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

The limit of the ratio of sine of an angle to the same angle is equal to one as the angle of a right triangle approaches zero. It is a most useful math property while finding the limit of any function in which the trigonometric function sine is involved. So, let us understand this limit rule in mathematical form.

- Denote an angle of a right triangle by a variable $x$.
- The ratio of the length of opposite side to the length of hypotenuse is represented by the sine of angle $x$. The sine of angle $x$ is written as $\sin{x}$ in trigonometry.

The ratio of the sine function $\sin{x}$ to the angle of a right-angled triangle $x$ is written as follows in mathematics.

$\dfrac{\sin{x}}{x}$

The limit of this rational function as the angle $x$ is closer to zero, is mathematically written as follows in calculus.

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

According to the calculus, the limit of the quotient sine of angle $x$ divided by the angle $x$ is one as the angle of a right triangle $x$ tends to zero.

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

The trigonometric limit rule in sine function is also popularly written in the following form in mathematics.

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

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

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

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

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