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

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

The limit of ≤ft(x) - ≤ft(x)x^3 as x → 0 is 12.

The limit of $\frac{\tan\left(x\right) - \sin\left(x\right)}{x^{3}}$ as $x \to 0$ is $\frac12$ .

\lim_{x \to 0} \frac{\tan\left(x\right) - \sin\left(x\right)}{x^{3}} = \frac12

Explain it step by step in chat Make a short video of it

Graph of (tan(x) - sin(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 1/2.

How to work it out

1
We want what (tan(x) - sin(x)) / $x^{3}$ approaches as x → 0.
$\lim_{x \to 0} \frac{\tan\left(x\right) - \sin\left(x\right)}{x^{3}}$
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.
$\lim \frac{\tan\left(x\right) - \sin\left(x\right)}{x^{3}} = \lim \frac{\frac{1}{\cos\left(x\right)^{2}} - \cos\left(x\right)}{3 x^{2}}$
4
Still 0/0 (both vanish again), so apply it once more.
$\lim \frac{\frac{1}{\cos\left(x\right)^{2}} - \cos\left(x\right)}{3 x^{2}} = \lim \frac{\frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - -\sin\left(x\right)}{6 x}$
5
Still 0/0 (both vanish again), so apply it once more.
$\lim \frac{\frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - -\sin\left(x\right)}{6 x} = \lim \frac{\frac{-\left(\left(2 -\sin\left(x\right) -\sin\left(x\right) + 2 \cos\left(x\right) -\cos\left(x\right)\right) \left(\cos\left(x\right)^{2}\right)^{2}\right) - -\left(2 \cos\left(x\right) -\sin\left(x\right) 2 \cos\left(x\right)^{2} 2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\left(\cos\left(x\right)^{2}\right)^{2}\right)^{2}} - -\cos\left(x\right)}{6}$
6
Now substitute x = 0: the denominator is not 0 any more.
$\frac{3}{6} = 0.5$
7
Method: series: tan x − sin x = x³/2 + ….
8
Result:
$\lim_{x \to 0} \frac{\tan\left(x\right) - \sin\left(x\right)}{x^{3}} = \frac12$

LaTeX source of the formulas on this page

\[ \lim_{x \to 0} \frac{\tan\left(x\right) - \sin\left(x\right)}{x^{3}} = \frac12 \]
\[ \lim_{x \to 0} \frac{\tan\left(x\right) - \sin\left(x\right)}{x^{3}} \]
\[ \lim \frac{\tan\left(x\right) - \sin\left(x\right)}{x^{3}} = \lim \frac{\frac{1}{\cos\left(x\right)^{2}} - \cos\left(x\right)}{3 x^{2}} \]
\[ \lim \frac{\frac{1}{\cos\left(x\right)^{2}} - \cos\left(x\right)}{3 x^{2}} = \lim \frac{\frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - -\sin\left(x\right)}{6 x} \]
\[ \lim \frac{\frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - -\sin\left(x\right)}{6 x} = \lim \frac{\frac{-\left(\left(2 -\sin\left(x\right) -\sin\left(x\right) + 2 \cos\left(x\right) -\cos\left(x\right)\right) \left(\cos\left(x\right)^{2}\right)^{2}\right) - -\left(2 \cos\left(x\right) -\sin\left(x\right) 2 \cos\left(x\right)^{2} 2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\left(\cos\left(x\right)^{2}\right)^{2}\right)^{2}} - -\cos\left(x\right)}{6} \]
\[ \frac{3}{6} = 0.5 \]

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

x	f(x)	Distance from the limit
0.1	0.50125544	1.26e-3
-0.1	0.50125544	1.26e-3
0.01	0.50001250	1.25e-5
-0.01	0.50001250	1.25e-5
0.001	0.50000013	1.25e-7
-0.001	0.50000013	1.25e-7
0.0001	0.50000000	2.00e-9
-0.0001	0.50000000	2.00e-9
0.00001	0.50000016	1.61e-7
-0.00001	0.50000016	1.61e-7

Questions

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

x^{3}

as x approaches 0?

The limit of $\frac{\tan\left(x\right) - \sin\left(x\right)}{x^{3}}$ as $x \to 0$ is $\frac12$ .

Which form is it?

0/0.

Limit of tan⁡(x)−sin⁡(x)x3\frac{\tan\left(x\right) - \sin\left(x\right)}{x^{3}}x3tan(x)−sin(x)​ as x→0x \to 0x→0

How to work it out

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

Questions

More well-known limits

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

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