5.10. Calculate the limit

$$
\lim _{x \rightarrow 0} \frac{\tan 6 x}{\sin 3 x}
$$
Solution. To calculate this limit, which represents an indeterminate form of the type "0/0", it is best to use the theorem on the replacement of infinitesimal functions with equivalent quantities.

According to this theorem, as $x \rightarrow 0, \operatorname{tg} 6 x \sim 6 x$, $\sin 3 x \sim 3 x$. Taking this into account, we have

$$
\lim _{x \rightarrow 0} \frac{\operatorname{tg} 6 x}{\sin 3 x}=\lim _{x \rightarrow 0} \frac{6 x}{3 x}=2
$$