![{\displaystyle \lim _{x\to 2}(4x^{2}-3x+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d8333705d1a4ce31c38e502026a6617df40152)
解答:![{\displaystyle 4(4)-2(3)+1=16-6+1=\mathbf {11} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7dbfa31b68d7c4dc03e7f6dea39148ef5663da8)
![{\displaystyle \lim _{x\to 5}x^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b34b994df019a5242e7d9292d4b4b8cdb999e1e3)
解答:![{\displaystyle 5^{2}=\mathbf {25} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe3d60ef58c68054c4a81c19a928e3b04707d2d)
![{\displaystyle \lim _{x\to 0^{-}}{\frac {x^{3}+x^{2}}{x^{3}+2x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae2769564285c1221588d7c06ae328bef69759a7)
解答:分解因式:
,可知
为一可去间断点,故极限为![{\displaystyle \mathbf {\frac {1}{2}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8640daa7253695c6103e64580ee7527f2891a94a)
![{\displaystyle \lim _{x\to 7^{-}}(|x^{2}+x|-x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/283abac8386d8c71af67a4473b699ab39500f39e)
解答:![{\displaystyle |7^{2}+7|-7=\mathbf {49} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/858bd6c36ca9aaac358c0a68c6952b912ccba22c)
![{\displaystyle \lim _{x\to -1^{+}}{\sqrt {1-x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a50407a352ba67aec4052040885bbe9b7bd2e74a)
解答:
在
时有意义,故极限为![{\displaystyle {\sqrt {1-1^{2}}}=\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/71981b87dbe5a9c6fcfcad111719a6eeccc9af6c)
![{\displaystyle \lim _{x\to -1^{-}}{\sqrt {1-x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce418fe37652f3ad7cd323f96509ba2e64ee99e8)
解答:
在
时无意义,故极限不存在
![{\displaystyle \lim _{x\to -1}{\frac {1}{x-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b28c946c47b2913b5952055f4d078c5de662caac)
解答:![{\displaystyle \mathbf {-{\frac {1}{2}}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7070d2da39d5f0bd8da5f1d678a92b70007f17)
![{\displaystyle \lim _{x\to 4}{\frac {1}{x-4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7365924df5ea38ce520505c2fb320b4e5522188)
解答:![{\displaystyle \lim _{x\to 4^{-}}{\frac {1}{x-4}}=-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d72bd7a5a89efd38b45d41cf5d3ebee2d718bdb3)
![{\displaystyle \lim _{x\to 4^{+}}{\frac {1}{x-4}}=+\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fade111dea26d5ce0f4f2c3f3d697b4bc0aae44)
极限不存在
![{\displaystyle \lim _{x\to 2}{\frac {1}{x-2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c2418a7e60196b58e0b4638177f987615fad60)
解答:![{\displaystyle \lim _{x\to 2^{-}}{\frac {1}{x-2}}=-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/480fbc0fa69edd54ad957b3524f2f2a77892a137)
![{\displaystyle \lim _{x\to 2^{+}}{\frac {1}{x-2}}=+\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0877ecfe2f1ea57ed69cbe401abd21a6b7834bd4)
极限不存在
![{\displaystyle \lim _{x\to -3}{\frac {x^{2}-9}{x+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ced70a1b83e9d5e9cc217d6f5fabbf8062ec1f6)
解答:![{\displaystyle \lim _{x\to -3}{\frac {(x+3)(x-3)}{x+3}}=\lim _{x\to -3}x-3=-3-3=\mathbf {-6} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f62eb46c56124187a099184726d705d4976c23d)
![{\displaystyle \lim _{x\to 3}{\frac {x^{2}-9}{x-3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/931ba8d10b86d47e16cec328f22e5a1ef4de4ce1)
解答:![{\displaystyle \lim _{x\to 3}{\frac {(x-3)(x+3)}{x-3}}=\lim _{x\to 3}x+3=3+3=\mathbf {6} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/25364c54cbb928b6c8f4f8fa998c16b659bbeff4)
![{\displaystyle \lim _{x\to -1}{\frac {x^{2}+2x+1}{x+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f447946be929ff3522921ab0f6471a6f1c59fdb9)
解答:![{\displaystyle \lim _{x\to -1}{\frac {(x+1)(x+1)}{x+1}}=\lim _{x\to -1}x+1=-1+1=\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d2778ce3d6b0aa072edb6fa96cebf2ed4962fb)
