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

## Formula

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

The limit of ratio of sin of angle to angle as the angle approaches zero is equal to one. This standard result is used as a rule to evaluate the limit of a function in which sine is involved.

### Introduction

$x$ is a variable and represents angle of a right triangle. The sine function is written as $\sin{x}$ as per trigonometry. The limit of quotient of $\sin{x}$ by $x$ as $x$ approaches zero is often appeared in calculus.

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

Actually, the limit of $\sin{(x)}/x$ as $x$ tends to $0$ is equal to $1$ and this standard trigonometric function result is used as a formula everywhere in calculus.

### Proof

There are two ways to prove this limit of trigonometric function property in mathematics.

#### Relation between Sine function and angle

It is derived on the basis of close relation between $\sin{x}$ function and angle $x$ as the angle $x$ closer to zero.

#### Taylor (or) Maclaurin Series Method

It can also be derived by the expansion of $\sin{x}$ function as per Taylor (or) Maclaurin series.

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