# $\displaystyle \lim_{x \to 0} \dfrac{\sin{x}}{x}$ formula

## Formula

$\displaystyle \large \lim_{x \,\to\, 0} \dfrac{\sin{x}}{x} = 1$

If $x$ is angle of a right angled triangle then sine function is written as $\sin{x}$. The value of ratio of sin function to angle as the angle approaches zero is expressed in the following mathematical form.

$\displaystyle \large \lim_{x \,\to\, 0} \dfrac{\sin{x}}{x}$

The value of ratio of sin function to angle is equal to one as the angle approaches zero. It is used as a formula in calculus and the limit rule is called limit of ratio of sine function to angle rule as angle approaches zero.

### Proof

There are two methods to prove this limit rule in calculus.

#### Relation of Sine function and angle

The limit of ratio of $\sin{x}$ to $x$ when $x$ tends to zero is derived in calculus on the basis of relation of sine function with angle of the right angled triangle.

#### Taylor or Maclaurin Series Method

The limit of ratio of $\sin{x}$ to $x$ as $x$ approaches to zero can also be derived in calculus as per Taylor or Maclaurin series.