![{\displaystyle \lim _{x\to -1}{\frac {x^{3}+1}{x+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02b7d99cd5de6dcdae73cbcef4175c0635f02c38)
解答:![{\displaystyle \lim _{x\to -1}{\frac {(x^{2}-x+1)(x+1)}{x+1}}=\lim _{x\to -1}x^{2}-x+1=(-1)^{2}-(-1)+1=1+1+1=\mathbf {3} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b00ea7eb4eb6ef4dceeda662bd25fadbe81f33)
![{\displaystyle \lim _{x\to 4}{\frac {x^{2}+5x-36}{x^{2}-16}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27fbd435f64546ccd54dc791eef728b6d97cfd57)
解答:![{\displaystyle \lim _{x\to 4}{\frac {(x-4)(x+9)}{(x-4)(x+4)}}=\lim _{x\to 4}{\frac {x+9}{x+4}}={\frac {4+9}{4+4}}=\mathbf {\frac {13}{8}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3214961a4127826889128b3cdce6e1421c55ded6)
![{\displaystyle \lim _{x\to 25}{\frac {x-25}{{\sqrt {x}}-5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3b9d6211ba6b3c3e5df9a7201ce514ef1392f2)
解答:![{\displaystyle \lim _{x\to 25}{\frac {({\sqrt {x}}-5)({\sqrt {x}}+5)}{{\sqrt {x}}-5}}=\lim _{x\to 25}({\sqrt {x}}+5)={\sqrt {25}}+5=5+5=\mathbf {10} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e86ce7db1a39f99a0b29003d916285efee1c6e)
![{\displaystyle \lim _{x\to 0}{\frac {\left|x\right|}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f88f007dca0c79b8754ac7b1ac512fb3fe80c8dc)
解答:![{\displaystyle \lim _{x\to 0^{-}}{\frac {\left|x\right|}{x}}=\lim _{x\to 0^{-}}{\frac {-x}{x}}=\lim _{x\to 0^{-}}-1=-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff84c461889692d3883923f3ecf06a37152bf127)
![{\displaystyle \lim _{x\to 0^{+}}{\frac {\left|x\right|}{x}}=\lim _{x\to 0^{+}}{\frac {x}{x}}=\lim _{x\to 0^{+}}1=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fddac7886085c746f2d1b9d31eddaf4c8501fb7)
极限不存在
![{\displaystyle \lim _{x\to 2}{\frac {1}{(x-2)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7dad377e5d512f16b16d312d1113d1c0088f09c)
解答:当
趋近于
时,分母趋近于
,故极限为![{\displaystyle \mathbf {+\infty } }](https://wikimedia.org/api/rest_v1/media/math/render/svg/06515e2e2d74dd50f7ea1db33fb18b11549a9f1a)
![{\displaystyle \lim _{x\to 3}{\frac {\sqrt {x^{2}+16}}{x-3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/173476674751ad858bee4cdda16b545aa3ee7e65)
解答:当
趋近于
时,分子趋近于
,分母趋近于
,但从左侧趋近时极限为
,从右侧趋近时极限为
,故极限不存在
![{\displaystyle \lim _{x\to -2}{\frac {3x^{2}-8x-3}{2x^{2}-18}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e381f1d61f8a0d9340f9a094865017becf7174cd)
解答:![{\displaystyle {\frac {3(-2)^{2}-8(-2)-3}{2(-2)^{2}-18}}={\frac {3(4)+16-3}{2(4)-18}}={\frac {12+16-3}{8-18}}={\frac {25}{-10}}=\mathbf {-{\frac {5}{2}}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd49f0a08ab682ce71ae7a5d1ff8fad6d9b43bb)
![{\displaystyle \lim _{x\to 2}{\frac {x^{2}+2x+1}{x^{2}-2x+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1581762f4b5ae45f0299168549612d0c2fd1da)
解答:![{\displaystyle {\frac {2^{2}+2(2)+1}{2^{2}-2(2)+1}}={\frac {4+4+1}{4-4+1}}={\frac {9}{1}}=\mathbf {9} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aff2c40d38904225377e3a14045309af3d400f57)
![{\displaystyle \lim _{x\to 3}{\frac {x+3}{x^{2}-9}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abe197967cbaf1e43eb33fd7192a832bf21f6776)
解答:![{\displaystyle \lim _{x\to 3}{\frac {x+3}{(x+3)(x-3)}}=\lim _{x\to 3}{\frac {1}{x-3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77d67f45d52b675ecfb4fed805289520612e17d7)
![{\displaystyle \lim _{x\to 3^{-}}{\frac {1}{x-3}}=-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e27b1dfe88f5295fd833606b5d25ec5baf8e2b40)
![{\displaystyle \lim _{x\to 3^{+}}{\frac {1}{x-3}}=+\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/707088d8b616cd55ef97c2aec5cf5197c2a02f5c)
