Evaluate the following limits:

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: $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$

So $\lim _{x \rightarrow 0} \frac{\tan 3 x}{\tan 5 x}=\lim _{x \rightarrow 0}\left(\frac{\tan 3 x}{3 x}\right) \times \frac{5 x}{\sin 5 x} \times \frac{3 x}{5 x}=\frac{3 x}{5 x}=\frac{3}{5}$

Therefore, $\lim _{x \rightarrow 0} \frac{\tan 3 x}{\tan 5 x}=\frac{3}{5}$