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« Limit of Irrational Functions
Lhospital Rule »
Limits: (lesson 4 of 5)

Limits of Trigonometric Functions

Important limits:

lim⁡x→0sin⁡xx=1Example: lim⁡x→0sin⁡3xx=lim⁡x→03sin⁡3x3x=3⋅1=3 \begin{aligned} &\color{blue}{\mathop {\lim }\limits_{x \to 0} \frac{\sin x}{x} = 1} \\ \text{Example:} \ &\mathop {\lim }\limits_{x \to 0} \frac{\sin 3x}{x} = \mathop {\lim }\limits_{x \to 0} \frac{3 \sin{3x}}{3x} = 3 \cdot 1 = 3 \\ \end{aligned}

lim⁡x→01−cos⁡xx=0 \color{blue}{\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos x}}{x} = 0}

Example

Find the limit:

lim⁡x→0tan⁡xx=0 \mathop {\lim }\limits_{x \to 0} \frac{{\tan x}}{x} = 0

Solution

Direct substitution gives the indeterminate form 00\frac{0}{0}. This problem can still be solved, however, by writing tan⁡x\tan x as sin⁡xcosx\frac{\sin x}{cos x}.

lim⁡x→0tan⁡xx=lim⁡x→0(sin⁡xx)(1cos⁡x)=lim⁡x→0sin⁡xx⋅lim⁡x→01cos⁡x=1 \begin{aligned} &\mathop {\lim }\limits_{x \to 0} \frac{{\tan x}}{x} = \mathop {\lim }\limits_{x \to 0} \left( {\frac{{\sin x}}{x}} \right)\left( {\frac{1}{{\cos x}}} \right) \\ &= \mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} \cdot \mathop {\lim }\limits_{x \to 0} \frac{1}{{\cos x}} = 1 \end{aligned}

Algebra

Calculus

Analytic geometry

Linear Algebra

Random Quote

Black holes result from God dividing the universe by zero.

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Random Quote

Black holes result from God dividing the universe by zero.

Author Unknown