题目

设 $f(x)=1-\cos{x}$，求极限 $\lim\limits_{x\to0}\dfrac{(1-\sqrt{\cos{x}})(1-\sqrt[3]{\cos{x}})(1-\sqrt[4]{\cos{x}})(1-\sqrt[5]{\cos{x}})}{f\{f[f(x)]\}}$

解答

当 $x \to 0$ 时：

$$
1-\sqrt{\cos x} \sim 1 - (1 - \frac{1}{2}x^2)^{\frac{1}{2}} \sim
-(-\frac{1}{4})x^2 = \frac{1}{4}x^2
$$

同理，由此推导可证得：$1 - \cos^\alpha x \sim \dfrac{1}{2} \alpha x^2$

$$
\begin{aligned}
\text{原式} &=
\frac
{\dfrac{1}{4} \cdot \dfrac{1}{6} \cdot \dfrac{1}{8} \cdot \dfrac{1}{10} \cdot x^8}
{1 - \cos(f[f(x)])} \\
&=
\frac
{\dfrac{x^8}{1920}}
{\dfrac{1}{2}[1 - \cos f(x)]^2} \\
&=
\frac
{\dfrac{x^8}{1920}}
{\dfrac{1}{2}[\dfrac{1}{2}(1 - \cos x)^2]^2}
\\
&=
\frac
{\dfrac{x^8}{1920}}
{\dfrac{1}{128} \cdot x^8}
\\
&=
\frac{1}{15}
\\
\end{aligned}
$$