使用等价无穷小求极限造成误差的讨论

Discussion on Errors Causing by Equivalent Infinitesimal to Find the Limits

现行高数教材普遍回避了,使用麦克劳林公式和等价无穷小替换这两种方法来求极限时,同是在使用等价无穷小作为近似值替换极限中某一项的方法来求极限的本质,以及若这么求极限,有时会产生错误的根本原因。探讨使用等价无穷小作为近似值,替换极限中某一项来求极限的本质,和使用这种方法可能会产生错误的根源,并提出对这种错误的彻底解决办法。

Many textbooks regard equivalent infinitesimal replacement and Maclauring Expansions as two methods to find the limits. This approach ignores the essence of finding limits （actually these two kinds of methods are the same way）, and misses the primacy reason that it will cause errors, which may lead to wrong values of the limits. In this paper, the essence of equivalent infinitesimal replacement including Maclaurin＇ s Expansions was discussed, also the reason for wrong values of the limits was found. A criterion of when the errors can lead to wrong value of the limits was given.