# What is the limit of sin(6x)/x as x approaches 0?

Jul 23, 2018

${\lim}_{x \to 0} \sin \frac{6 x}{x} = 6$

#### Explanation:

Let ,
$L = {\lim}_{x \to 0} \sin \frac{6 x}{x} = {\lim}_{x \to 0} \sin \frac{6 x}{6 x} \times 6$

Subst. $6 x = \theta \implies x \to 0 , t h e n , \theta \to 0$

So.

$L = {\lim}_{\theta \to 0} \frac{\sin \theta}{\theta} \times 6 = \left(1\right) \times 6 = 6$

$6$

#### Explanation:

$\setminus {\lim}_{x \setminus \to 0} \setminus \frac{\setminus \sin \left(6 x\right)}{x}$

$= \setminus {\lim}_{x \setminus \to 0} \setminus \frac{6 \setminus \sin \left(6 x\right)}{6 x}$

$= 6 \setminus {\lim}_{x \setminus \to 0} \setminus \frac{\setminus \sin \left(6 x\right)}{\left(6 x\right)}$

$= 6 \setminus \cdot 1 \setminus \quad \left(\setminus \because \setminus {\lim}_{t \setminus \to 0} \setminus \frac{\setminus \sin t}{t} = 1\right)$

$= 6$

Jul 23, 2018

${\lim}_{x \to 0} \frac{\sin \left(6 x\right)}{x} = 6$

#### Explanation:

${\lim}_{x \to 0} \frac{\sin \left(6 x\right)}{x}$

Let $y = 6 x$
$\frac{y}{6} = x$
As $x$ approaches $0$, $y$ also approaches $0$.

$\therefore {\lim}_{y \to 0} \frac{\sin \left(y\right)}{\frac{y}{6}}$
$= {\lim}_{y \to 0} \frac{6 \sin \left(y\right)}{y}$
$= 6 {\lim}_{y \to 0} \sin \frac{y}{y}$
$= 6$

Note: ${\lim}_{a \to 0} \sin \frac{a}{a} = 1$ is a common limit and has been proven countless times.