Limits of Trigonometric functions

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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