page

\[ \lim_{x \to 0} \frac{1 - \cos\left(x\right)}{x^{2}} = \frac12 \]
\[ \lim_{x \to 0} \frac{1 - \cos\left(x\right)}{x^{2}} \]
\[ \lim \frac{1 - \cos\left(x\right)}{x^{2}} = \lim \frac{\sin\left(x\right)}{2 x} \]
\[ \lim \frac{\sin\left(x\right)}{2 x} = \lim \frac{\cos\left(x\right)}{2} \]
\[ \frac{1}{2} = 0.5 \]

Values of (1 - cos(x)) / $x^{2}$ as x → 0 (computed)

x	f(x)	Distance from the limit
0.1	0.49958347	4.17e-4
-0.1	0.49958347	4.17e-4
0.01	0.49999583	4.17e-6
-0.01	0.49999583	4.17e-6
0.001	0.49999996	4.17e-8
-0.001	0.49999996	4.17e-8
0.0001	0.50000000	3.04e-9
-0.0001	0.50000000	3.04e-9
0.00001	0.50000004	4.14e-8
-0.00001	0.50000004	4.14e-8

Questions

What is the limit of (1 - cos(x)) /

x^{2}

as x approaches 0?

The limit of $\frac{1 - \cos\left(x\right)}{x^{2}}$ as $x \to 0$ is $\frac12$ .

Which form is it?

0/0.

Limit of 1−cos⁡(x)x2\frac{1 - \cos\left(x\right)}{x^{2}}x21−cos(x)​ as x→0x \to 0x→0

How to work it out

Values of (1 - cos(x)) / x2x^{2}x2 as x → 0 (computed)

Questions

More well-known limits

Limit of $\frac{1 - \cos\left(x\right)}{x^{2}}$ as $x \to 0$

Values of (1 - cos(x)) / $x^{2}$ as x → 0 (computed)