Question:
Evaluate the following limits:
$\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 8 x}$
Solution:
To Find: Limits
NOTE: First Check the form of imit. Used this method if the limit is satisfied any one from 7 indeterminate form.
In this Case, indeterminate Form is $\frac{0}{0}$
Formula used: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
So $\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 8 x}=\lim _{x \rightarrow 0}\left(\frac{\sin 5 x}{5 x}\right) \times \frac{8 x}{\sin 8 x} \times \frac{5 x}{8 x}=\frac{5 x}{8 x}=\frac{5}{8}$
Therefore, $\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 8 x}=\frac{5}{8}$