關(guān)鈺淋

摘 要 利用等價(jià)無(wú)窮小代換法求函數(shù)極限是數(shù)學(xué)分析中的重要方法之一。由于這種方法可以大大簡(jiǎn)化一些函數(shù)極限的計(jì)算，因而備受廣大同學(xué)的青睞。然而這種方法并非在任何情況下都可以使用，使用時(shí)稍有不慎就會(huì)產(chǎn)生意想不到的錯(cuò)誤。本文對(duì)導(dǎo)師在課堂上布置的討論課題“利用等價(jià)無(wú)窮小求極限應(yīng)注意哪些問(wèn)題”進(jìn)行了較為深入的研究，給出了幾個(gè)關(guān)于無(wú)窮小量的加、減、乘、除的極限以及復(fù)合函數(shù)中使用等價(jià)無(wú)窮小代換法的條件，并給出了證明及應(yīng)用舉例。

關(guān)鍵詞 數(shù)學(xué)分析 無(wú)窮小量 等價(jià)代換 極限 洛必達(dá)法則

中圖分類號(hào)：O211.4 文獻(xiàn)標(biāo)識(shí)碼：A DOI：10.16400/j.cnki.kjdkx.2018.07.015

Abstract The use of the equivalent infinity substituting method to find the function limit is one of the important methods in mathematical analysis. Because this method can greatly simplify the calculation of some function limits， it is greatly favored by the majority of students. However， this method cannot be used under any circumstances， and a slight mistake in use can produce unexpected errors. In this paper， the discussion topic of the tutor in the classroom， "What problems should be paid attention to by using the equivalent infinitesimal minimum limit" is studied in depth， and several limits on the addition， subtraction， multiplication and division of infinitesimal quantities and compound functions are given. The conditions of the equivalent infinitesimal substitution method are used， and the proof and application examples are given.

Keywords mathematics analysis； infinitesimal； equivalent substitution； limit； L'Opida Rule

參考文獻(xiàn)

[1] 華東師范大學(xué)數(shù)學(xué)系編.數(shù)學(xué)分析（上）（第四版）[M].北京：高等教育出版社，2017.

[2] 陳新明.用等價(jià)無(wú)窮小代換法求極限的一些問(wèn)題[J].高等數(shù)學(xué)研究，2008.11（5）：56-58.

[3] 同濟(jì)大學(xué)數(shù)學(xué)教研室編.高等數(shù)學(xué)（上）（第七版）[M].北京：高等教育出版社，2017.