The trigonometric functions appear in some cases while finding the limits of the functions in calculus. Some special limit laws are required to find the limits of the functions in which trigonometric functions are involved and they are called the trigonometric limits rules.

There are two types of trigonometric limit properties in calculus and they are used as formulas, while evaluating the limits of functions in which trigonometric functions are involved.

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

The limit of sine of angle $x$ divided by $x$ as the value of $x$ approaches to $0$ is equal to one. It is called the trigonometric limit rule in sine function.

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

The limit of tangent of angle $x$ divided by $x$ as the value of $x$ tends to $0$ is equal to one. It is called the trigonometric limit rule in tan function.

$(1).\,\,$ Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{3x}}{\sin{4x}}}$

$(2).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1-\cos{(2x)}}{x^2}}$

$(3).\,\,$ Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \Big(\dfrac{\sin{x}}{x}\Big)^{\dfrac{1}{x^2}}}$

You have learned the trigonometric limits formulas with proofs. It is time to learn how to use them in limit questions in which trigonometric functions are involved.

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