Limit $\dfrac{\sin x}{x}$ is $1$ when $x$ tends to zero


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


$x$ is a literal and represents an angle of the right angled triangle and $\sin x$ is the sine function. The ratio of $\sin x$ to $x$ is expressed as $\dfrac{\sin x}{x}$. The value of ratio of $\sin x$ to $x$ as $x$ approaches zero is expressed in mathematical form in limit form.

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

The range of the $\sin x$ function is $[-1, 1]$. It is evident that the values of $\sin x$ function lies from $-1$ to $1$. A special property of the sine function is revealed when you study its functionality closely for angles which are very close to zero. In other words, the values of $\sin x$ function is approximately equals to angles when the angles tend to zero.


$(1) \,\,\,\,\,$ $x = 0.176598 \implies \sin 0.176598$ $=$ $0.1756815076\cdots$ $\approx$ $0.176598$

$(2) \,\,\,\,\,$ $x = 0.053874 \implies \sin 0.053874$ $=$ $0.0538479431\cdots$ $\approx$ $0.053874$

$(3) \,\,\,\,\,$ $x = 0.001234 \implies \sin 0.001234$ $=$ $0.0012339996\cdots$ $\approx$ $0.001234$

$(4) \,\,\,\,\,$ $x = 0.000235 \implies \sin 0.000235$ $=$ $0.0002349999\cdots$ $\approx$ $0.000235$

$(5) \,\,\,\,\,$ $x = 0.000056 \implies \sin 0.000056$ $=$ $0.0000559999\cdots$ $\approx$ $0.000056$

The examples clear that the value of $\sin x$ is approximately equal to the angle. So, it is expressed as $\sin x \approx x$.

$$\implies \lim_{x \to 0} \dfrac{\sin x}{x} = \lim_{x \to 0} \dfrac{x}{x}$$

$$\require{cancel} \implies \lim_{x \to 0} \dfrac{\sin x}{x} = \lim_{x \to 0} \dfrac{\cancel{x}}{\cancel{x}}$$

$$\implies \lim_{x \to 0} \dfrac{\sin x}{x} = \lim_{x \to 0} 1$$

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

Therefore, the identity is evident that the value of ratio of $\sin x$ to $x$ is one when limit $x$ tends to zero.

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