Limit of $\frac{\sin\left(4 x\right)}{x}$ as $x \to 0$

Q: What is the limit of sin(4x) / x as x approaches 0?

The limit of (≤ft(4 x))/(x) as x → 0 is 4.

The limit of $\frac{\sin\left(4 x\right)}{x}$ as $x \to 0$ is $4$ .

\lim_{x \to 0} \frac{\sin\left(4 x\right)}{x} = 4

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Graph of sin(4x) / x near x = 0. The open circle marks the point x = 0, where the function may be undefined; the curve still heads to 4.

How to work it out

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

LaTeX source of the formulas on this page

\[ \lim_{x \to 0} \frac{\sin\left(4 x\right)}{x} = 4 \]
\[ \lim_{x \to 0} \frac{\sin\left(4 x\right)}{x} \]
\[ \lim \frac{\sin\left(4 x\right)}{x} = \lim \frac{4 \cos\left(4 x\right)}{1} \]
\[ \frac{4}{1} = 4 \]

Values of sin(4x) / x as x → 0 (computed)

x	f(x)	Distance from the limit
0.1	3.8941834	1.06e-1
-0.1	3.8941834	1.06e-1
0.01	3.9989334	1.07e-3
-0.01	3.9989334	1.07e-3
0.001	3.9999893	1.07e-5
-0.001	3.9999893	1.07e-5
0.0001	3.9999999	1.07e-7
-0.0001	3.9999999	1.07e-7
0.00001	4.0000000	1.07e-9
-0.00001	4.0000000	1.07e-9

Questions

What is the limit of sin(4x) / x as x approaches 0?

The limit of $\frac{\sin\left(4 x\right)}{x}$ as $x \to 0$ is $4$ .

Which form is it?

0/0.

Limit of sin⁡(4x)x\frac{\sin\left(4 x\right)}{x}xsin(4x)​ as x→0x \to 0x→0

How to work it out

Values of sin(4x) / x as x → 0 (computed)

Questions

More well-known limits

Limit of $\frac{\sin\left(4 x\right)}{x}$ as $x \to 0$