There are two standard results in which trigonometric functions are involved and they’re used as rules to find the limits of functions in which trigonometric functions are appeared.

If $x$ is used to represent angle of right triangle, then the trigonometric functions sine and tangent are written as $\sin{x}$ and $\tan{x}$ respectively.

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

The limit of quotient of $\sin{x}$ by $x$ as $x$ approaches $0$ is equal to $1$.

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\tan{x}}{x}} \,=\, 1$

The limit of ratio of $\tan{x}$ to $x$ as $x$ tends to $0$ is equal to $1$.

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