等价无穷小的“串联”使用

题目 1

$$
I_{1} = \lim_{x \rightarrow 0} \frac{1 – \cos x \cos 2x}{x^{2}} = ?
$$

解析 1

根据常用的等价无穷小公式，我们有：

$$
\begin{aligned}
I_{1} & = \lim_{x \rightarrow 0} \frac{1 – \cos x \cos 2x}{x^{2}} \\ \\
& = \lim_{x \rightarrow 0} \frac{1 – \cos x + \cos x – \cos x \cos 2x}{x^{2}} \\ \\
& = \lim_{x \rightarrow 0} \frac{1 – \cos x}{x^{2}} + \lim_{x \rightarrow 0} \frac{\cos x – \cos x \cos 2x}{x^{2}} \\ \\
& = \lim_{x \rightarrow 0} \frac{1 – \cos x}{x^{2}} + \lim_{x \rightarrow 0} \cos x \frac{1 – \cos 2x}{x^{2}} \\ \\
& = \lim_{x \rightarrow 0} \frac{\frac{1}{2} x^{2}}{x^{2}} + \lim_{x \rightarrow 0} \frac{\frac{1}{2} \left( 2x \right)^{2}}{x^{2}} \\ \\
& = \frac{1}{2} + 2 \\ \\
& = \textcolor{lightgreen}{ \frac{5}{2} }
\end{aligned}
$$

题目 2

$$
I_{2} = \lim_{x \rightarrow 0} \frac{\tan (\sin x) – x \cos x}{x^{3}} = ?
$$

解析 2

根据常用的等价无穷小公式，我们有：

$$
\begin{aligned}
I_{2} & = \lim_{x \rightarrow 0} \frac{\tan (\sin x) – x \cos x}{x^{3}} \\ \\
& = \lim_{x \rightarrow 0} \frac{\tan (\sin x) – \sin x + \sin x – x \cos x}{x^{3}} \\ \\
& = \lim_{x \rightarrow 0} \frac{\tan (\sin x) – \sin x}{x^{3}} + \lim_{x \rightarrow 0} \frac{\sin x – x \cos x}{x^{3}} \\ \\
& = \lim_{x \rightarrow 0} \frac{\tan (\sin x) – \sin x}{\sin ^{3} x} + \lim_{x \rightarrow 0} \frac{x – x \cos x}{x^{3}} \\ \\
& = \lim_{k \rightarrow 0} \frac{\tan k – k}{k ^{3}} + \lim_{x \rightarrow 0} \frac{1 – \cos x}{x^{2}} \\ \\
& = \lim_{k \rightarrow 0} \frac{\frac{1}{3} k^{3}}{k ^{3}} + \lim_{x \rightarrow 0} \frac{\frac{1}{2} x^{2}}{x^{2}} \\ \\
& = \frac{1}{3} + \frac{1}{2} \\ \\
& = \textcolor{lightgreen}{ \frac{5}{6} }
\end{aligned}
$$

题目 3

根据常用的等价无穷小公式，我们有：

$$
I_{3} = \lim_{x \rightarrow 0} \frac{\tan (\sin x) – \arcsin x}{x^{3}} = ?
$$

解析 3

$$
\begin{aligned}
I_{3} & = \lim_{x \rightarrow 0} \frac{\tan (\sin x) – \arcsin x}{x^{3}} \\ \\
& = \lim_{x \rightarrow 0} \frac{\tan (\sin x) – \sin x + \sin x – x + x – \arcsin x}{x^{3}} \\ \\
& = \lim_{x \rightarrow 0} \frac{\tan (\sin x) – \sin x}{x^{3}} + \lim_{x \rightarrow 0} \frac{\sin x – x}{x^{3}} + \lim_{x \rightarrow 0} \frac{x – \arcsin x}{x^{3}} \\ \\
& = \lim_{x \rightarrow 0} \frac{\tan (\sin x) – \sin x}{\sin ^{3} x} + \lim_{x \rightarrow 0} \frac{\sin x – x}{x^{3}} + \lim_{x \rightarrow 0} \frac{x – \arcsin x}{x^{3}} \\ \\
& = \lim_{k \rightarrow 0} \frac{\tan k – k}{k ^{3}} + \lim_{x \rightarrow 0} \frac{\sin x – x}{x^{3}} + \lim_{x \rightarrow 0} \frac{x – \arcsin x}{x^{3}} \\ \\
& = \lim_{k \rightarrow 0} \frac{\frac{1}{3} k^{3}}{k ^{3}} + \lim_{x \rightarrow 0} \frac{\frac{-1}{6} x^{3}}{x^{3}} + \lim_{x \rightarrow 0} \frac{\frac{-1}{6} x^{3}}{x^{3}} \\ \\
& = \frac{1}{3} – \frac{1}{6} – \frac{1}{6} \\ \\
& = \textcolor{lightgreen}{ 0 }
\end{aligned}
$$

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