无穷小的比较
定理1
\(\beta\) 和 \(\alpha\) 是等价无穷小的充分必要条件为$$\beta=\alpha+o(\alpha).$$
证:必要性:设 \(\alpha\sim\beta\),则$$\lim\frac{\beta-\alpha}{\alpha}=\lim(\frac{\beta}{\alpha}-1)=\lim\frac{\beta}{\alpha}-1=0,$$因此 \(\beta-\alpha=o(\alpha)\),即 \(\beta=\alpha+o(\alpha)\).
充分性:设 \(\beta=\alpha+o(\alpha)\),则$$\lim\frac{\beta}{\alpha}=\lim\frac{\alpha+o(\alpha)}{\alpha}=lim(1+\frac{o(\alpha)}{\alpha})=1,$$因此 \(\alpha\sim\beta\).
定理2
设 \(\alpha\sim\widetilde{\alpha},\beta\sim\widetilde{\beta}\),且 \(\lim\frac{\widetilde{\beta}}{\widetilde{\alpha}}\) 存在,则$$\lim\frac{\beta}{\alpha}=\lim\frac{\widetilde{\beta}}{\widetilde{\alpha}}.$$
证:\(\lim\frac{\beta}{\alpha}=\lim(\frac{\beta}{\widetilde{\beta}}\cdot\frac{\widetilde{\beta}}{\widetilde{\alpha}}\cdot\frac{\widetilde{\alpha}}{\alpha})=\lim\frac{\beta}{\widetilde{\beta}}\cdot\lim\frac{\widetilde{\beta}}{\widetilde{\alpha}}\cdot\lim\frac{\widetilde{\alpha}}{\alpha}=\lim\frac{\widetilde{\beta}}{\widetilde{\alpha}}\).
定理2表明,求两个无穷小之比的极限时,分子和分母都可以用等价无穷小来代替.因此,如果用来代替的无穷小选得合适的话,就可以简化计算.