Limit of $\frac{\sin\left(2 x\right)}{x}$ as $x \to 0$

Q: What is the limit of sin(2x) / x as x approaches 0?

The limit of (≤ft(2 x))/(x) as x → 0 is 2.

The limit of $\frac{\sin\left(2 x\right)}{x}$ as $x \to 0$ is $2$ .

\lim_{x \to 0} \frac{\sin\left(2 x\right)}{x} = 2

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

How to work it out

1
We want what sin(2x) / x approaches as x → 0.
$\lim_{x \to 0} \frac{\sin\left(2 x\right)}{x}$
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.
$\lim \frac{\sin\left(2 x\right)}{x} = \lim \frac{2 \cos\left(2 x\right)}{1}$
4
Now substitute x = 0: the denominator is not 0 any more.
$\frac{2}{1} = 2$
5
Method: write sin(2x)/x = 2 · sin(2x)/(2x) and use sin u / u → 1.
6
Result:
$\lim_{x \to 0} \frac{\sin\left(2 x\right)}{x} = 2$

LaTeX source of the formulas on this page

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

Values of sin(2x) / x as x → 0 (computed)

x	f(x)	Distance from the limit
0.1	1.9866933	1.33e-2
-0.1	1.9866933	1.33e-2
0.01	1.9998667	1.33e-4
-0.01	1.9998667	1.33e-4
0.001	1.9999987	1.33e-6
-0.001	1.9999987	1.33e-6
0.0001	2.0000000	1.33e-8
-0.0001	2.0000000	1.33e-8
0.00001	2.0000000	1.33e-10
-0.00001	2.0000000	1.33e-10

Questions

What is the limit of sin(2x) / x as x approaches 0?

The limit of $\frac{\sin\left(2 x\right)}{x}$ as $x \to 0$ is $2$ .

Which form is it?

0/0.

Limit of sin⁡(2x)x\frac{\sin\left(2 x\right)}{x}xsin(2x)​ as x→0x \to 0x→0

How to work it out

Values of sin(2x) / x as x → 0 (computed)

Questions

More well-known limits

Limit of $\frac{\sin\left(2 x\right)}{x}$ as $x \to 0$