Limit of $\frac{\sin\left(x\right) - x}{x^{3}}$ as $x \to 0$

Q: What is the limit of (sin(x) - x) / x^3 as x approaches 0?

The limit of ≤ft(x) - xx^3 as x → 0 is -16.

The limit of $\frac{\sin\left(x\right) - x}{x^{3}}$ as $x \to 0$ is $-\frac16$ .

\lim_{x \to 0} \frac{\sin\left(x\right) - x}{x^{3}} = -\frac16

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Graph of (sin(x) - x) / x^3 near x = 0. The open circle marks the point x = 0, where the function may be undefined; the curve still heads to −1/6.

How to work it out

1
We want what (sin(x) - x) / $x^{3}$ approaches as x → 0.
$\lim_{x \to 0} \frac{\sin\left(x\right) - x}{x^{3}}$
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.
$\lim \frac{\sin\left(x\right) - x}{x^{3}} = \lim \frac{\cos\left(x\right) - 1}{3 x^{2}}$
4
Still 0/0 (both vanish again), so apply it once more.
$\lim \frac{\cos\left(x\right) - 1}{3 x^{2}} = \lim \frac{-\sin\left(x\right)}{6 x}$
5
Still 0/0 (both vanish again), so apply it once more.
$\lim \frac{-\sin\left(x\right)}{6 x} = \lim \frac{-\cos\left(x\right)}{6}$
6
Now substitute x = 0: the denominator is not 0 any more.
$\frac{-1}{6} = -0.16666667$
7
Method: series: sin x = x − x³/6 + ….
8
Result:
$\lim_{x \to 0} \frac{\sin\left(x\right) - x}{x^{3}} = -\frac16$

LaTeX source of the formulas on this page

\[ \lim_{x \to 0} \frac{\sin\left(x\right) - x}{x^{3}} = -\frac16 \]
\[ \lim_{x \to 0} \frac{\sin\left(x\right) - x}{x^{3}} \]
\[ \lim \frac{\sin\left(x\right) - x}{x^{3}} = \lim \frac{\cos\left(x\right) - 1}{3 x^{2}} \]
\[ \lim \frac{\cos\left(x\right) - 1}{3 x^{2}} = \lim \frac{-\sin\left(x\right)}{6 x} \]
\[ \lim \frac{-\sin\left(x\right)}{6 x} = \lim \frac{-\cos\left(x\right)}{6} \]
\[ \frac{-1}{6} = -0.16666667 \]

Values of (sin(x) - x) / $x^{3}$ as x → 0 (computed)

x	f(x)	Distance from the limit
0.1	-0.16658335	8.33e-5
-0.1	-0.16658335	8.33e-5
0.01	-0.16666583	8.33e-7
-0.01	-0.16666583	8.33e-7
0.001	-0.16666666	8.33e-9
-0.001	-0.16666666	8.33e-9
0.0001	-0.16666666	5.18e-9
-0.0001	-0.16666666	5.18e-9
0.00001	-0.16666728	6.18e-7
-0.00001	-0.16666728	6.18e-7

Questions

What is the limit of (sin(x) - x) /

x^{3}

as x approaches 0?

The limit of $\frac{\sin\left(x\right) - x}{x^{3}}$ as $x \to 0$ is $-\frac16$ .

Which form is it?

0/0.

Limit of sin⁡(x)−xx3\frac{\sin\left(x\right) - x}{x^{3}}x3sin(x)−x​ as x→0x \to 0x→0

How to work it out

Values of (sin(x) - x) / x3x^{3}x3 as x → 0 (computed)

Questions

More well-known limits

Limit of $\frac{\sin\left(x\right) - x}{x^{3}}$ as $x \to 0$

Values of (sin(x) - x) / $x^{3}$ as x → 0 (computed)