« Limit of Irrational Functions

Lhospital Rule »

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 $\frac{\sin x}{cos x}$.

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