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

Formula

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

The limit of quotient of sin of an angle by angle as the angle approaches zero is equal to one. It is a most useful limit rule for dealing trigonometric functions especially sine in calculus.

Proof

$x$ is a variable and also represents angle of the right triangle. The limit of quotient of sin of angle $x$ by angle $x$ as the angle $x$ tends to zero is written mathematically in calculus as follows.

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

The trigonometric limit property can be derived in mathematics in two different methods.

Relation of 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 sinx function as per Taylor (or) Maclaurin series.