Example 1.

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Let $y=\sqrt[3]{1+3 x}, x=\frac{1}{3}\left(y^{3}-1\right)$, when $x \rightarrow 0$, $y \rightarrow 1$.

Original expression $=\frac{2}{3} \lim _{y \rightarrow 1} \frac{y^{3}-2 y^{3}+1}{y^{3}-3 y+2}$ is still of the form $\frac{0}{0}$. Let $z=y-1$, then,
Original expression $=\frac{2}{3} \lim _{z \rightarrow 0} \frac{\left(z^{2}+3 z+3\right)}{z+3}=2$.
Second, for indeterminate forms of rational expressions in trigonometric functions of the form $\lim _{x \rightarrow x_{0}} R(\sin x, \cos x)$, the universal substitution can be used.