几个等价无穷小及证明

\(\tan x \sim x\)

\[\lim\limits_{x\to 0 } \dfrac{\tan x}{x}=\lim\limits_{x \to 0 } \dfrac{\sin x}{x\cos x}=\lim \limits_{x\to 0 } \dfrac{\sin x}{x} \cdot \lim \limits_{x \to 0 } \dfrac{1}{\cos x}=1\]

\(1-\cos x \sim \dfrac{1}{2}x^2\)

\[\lim \limits_{x \to 0 } \dfrac{1-\cos x}{x^{2}}=\lim \limits_{x \to 0 } \dfrac{2 \sin ^{2} \dfrac{x}{2}}{4 \cdot\left(\dfrac{x}{2}\right)^{2}}=\dfrac{1}{2} \cdot\left(\lim \limits_{x \to 0 } \dfrac{\sin \dfrac{x}{2}}{\dfrac{x}{2}}\right)^{2}=\dfrac{1}{2}\]

另解: \[\lim \limits_{x \to 0 } \dfrac{1-\cos x}{x^{2}}=\lim \limits_{x \to 0 }\left(\dfrac{\sin ^{2} x}{x^{2}} \cdot \dfrac{1}{1+\cos x}\right)=\dfrac{1}{2}\]

\(\arcsin x \sim x\)

\(\arctan x \sim x\)

令\(t=\arctan x\), 则\(x=\tan t\)，当\(x \rightarrow 0\) 时，\(t \rightarrow 0\).

\[\lim \limits_{x \to 0 } \dfrac{\arctan x}{x}=\lim \limits_{t \to 0 } \dfrac{t}{\tan t}=\lim \limits_{t \to 0 } \dfrac{t \cos t}{\sin t}=\lim \limits_{t \to 0 } \dfrac{t}{\sin t} \cdot \lim \limits_{t \to 0 } \cos t=1\]

\(\sec x-1 \sim \dfrac{x^{2}}{2}\)

\[\lim \limits_{x \to 0 } \dfrac{\dfrac{1}{\cos x}-1}{x^{2}}=\lim \limits_{x \to 0 } \dfrac{1-\cos x}{x^{2} \cdot \cos x}=\dfrac{\dfrac{1}{2} x^{2}}{x^{2} \cdot \cos x}=\dfrac{1}{2}\]

\(\tan x-\sin x \sim \dfrac{1}{2} x^{3}\)

\[\lim \limits_{x \to 0 } \dfrac{\tan x-\sin x}{\dfrac{1}{2} x^{3}}=\lim \limits_{x \to 0 } \dfrac{\tan x}{x} \cdot \dfrac{1-\cos x}{\dfrac{1}{2} x^{2}}=\lim \limits_{x \to 0 } \dfrac{\tan x}{x} \cdot \lim \limits_{x \to 0 } \dfrac{1-\cos x}{\dfrac{1}{2} x^{2}}=1\]