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[高数145]求极限|取整函数相关的极限问题
高等数学习题讲解 第145题
高等数学(上)习题集127/264
![[高数144]求极限|利用欧拉常数相关极限和Stirling公式](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/ecdd2e9f0f8f426fb1d02d57c4648262~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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=yaR93YIG88rUWCUk7DW6UREqQ8g%3D)
App09:18![[高数145]求极限|取整函数相关的极限问题](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/3934a7553c934b69acdb94d4fa4db1a9~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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=MpZLSqYD7vjli3FQdTrh4Zv6Za4%3D)
App02:16![[高数146]求极限|两种方法求和式的极限问题](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/aabd864dab8946d1bab91a70f6d81834~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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=Sl64FqIgAN1ztFzf7imuinywmRk%3D)
App06:43![[高数147]求极限|奇数相乘/偶数相乘的极限|巧用均值不等式](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/2e3c6b47f5b44e1eabfa1de9d9522755~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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=C0PuMP0sQ3x14vr5mr1BtVoQYEE%3D)
App05:16![[高数148]求极限|两边夹准则求极限](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/1549ec15ebed4c11b85346d4aaa2f673~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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=RkZGJqiKaJRVKjDIg7MCPKJNmco%3D)
App04:07![[高数149]求极限|两边夹准则+已知极限](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/56b1ae4dabf7456aa5732b982835f94f~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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=tzyD7mf24eE%2F2O1S3dpZEQezDzY%3D)
App03:26![[高数150]求极限|积分上限函数极限问题|洛必达法则](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/9a060b86a0734da799c31304021291f6~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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=ceXjqaPA8i2%2Bw%2FfCg7IP%2BG0PbaY%3D)
App03:38![[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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=IZdBzwWc2I6WJdS5xAxxvs6HtZQ%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://p3-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=mvAFpo7aBrobCEb4a%2BXF67im6iI%3D)
App04:22![[高数153]求极限|抽象函数的极限问题](http://p26-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=5dndrkHU5u0vbwU5S7yHl0%2FWTd4%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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=RGdl1s0ajPfPrOfXnz1DApLPZYE%3D)
App07:18![[高数155]求极限|利用Taylor公式求极限](http://p26-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=%2FiNhXRGnViNkQHTLR3wn1KeJ0aU%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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=TPMt2CkE3BKTaUbZE%2FvSPYLJ3p8%3D)
App05:56![[高数157]求极限问题|利用Taylor公式求极限(3)](http://p26-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=dDQdzpUu38PHwlI01GZFsjOYr5Y%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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=eLxPNFg33Bsi3JU785%2FgKhkXHMU%3D)
App05:27![[高数159]求极限问题|利用定积分的定义(2)](http://p26-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=vGPqgyBu%2F9eji9fjnWecydhkKAA%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://p3-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=8yE1qFSSODZqLSq%2BMp5L5zaR05U%3D)
App05:55![[高数161]一道利用不定积分求函数表达式的题目](http://p26-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=ofNoziiZoyoNRvO3GalZE677xZQ%3D)
App07:13![[高数162]被积函数是分段函数的不定积分问题](http://p26-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=KcvAm6N5AIdbv8m%2FEsLzYs09TjI%3D)
App04:09![[高数163]被积函数带绝对值的不定积分问题](http://p3-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=clZ%2F7WPLB2EoCstW7%2FMHK8X2zxo%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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=5M6o%2BklpATzapYxawr2Wb8hTvN8%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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=Tpvi1rgrQbgYNaFGde%2BLFXAf%2Btg%3D)
App04:31![[高数166]未知函数的不定积分](http://p3-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=Ua11o6E552t3EFztJLggtWgG8q4%3D)
App03:09![[高数167]这种凑微分的方法也是少见](http://p26-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=yWtGQHv2l7GwaW9s6bl3Gpt5b5I%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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=iEdlrkQ079ktW0R4jqoMaThovN8%3D)
App08:08![[高数169]先倒代换后根式代换](http://p26-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=kbZHu4Gl4aw%2BLdo4OGajcNdyUX0%3D)
App07:09![[高数170]题目简洁但解答过程复杂](http://p26-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=AAeeOoHeB4LDfEWvNZmQiUGOIl0%3D)
App11:20![[高数171]注意观察被积函数的特点](http://p3-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=7YRSwCXVo9%2B24ICW73236Kmelpo%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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=3hfF%2FKiVpqb9lBmTjgNspU1sTTA%3D)
App05:34![[高数173]这样子的被积函数可以使用根式代换](http://p3-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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=BP3zwN%2FjDOxRXZbsLdhDr41QUMQ%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=2025030922415302473155D46F7585FAB9&requestFrom=704&version_code=80402&x-expires=1773067313&x-signature=VvLapBfLSlXpweDUeS9ZxHTTt0I%3D)
App02:20
