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[高数152]求极限|适当的化简可以减少洛必达法则的求导运算量
高等数学习题讲解 第152题
高等数学(上)习题集134/264
![[高数151]求极限|等价无穷小替换+洛必达法则](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/cdbce3ca6c364027a11196fbe56ddde3~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=3a72On2DxwoBwZE31StTEzReAXY%3D)
App03:46![[高数152]求极限|适当的化简可以减少洛必达法则的求导运算量](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/b9ad766927164691a886ad1d1903f452~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=2u8GML%2FVeL46ijsfknf6s0WhHlk%3D)
App04:22![[高数153]求极限|抽象函数的极限问题](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/510026c31de548deada355844c5c465f~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=517FYw3LQ2RrECS4ekM5RGrUnaM%3D)
App04:52![[高数154]求极限|抽象函数的极限问题|洛必达法则+微分中值定理](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/849eb0a500fc4c6380b30dd9b4e0f6c3~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=Bn7fI%2ByiPSLvkAJbnKXjjAjPU%2Fk%3D)
App07:18![[高数155]求极限|利用Taylor公式求极限](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/cd0bd55c4d87413e9d184e3079166694~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=y173ZrwoAnXsLj2u7KuML%2FIHG%2B4%3D)
App08:49![[高数156]求极限问题|利用Taylor公式求极限(2)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/98b404e2f7f1445893bbecb05bfe681d~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=77sl2HFK5lvpvjUDP3y2IBiSjd0%3D)
App05:56![[高数157]求极限问题|利用Taylor公式求极限(3)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/fb0283b451834da1b9c47bfda4440861~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=mz6IBFuWhN6Wj5Vv4PApfOV1JU8%3D)
App05:04![[高数158]求极限问题|利用定积分的定义(1)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/2f6e428f68e24585a7da00dfdca6b1c7~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=4Jk5CD3HGMGEqXBhF%2B1TpaAXIGM%3D)
App05:27![[高数159]求极限问题|利用定积分的定义(2)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/dc1310def2a0443fafb523f9aaf41976~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=6bKw5IM3eVEB0p1eXub2yl4MHPw%3D)
App04:55![[高数160]求极限问题|利用定积分的定义+两边夹准则](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/f96cf433f4544601b0151dfae40065de~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=Ug8ie7T%2F4U2HPocdCtNcI%2F0gxtY%3D)
App05:55![[高数161]一道利用不定积分求函数表达式的题目](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/bd66185e4f5e400d81d79608a7a06c09~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=J%2FSQjHCf0jx%2BEebpJYgVuBIAirU%3D)
App07:13![[高数162]被积函数是分段函数的不定积分问题](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/36c2dcd42eb54ed6b3947e1e3cb82a71~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=n0DX48RITzj0pVwO3qnJNzo%2F0%2BE%3D)
App04:09![[高数163]被积函数带绝对值的不定积分问题](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/1e7bd3e72231482eb40ce7bc023a884c~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=6fgUKfONlOwOWG2lynN8iOwlMEU%3D)
App03:49![[高数164]隐函数的不定积分问题(1)](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/16d57214be694c16b18ddcb17ab232b5~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=cjz%2BAC%2Ft5C7sa7cmbAH4VaPiOWY%3D)
App04:54![[高数165]隐函数的不定积分问题(2)](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/bfeb34fd42b245c2b43bf283fabaac14~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=CiHLVEuLrKb1U9jcvdV%2FsP%2B3VaI%3D)
App04:31![[高数166]未知函数的不定积分](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/ddd321a2c39a446cb223b4153af9aa4a~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=aYUz7lKK9PbOvNtK85jI3CExHkM%3D)
App03:09![[高数167]这种凑微分的方法也是少见](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/6420bd1330fd4bb280024435e2b370ab~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=aNQiVh7DZEuKM8ZNL0Y5RdOx5b4%3D)
App04:48![[高数168]有道不难但复杂的题](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/94e15bf1e7884823982cd17080a8ec4b~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=069PHF9eM965qYfMGHSWRULHYBE%3D)
App08:08![[高数169]先倒代换后根式代换](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/76d32cdc203244d4bfe117ab240d0cf6~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=A0%2BkehXcPb%2BnPPxJhh3ynqgHQQk%3D)
App07:09![[高数170]题目简洁但解答过程复杂](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/0e5398a781cf4715ae99c4bdb49de1b2~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=wNLruCrsHXKMFXeFpXtxhBSmPYE%3D)
App11:20![[高数171]注意观察被积函数的特点](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/8376261b53c44289aa4bb0bbf724d157~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=TN%2FobSmjbIDgZcexd5QcI2KxKRc%3D)
App02:39![[高数172]见招拆招|遇水搭桥|逢山开路](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/8afdf78dcc004f0686158fab4374e3b1~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=i2zu%2Bb68bUHz%2BFeRxA8g%2B0SXw2U%3D)
App05:34![[高数173]这样子的被积函数可以使用根式代换](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/9c1d68c58ab84829834200c0a438e2aa~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=%2Fw1oM9EekwCghEKEIY5qQCeXyS8%3D)
App08:42![[高数174]三角函数的不定积分|三角公式很重要 (1)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/008446c617d34ec7a4c7a1e5055448e0~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=HHz9ucnZA47f%2FPu2Hdf3PdOkGZI%3D)
App02:20![[高数175]三角函数的不定积分|三角公式很重要 (2)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/5af14b80a1114bb4a196fdc8d4aba75d~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=jzUXqH5rBuEqQqjMfzYhWCvLfmE%3D)
App01:57![[高数176]三角函数的不定积分|三角公式很重要 (3)](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/07dde814583b433bbfcd623611082387~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=uE0EOfqEtWzvlrwyM3Ou0iCh65E%3D)
App03:06![[高数177]三角函数的不定积分|三角公式很重要 (4)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/6c2323be51094d69b65c6573a83c6e33~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=JVgotsim%2Fy3CsgsyQskIJ46RScg%3D)
App04:16![[高数178]三角函数的不定积分|三角公式很重要 (5)](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/7299bb1d65a147968201d509883b6d4e~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=qdHdAckwcwAszLn%2BGFwgI%2FFR4%2FE%3D)
App05:22![[高数179]三角函数的不定积分|三角公式很重要 (6)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/0921484550ba48ba980761c07fa2861b~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=JAlpVY4x5RtaSCN1jHPqxPU6Emk%3D)
App04:38![[高数180]不看答案我得想很久](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/c54177f0ac484b668b99c9704c9d92b6~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2025030923104016BE2971707D77C20F81&requestFrom=704&version_code=80402&x-expires=1773069040&x-signature=Rmd6dMgApDEpMjArqnsSgPyi%2BS4%3D)
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