Limits: (lesson 4 of 5)

## Limits of Trigonometric Functions

### Important limits:

$$
\begin{aligned}
&\color{blue}{\mathop {\lim }\limits_{x \to 0} \frac{\sin x}{x} = 1} \\
\text{Example:} \ &\mathop {\lim }\limits_{x \to 0} \frac{\sin 3x}{x} = \mathop {\lim }\limits_{x \to 0} \frac{3 \sin{3x}}{3x} = 3 \cdot 1 = 3 \\
\end{aligned}
$$

$$
\color{blue}{\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos x}}{x} = 0}
$$

Example

Find the limit:

$$ \mathop {\lim }\limits_{x \to 0} \frac{{\tan x}}{x} = 0 $$

Solution

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

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