Example 1. Find $\lim _{x \rightarrow 0} \frac{\ln (1+\alpha x)}{\sin \beta x}$.
Since $\ln (1+\alpha x) \sim \alpha x, \sin \beta x \sim$ $\beta x(x \rightarrow 0$ when $)$, the original expression $=\lim _{x \rightarrow 0}-\frac{\sim x}{\beta^{3} x}=\frac{\alpha}{\beta}$.