题目

$$
\text{求极限 }
\lim_{x\to0}\frac{\sqrt[4]{1-\sqrt[3]{1-\sqrt{1-x}}} - 1}
{(1+x)^\frac{1}{\sqrt[3]{x^2}} - 1}
$$

解答

$$
\sqrt[4]{1-\sqrt[3]{1-\sqrt{1-x}}} - 1
\sim
\sqrt[4]{1-\sqrt[3]{\frac{1}{2}x}} - 1
\sim
-\frac{1}{2^{\frac{7}{3}}}x^{\frac{1}{3}}
$$

$$
(1+x)^\frac{1}{\sqrt[3]{x^2}} - 1 \sim x^{\frac{1}{3}}
$$

$$
\text{原式} = \lim_{x\to0}\frac{-\dfrac{1}{2^{\frac{7}{3}}}x^{\frac{1}{3}}}{x^{\frac{1}{3}}} =
-\frac{1}{2^{\frac{7}{3}}}
$$