导数公式和洛必达法则

雨点打透心脏的1/2处 2023-06-14 10:56 31阅读 0赞

### **导数的公式** ###

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### **等价替换公式** ###

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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）。** ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3l3bDQ3MDgxMjA4Nw_size_16_color_FFFFFF_t_70 3][]

### 洛必达法则（情形一：x→a时的洛必达法则） ###

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### **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。）** ###

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### **x→a情形的几个简单例题。** ###

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### **情形二：x→∞时的洛必达法则。** ###

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### x→∞情形的几个简单例题。 ###

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### **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：

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