« Limit of Irrational Functions |
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 $\frac{\sin x}{cos x}$.
$$ \begin{aligned} &\mathop {\lim }\limits_{x \to 0} \frac{{\tan x}}{x} = \mathop {\lim }\limits_{x \to 0} \left( {\frac{{\sin x}}{x}} \right)\left( {\frac{1}{{\cos x}}} \right) \\ &= \mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} \cdot \mathop {\lim }\limits_{x \to 0} \frac{1}{{\cos x}} = 1 \end{aligned} $$