定義
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等價無窮小的定義:設當 時, 和 均為無窮小量。若 ,則稱 和 是等價無窮小量,記作 。
![等價無窮小](/img/2/775/wZwpmL1cjN1MTN1UTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1EzLzAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
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例如:由於 ,故有 。
等價無窮小替換是計算未定型極限的常用方法,它可以使求極限問題化繁為簡,化難為易。
求極限時,使用等價無窮小的條件 :
被代換的量,在取極限的時候極限值為0;
被代換的量,作為被乘或者被除的元素時可以用等價無窮小代換,但是作為加減的元素時就不可以。
1.被代換的量,在取極限的時候極限值為0;
2.被代換的量,作為被乘或者被除的元素時可以用等價無窮小代換,但是作為加減的元素時就不可以。
定理
無窮小等價替換定理
![等價無窮小](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
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![等價無窮小](/img/8/3ed/wZwpmLxADN2QTO4MzNxADN0UTMyITNykTO0EDMwAjMwUzLzczL0YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
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設函式 , , ,在 內有定義,且有
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(1)若 ,則 ;
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(2)若 ,則 。
證明:
![等價無窮小](/img/d/d45/wZwpmLzYzN5EzM0QzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL4gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(1) 。
![等價無窮小](/img/d/afe/wZwpmL2cTOwUDOxUzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1MzLwMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
(2)。
例如:利用等價無窮小量代換求極限
![等價無窮小](/img/c/f36/wZwpmL3EDN5MDN1EjMzEzM1UTM1QDN5MjM5ADMwAjMwUzLxIzL3YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
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而,,,
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故有。
注意:等價無窮小一般只能在乘除中替換,在加減中替換有時會出錯(加減時可以整體代換,不一定能隨意 單獨代換或分別代換) 。如在上例中:
![等價無窮小](/img/0/3c5/wZwpmLwMjMyQTMwEzMzEzM1UTM1QDN5MjM5ADMwAjMwUzLxMzL2gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
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若因有,,而推出,則得到的是 錯誤的結果。
註:可直接等價替換的類型
![等價無窮小](/img/8/090/wZwpmLxUTNwUjM2YjMzEzM1UTM1QDN5MjM5ADMwAjMwUzL2IzL4IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![等價無窮小](/img/f/418/wZwpmL1EDO4ITOwMzMzEzM1UTM1QDN5MjM5ADMwAjMwUzLzMzL2IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![等價無窮小](/img/1/10e/wZwpmL2YTM1ITMwkDNzEzM1UTM1QDN5MjM5ADMwAjMwUzL5QzLxAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![等價無窮小](/img/a/a88/wZwpmLzQTM0UzN2IzMzEzM1UTM1QDN5MjM5ADMwAjMwUzLyMzL0czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(以上幾個性質可以用來化簡一些未定式以方便運用洛必達法則)
需要滿足一定條件才能替換的類型
![等價無窮小](/img/f/e78/wZwpmL0YjN5gDNxgTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL4EzL3czLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![等價無窮小](/img/1/dae/wZwpmLygTNxYTM3MTMzEzM1UTM1QDN5MjM5ADMwAjMwUzLzEzLygzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
若 ,則
(該條性質非常重要,這是判斷在加減法中能否分別等價替換的重要依據)
變上限積分函式(積分變限函式)也可以用等價無窮小進行替換。
公式
![常見等價無窮小](/img/9/725/wZwpmL4YDO1ATMwQDNzEzM1UTM1QDN5MjM5ADMwAjMwUzL0QzLyczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![等價無窮小](/img/2/a78/wZwpmL4YDM5ITN3EzMzEzM1UTM1QDN5MjM5ADMwAjMwUzLxMzLyEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
當 時,
![等價無窮小](/img/a/ab2/wZwpmL1IjN4ADNwIjMzEzM1UTM1QDN5MjM5ADMwAjMwUzLyIzLxYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![等價無窮小](/img/2/2f4/wZwpmL4ADNzYTO2kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzLzUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![等價無窮小](/img/a/f9c/wZwpmLxYzNyETO3UjMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1IzL0czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
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![等價無窮小](/img/a/2ea/wZwpmL1gDOxADM5QTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL2IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![等價無窮小](/img/9/dc7/wZwpmL3MjN2kTN0YTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL2EzL4czLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
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![等價無窮小](/img/9/5dd/wZwpmL1AjNygzN5gDNzEzM1UTM1QDN5MjM5ADMwAjMwUzL4QzLyczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![等價無窮小](/img/0/ec2/wZwpmLyQTO3YzM3ATMzEzM1UTM1QDN5MjM5ADMwAjMwUzLwEzL2EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![等價無窮小](/img/6/122/wZwpmLycjM3ITM5MTMzEzM1UTM1QDN5MjM5ADMwAjMwUzLzEzL4AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![等價無窮小](/img/a/1ff/wZwpmL0UTN2gTN2YzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLzgzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![等價無窮小](/img/b/72d/wZwpmLxYjM4UzN3UzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1MzLwYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
註:以上各式可通過泰勒展開式推導出來。
推導過程
![等價無窮小](/img/a/76b/wZwpmLxcjM2QDNzIDNzEzM1UTM1QDN5MjM5ADMwAjMwUzLyQzLxAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
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![等價無窮小](/img/7/627/wZwpmL2AjNxgTO4MzNxADN0UTMyITNykTO0EDMwAjMwUzLzczL0MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
α和β都是無窮小,且 , 存在(或 ),則有
![等價無窮小](/img/b/d2c/wZwpmLwUzN1kTO2QTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzLwYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
極限
數學分析的基礎概念。它指的是變數在一定的變化過程中,從總的來說逐漸穩定的這樣一種變化趨勢以及所趨向的數值(極限值)。極限方法是數學分析用以研究函式的基本方法,分析的各種基本概念(連續、微分、積分和級數)都是建立在極限概念的基礎之上,然後才有分析的全部理論、計算和套用.所以極限概念的精確定義是十分必要的,它是涉及分析的理論和計算是否可靠的根本問題。歷史上是柯西(Cauchy,A.-L.)首先較為明確地給出了極限的一般定義。他說,“當為同一個變數所有的一系列值無限趨近於某個定值,並且最終與它的差要多小就有多小”(《分析教程》,1821),這個定值就稱為這個變數的極限.其後,外爾斯特拉斯(Weierstrass,K.(T.W.))按照這個思想給出嚴格定量的極限定義,這就是現在數學分析中使用的ε-δ定義或ε-Ν定義等。從此,各種極限問題才有了切實可行的判別準則。在分析學的其他學科中,極限的概念也有同樣的重要性,在泛函分析和點集拓撲等學科中還有一些推廣。