Example 5. Find $\lim _{x \rightarrow 0} \frac{x \sin 6 x}{(\operatorname{arctg} 2 x)^{2}}$.
Solution. Taking into account that

$$
\sin c x \sim c x \text { and } \operatorname{arctg} c x \sim c x
$$

(see examples 3 and 4), we get

$$
\lim _{x \rightarrow 0} \frac{x \sin c x}{(\operatorname{arctg} 2 x)^{2}}=\lim _{x \rightarrow 0} \frac{x}{2 x} \cdot \frac{6 x}{2 x}=\frac{1}{2} \cdot 3=\frac{3}{2}
$$