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洛必达法则是个好法则,我们都很喜欢用它,但稍不注意可能就落入陷阱了,尤其是考研的同学,出题老师可能会故意在细节上考察大家。所以本篇讲一下洛必达法则3大陷阱,提防着点总是好的!
<>1.洛必达法则内容:
<>2.粗略证明
注意:不是严谨证明,主要理解思路,严格证明用柯西中值定理,大家去看书。
从0/0型讲起,
无穷/无穷型我就不写了,因为你知道了0/0型,将分子分母颠倒便可用同样的道理得到相同结论了。
<>3.陷阱
利用洛必达法则注意以下陷阱:
<>3.1要求右侧极限存在
洛必达使用逻辑是有点诡异的,右侧极限存在,回推原极限存在,注意这里的存在包括无穷。那么不存在的情况,我们目前接触的应该是震荡的情况,需要找其他方法,通常比洛必达还要简单。举例:
<>3.2时刻检查是否满足0/0或无穷/无穷
通常用洛必达法则,第一步大家使用的时候,应该都会check是否满足条件,但是多次使用洛必达的时候一定注意别忘了check。
举例:
<>3.3求导后函数要简化
有些函数求导后会更加复杂,或者我们在选取分子分母的时候要比较细心,如果发现很难算,一定记得回头,调换分子分母试一下或者另谋它法。
注意使用条件和这些陷阱的话,洛必达法则还是很好用的。
也有网友建议:
洛必达有啥好的…用前还要判断适用条件…烦死了烦死了…
还不如无脑上泰勒…凡是洛必达能解的,泰勒必定能解,洛必达失效时
泰勒还是能用…
没有什么题是一发三阶泰勒搞不定的,如果有,那就五阶…