无穷小与有理化、积分、中值定理相结合的一道题目

一、题目

当 $x \rightarrow 0$ 时，无穷小量：

$$
\begin{aligned}
& \alpha = \sqrt{1 + x \cos x} – \sqrt{1 + \sin x} \\
& \beta = \int _{0}^{e^{2x} – 1} \frac{\sin ^{2} t}{t} \mathrm{~d} t \\
& \gamma = \cos (\tan x) – \cos x
\end{aligned}
$$

的阶数由高到低次序为 ($\quad$)

难度评级：

二、解析

1. $\alpha$

$$
\begin{aligned}
\alpha & = \sqrt{1 + x \cos x} – \sqrt{1 + \sin x} \\ \\
& = \frac{ (\sqrt{1 + x \cos x} – \sqrt{1 + \sin x}) (\sqrt{1 + x \cos x} + \sqrt{1 + \sin x})}{\sqrt{1 + x \cos x} + \sqrt{1 + \sin x}} \\ \\
& = \frac{x \cos x – \sin x}{\sqrt{1 + x \cos x} + \sqrt{1 + \sin x}} \\ \\
& \sim \frac{1}{2} (x \cos x – \sin x)
\end{aligned}
$$

又因为：

$$
\begin{aligned}
x \cos x – \sin x \\ \\
& = \frac{x \cos x – \sin x}{x^{3}} \Rightarrow \text{ 洛必达运算 } \\ \\
& = \frac{\cos x – x \sin x – \cos x}{3x^{2}} \\ \\
& = \frac{-x \sin x}{3x^{2}} = \frac{-1}{3}
\end{aligned}
$$

于是：

$$
\begin{aligned}
\textcolor{springgreen}{\alpha} \\ \\
& \sim \frac{1}{2} (x \cos x – \sin x) \\ \\
& \sim \frac{1}{2} (\frac{-1}{3} x^{3}) \\ \\
& \sim \textcolor{springgreen}{\frac{-1}{6} x^{3}}
\end{aligned}
$$

错误的解法：

$$
\textcolor{orangered}{
\bcancel{
\begin{aligned}
\frac{1}{2} (x \cos x – \sin x) \\
& \sim \frac{1}{2} (x \cos x – x) \\
& \sim \frac{1}{2} x (\cos x – 1) \\
& \sim \frac{1}{2} x \cdot (- \frac{1}{2} x^{2}) \\
& \sim \frac{-1}{4} x^{3}
\end{aligned}
}
}
$$

2. $\beta$

$$
\begin{aligned}
\textcolor{springgreen}{\beta} \\ \\
& = \int _{0}^{e^{2x} – 1} \frac{\sin ^{2} t}{t} \mathrm{~d} t \\ \\
& = \int _{0}^{2x} \frac{t^{2}}{t} \mathrm{~d} t \\ \\
& = \int _{0}^{2x} t \mathrm{~d} t \\ \\
& = \frac{1}{2} t^{2} \Big |_{0}^{2x} = \textcolor{springgreen}{2x^{2}}
\end{aligned}
$$

3. $\gamma$

若令 $f(t) = \cos t$, 则 $f^{\prime} (t) = – \sin t$. 于是，根据中值定理可得：

$$
\frac{f(\tan x) – f(x)}{\tan x – x} = f^{\prime}(\xi) \Rightarrow
$$

$$
f(\tan x) – f(x) = f^{\prime}(\xi) (\tan x – x) \Rightarrow
$$

$$
\begin{aligned}
\textcolor{springgreen}{\cos (\tan x) – \cos x} \\ \\
& = (- \sin \xi) (\tan x – x) \\ \\
& \textcolor{springgreen}{\sim \frac{-1}{3} x^{4}}
\end{aligned}
$$

当 $x \rightarrow 0$ 时，无穷小量，的阶数由高到低得次序为：

$$
\textcolor{springgreen}{
\boldsymbol{
\gamma, \quad \alpha, \quad \beta
}
}
$$

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