1 . ①在高等数学中,关于极限的计算,常会用到:i)四则运算法则:如果
,
,则
,
,若
,则
;ii)洛必达法则1:若函数
,
的导函数分别为
,
,且
,
则
;②设
,k是大于1的正整数,若函数
满足:对
,均有
成立,则称函数
为区间
上的k阶无穷递降函数.结合以上两个信息,回答下列问题:
(1)计算:①
;
②
;
(2)试判断
是否为区间
上的2阶无穷递降函数;并证明:
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b89cbc2f466f483804905ea82d3faa5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e137fd4fc65a5c3992e32db3060fa46.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090a91e4f3c8930674f98a9fa527709b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/783c88951a458d5862557f2a041f817a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3927e9f1e25bfe84d4d03caa53d80196.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f021edb6f6d256b0837b365db77b56d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0feda45cb840b1f30f3241998d82e5a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51e1f6cf8990c217af7e109120f35cc3.png)
(1)计算:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7529d1357e6d9e2343b2bb7fcb9aaf55.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15d5f4feda6efd2af4f1be9180ae504.png)
(2)试判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e40e451ed5164bec837f978fddd6413.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e88cf1590641c7a45d48dfcccad70e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/664710185651593168e63b90188b718a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5be1d8c6384d7fabddb693b2b7fcdf4a.png)
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名校
解题方法
2 . 函数极限是现代数学中非常重要的概念,函数
在
处的极限定义如下:
,存在正数
,当
时,均有
,则称
在
处的极限为A,记为
,例如:
在
处的极限为2,理由是:
,存在正数
,当
时,均有
,所以
.已知函数
,
,(
,
为自然对数的底数).
(1)证明:
在
处的极限为
;
(2)若
,
,
,求
的最大值;
(3)若
,用函数极限的定义证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11abb76da45ffa52b47c3a6b9a03ac7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c9705e4d8649224c47228f0d1d0f2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/700458c01a7ad031e27d80ed43e9e882.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09806564a0244b420341e5366f136f58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/761be12e359f89c7627eb9200ba0912b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11abb76da45ffa52b47c3a6b9a03ac7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8abb00b0020eb89f4d18d1a5903f8a32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d3cc66b811ad2395efe04d93b61c711.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c9705e4d8649224c47228f0d1d0f2e.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/187c21027ff08411931d32c530b64fd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bd1d338bd463d522aafd98357c4c012.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57a9bb26472ca40b8a619bfd9ea06a9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2492d486aef92677bc4d9c88c28b6845.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90132e65026968c74776c719242ecd0c.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b85c1784366cf7f60aa01dd62e529d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c58fea3170ce3e4fabed81babd54de1.png)
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3 . ①在微积分中,求极限有一种重要的数学工具——洛必达法则,法则中有一结论:若函数
,
的导函数分别为
,
,且
,则
;
②设
,k是大于1的正整数,若函数
满足:对任意
,均有
成立,且
,则称函数
为区间
上的k阶无穷递降函数.
结合以上两个信息,回答下列问题:
(1)证明
不是区间
上的2阶无穷递降函数;
(2)计算:
;
(3)记
,
;求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22add663bd26e87d972a10dc5fd9ada1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62ceac3910b9f134bab0b92e8d9a9eb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74acc4d2f565d7088e8d737718e89602.png)
②设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73e0c1abf0378a7f5d79672f622b275e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e54d86850a733707433da2e423a5c81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/580f20b900b6d8c9e90c84a0588ae74d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c3e441923ed3c1a32720d6aeac2f599.png)
结合以上两个信息,回答下列问题:
(1)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64d1f6f459292de1002f863203ce91a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fab11f38ab8593932082ec4d9c8c91f.png)
(2)计算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8063898825e02107b7e04f6eba28cb8c.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/602d05de8ada4a6f4d53bab28430f684.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d40b0c4fd043d372c463db08659e779.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caea9a696f22c76f8f4563ac45d124b1.png)
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2024-04-18更新
|
452次组卷
|
6卷引用:专题14 洛必达法则的应用【练】
(已下线)专题14 洛必达法则的应用【练】广东省广州市天河中学高中部2023-2024学年高二下学期基础测试数学试题(已下线)模块五 专题5 全真拔高模拟5(人教B版高二期中研习)四川省广安市华蓥中学2023-2024学年高二下学期4月月考数学试题广东省广州市天河中学2023-2024学年高二下学期第二次月考数学试题黑龙江省哈尔滨市双城区兆麟中学2023-2024学年高二下学期5月期中考试数学试题
4 . 在数列的极限一节,课本中给出了计算由抛物线
、
轴以及直线
所围成的曲边区域面积
的一种方法:把区间
平均分成
份,在每一个小区间上作一个小矩形,使得每个矩形的左上端点都在抛物线
上(如图),则当
时,这些小矩形面积之和的极限就是
.已知
.利用此方法计算出的由曲线
、
轴以及直线
所围成的曲边区域的面积为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/23/3e3b6cc0-af8a-4a20-8adf-f85724b50c7d.png?resizew=109)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/644ede054b3087f93bd2c65683731984.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6921dc242c40a1d342e3b033fc3aa9c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aef469c7b7cb9945b984222381b9c000.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/23/3e3b6cc0-af8a-4a20-8adf-f85724b50c7d.png?resizew=109)
A.![]() | B.![]() | C.![]() | D.![]() |
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真题
5 . 已知
函数
.
(Ⅰ)求函数
的定义域,并判断
的单调性;
(Ⅱ)若
;
(Ⅲ)当
(
为自然对数的底数)时,设
,若函数
的极值存在,求实数
的取值范围以及函数
的极值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0be31e9c40181983d5a380b75186d703.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b79223f97c89a0b944f17e362a0dc51.png)
(Ⅰ)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(Ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb6b6a1b7ccaaedc0a534bb03dfd143a.png)
(Ⅲ)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b7393fc425948d4261bb6c7d67f88e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e27cc47df337277d4142474c50155958.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
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