设 $\alpha(x),\beta(x)$ 是 $x\rightarrow x_0$ 时的无穷小量, $\alpha_1(x),\beta_1(x)$ 也是 $x\rightarrow x_0$ 时的无穷小量. 并且当 $x\rightarrow x_0$ 时, $\alpha(x)\sim\alpha_1(x)$, $\beta(x)\sim\beta_1(x)$.
问什么时候有 $\alpha(x)-\beta(x)\sim\alpha_1(x)-\beta_1(x)$?
A. 当 $\alpha(x)$ 与 $\beta(x)$ 不等价时, 可以这样代换.
1
为简单起见, 以下将 $\alpha(x), \beta(x)$ 简记为 $\alpha,\beta$, 其余类似.
(1) 若 $\alpha=o(\beta)$, 则 $\alpha_1=o(\beta_1)$. 从而有
\[
\lim_{x\rightarrow x_0}\frac{\alpha-\beta}{\alpha_1-\beta_1}=\lim_{x\rightarrow x_0}\frac{\beta_1}{\beta}\cdot\frac{\alpha-\beta}{\alpha_1-\beta_1}=\lim_{x\rightarrow x_0}\frac{\frac{\alpha}{\beta}-1}{\frac{\alpha_1}{\beta_1}-1}=\frac{0-1}{0-1}=1
\]
(2) 若 $\beta=o(\alpha)$, 则 $\beta_1=o(\alpha_1)$. 类似上面可证明
\[
\lim_{x\rightarrow x_0}\frac{\alpha-\beta}{\alpha_1-\beta_1}=1.
\]
(3) 若 $\alpha$ 与 $\beta$ 是同阶非等价无穷小, 则 $\alpha_1$ 与 $\beta_1$ 也是同阶非等价无穷小. 不妨设 $\lim\limits_{x\rightarrow x_0}\frac{\alpha}{\beta}=k$($k\neq 1$), 则 $\lim\limits_{x\rightarrow x_0}\frac{\alpha_1}{\beta_1}=k$. 于是
\[
\begin{split}
\lim_{x\rightarrow x_0}\frac{\alpha-\beta}{\alpha_1-\beta_1}&=\lim_{x\rightarrow x_0}(\frac{\alpha}{\alpha_1-\beta_1}-\frac{\beta}{\alpha_1-\beta_1})\\
&=\lim_{x\rightarrow x_0}\frac{\alpha}{\alpha_1-\beta_1}-\lim_{x\rightarrow x_0}\frac{\beta}{\alpha_1-\beta_1}\\
&=\lim_{x\rightarrow x_0}\frac{\frac{\alpha}{\alpha_1}}{1-\frac{\beta_1}{\alpha_1}}-\lim_{x\rightarrow x_0}\frac{\frac{\beta}{\beta_1}}{\frac{\alpha_1}{\beta_1}-1}\\
&=\frac{1}{1-\frac{1}{k}}-\frac{1}{k-1}\\
&=\frac{k}{k-1}-\frac{1}{k-1}\\
&=1.
\end{split}
\]