AM@等价无穷小概念@原理@应用

设 α ∼ α ~ \alpha\sim{\widetilde{\alpha}} α∼α , β ∼ β ~ \beta\sim{\widetilde{\beta}} β∼β ​,且 lim ⁡ β ~ α ~ \lim\frac{\widetilde{\beta}}{\widetilde{\alpha}} limα β ​​存在,则 lim ⁡ β α \lim\frac{\beta}{\alpha} limαβ​= lim ⁡ β ~ α ~ \lim\frac{\widetilde{\beta}}{\widetilde{\alpha}} limα β ​​= A A A

证明: lim ⁡ β α \lim\frac{\beta}{\alpha} limαβ​= lim ⁡ ( β α ~ ⋅ β ~ α ~ ⋅ α ~ α ) \lim(\frac{{\beta}}{\widetilde{\alpha}}\cdot\frac{\widetilde{\beta}}{\widetilde{\alpha}}\cdot{\frac{\widetilde{\alpha}}{{\alpha}}}) lim(α β​⋅α β ​​⋅αα ​)= lim ⁡ β α ~ \lim\frac{{\beta}}{\widetilde{\alpha}} limα β​ ⋅ \cdot ⋅ lim ⁡ β ~ α ~ \lim\frac{\widetilde{\beta}}{\widetilde{\alpha}} limα β ​​ ⋅ \cdot ⋅ lim ⁡ α ~ α \lim{\frac{\widetilde{\alpha}}{{\alpha}}} limαα ​= 1 × lim ⁡ β ~ α ~ × 1 1\times{\lim\frac{\widetilde{\beta}}{\widetilde{\alpha}}}\times{1} 1×limα β ​​×1= lim ⁡ β ~ α ~ \lim\frac{\widetilde{\beta}}{\widetilde{\alpha}} limα β ​​

定理表明,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代替

适当的代替无穷小,可以这类极限计算问题得到简化

但要注意,等价无穷小的应用时有严格要求的,要特别注意自变量的变化过程,而不是单看分子分母解析式

例如: A = lim ⁡ x → 0 tan ⁡ 2 x sin ⁡ 5 x A=\lim\limits_{x\to{0}}\frac{\tan{2x}}{\sin{5x}} A=x→0lim​sin5xtan2x​