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导数的公式
等价替换公式
引言:中值定理的“实用性”。
微分中值定理是微积分理论的基础,其理论意义我们在今后的学习中会逐步加深体会。但由于中值定理的特点(仅能保证中值存在,无法具体求出),初学者往往对其“实用性”表示怀疑。本节我们就来介绍柯西中值定理的一个重要应用:求未定式极限的洛必达法则。
求极限是高等数学课程中第一个重要的计算,且贯穿高等数学的始终。应该指出,极限计算从来就不是一个简单的问题(回忆我们在第一章中练习过的题目),这种情况直到学习了洛必达法则才算有了本质的转变。
从柯西中值定理到洛必达法则(洛必达法则的证明概要)。
洛必达法则(情形一:x→a时的洛必达法则)
对定理一的评注。(一,只有未定式才能用洛必达法则;二,洛必达法则可以连用。)
x→a情形的几个简单例题。
情形二:x→∞时的洛必达法则。
x→∞情形的几个简单例题。
拓展阅读:无穷大的比较。
关于无穷大量的增长速度及用极限定义无穷大量的阶,详细介绍见下文: