「高等数学学习笔记 DAY17」

无穷小的比较

定理1

(eta) 和 (alpha) 是等价无穷小的充分必要条件为$$eta=alpha+o(alpha).$$

证:必要性:设 (alphasimeta),则$$limfrac{eta-alpha}{alpha}=lim(frac{eta}{alpha}-1)=limfrac{eta}{alpha}-1=0,$$因此 (eta-alpha=o(alpha)),即 (eta=alpha+o(alpha)).

充分性:设 (eta=alpha+o(alpha)),则$$limfrac{eta}{alpha}=limfrac{alpha+o(alpha)}{alpha}=lim(1+frac{o(alpha)}{alpha})=1,$$因此 (alphasimeta).

定理2

设 (alphasimwidetilde{alpha},etasimwidetilde{eta}),且 (limfrac{widetilde{eta}}{widetilde{alpha}}) 存在,则$$limfrac{eta}{alpha}=limfrac{widetilde{eta}}{widetilde{alpha}}.$$

证:(limfrac{eta}{alpha}=lim(frac{eta}{widetilde{eta}}cdotfrac{widetilde{eta}}{widetilde{alpha}}cdotfrac{widetilde{alpha}}{alpha})=limfrac{eta}{widetilde{eta}}cdotlimfrac{widetilde{eta}}{widetilde{alpha}}cdotlimfrac{widetilde{alpha}}{alpha}=limfrac{widetilde{eta}}{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.