Question:
Evaluate the following limits:
$\lim _{x \rightarrow 0} \frac{\sin 4 x}{\tan 7 x}$
Solution:
To Find: Limits
NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form.
In this Case, indeterminate Form is $\frac{0}{0}$
Formula used: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ and $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$
So $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\tan 7 x}=\lim _{x \rightarrow 0}\left(\frac{\sin 4 x}{4 x}\right) \times \frac{7 x}{\sin 7 x} \times \frac{4 x}{7 x}=\frac{4 x}{7 x}=\frac{4}{7}$
Therefore, $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\tan 7 x}=\frac{4}{7}$