极限不存在
![{\displaystyle \lim _{x\to -1}{\frac {x+1}{x^{2}+x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62d413bcb6034df0312c14758fc25e569101dab7)
解答:![{\displaystyle \lim _{x\to -1}{\frac {x+1}{x(x+1)}}=\lim _{x\to -1}{\frac {1}{x}}={\frac {1}{-1}}=\mathbf {-1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/30db8f7beb4efb2a74f00f10a0c12b18aa8a7295)
![{\displaystyle \lim _{x\to 1}{\frac {1}{x^{2}+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89fe6c98e7fc60c94154175c2a3a316888a6efd6)
解答:![{\displaystyle {\frac {1}{1^{2}+1}}={\frac {1}{1+1}}=\mathbf {\frac {1}{2}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d846f1bf4adae0b610682274c566e996b083a14)
![{\displaystyle \lim _{x\to 1}x^{3}+5x-{\frac {1}{2-x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57a8d109e8d0486a6fb3abd76b12b7a0f379bcbb)
解答:![{\displaystyle 1^{3}+5(1)-{\frac {1}{2-1}}=1+5-{\frac {1}{1}}=6-1=\mathbf {5} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4227745371e8ee46d73f6628c3cd5dd35faa32)
![{\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x^{2}+2x-3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b62d7e448c9dcbee3a1c7eaa944d375dfd41b1)
解答:![{\displaystyle \lim _{x\to 1}{\frac {(x-1)(x+1)}{(x-1)(x+3)}}=\lim _{x\to 1}{\frac {x+1}{x+3}}={\frac {1+1}{1+3}}={\frac {2}{4}}=\mathbf {\frac {1}{2}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b49b527a121da5d06744e9d85cfe2d6e6f7e3b1)
![{\displaystyle \lim _{x\to 1}{\frac {5x}{x^{2}+2x-3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc034a8d8dab09cbcd5888f3b3a9c3f3fde13191)
解答:当
趋近于
时,分子趋近于
,分母趋近于
,但从左侧趋近时极限为
,从右侧趋近时极限为
,故极限不存在
![{\displaystyle \lim _{x\to \infty }{\frac {-x+\pi }{x^{2}+3x+2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e317c1ab278f9895e1dcc01904dbcd74b8795553)
解答:分母比分子高阶,故极限为![{\displaystyle \mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d)
![{\displaystyle \lim _{x\to -\infty }{\frac {x^{2}+2x+1}{3x^{2}+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa12d9fa636895959ebfd3c1edd5770f83d65e70)
解答:分子与分母同阶,故极限为最高次项系数之比,即![{\displaystyle \mathbf {\frac {1}{3}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/23985a284c7c9f6945e0a4fe54826edcb71dd11d)
![{\displaystyle \lim _{x\to -\infty }{\frac {3x^{2}+x}{2x^{2}-15}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e72c278b9d8b1d3db42d85e1fbf25aae5d4b8c9)
解答:分子与分母同阶,故极限为最高次项系数之比,即![{\displaystyle \mathbf {\frac {3}{2}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc439bf9abe6861856c1679399e7a455ac8151fb)
![{\displaystyle \lim _{x\to -\infty }3x^{2}-2x+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5f166d111e22c132ccbc64e2369a295f8ef59f)
解答:极限为![{\displaystyle \mathbf {+\infty } }](https://wikimedia.org/api/rest_v1/media/math/render/svg/06515e2e2d74dd50f7ea1db33fb18b11549a9f1a)
![{\displaystyle \lim _{x\to \infty }{\frac {2x^{2}-32}{x^{3}-64}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7da8f3aa5501ea70773d64ac2de71a6fbe5800f2)
解答:分母比分子高阶,故极限为![{\displaystyle \mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d)
![{\displaystyle \lim _{x\to \infty }6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f78c9ab92f0d8212cb23dc0a0d54150975e5bf4e)
解答:极限为![{\displaystyle \mathbf {6} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee8a5cef3798b0ef8704ea003512a42028b3cfa8)
![{\displaystyle \lim _{x\to \infty }{\frac {3x^{2}+4x}{x^{4}+2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/118e1c7d8f08392abb3c82d164eaae9c4392f837)
解答:分母比分子高阶,故极限为![{\displaystyle \mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d)
![{\displaystyle \lim _{x\to -\infty }{\frac {2x+3x^{2}+1}{2x^{2}+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/178b3408ab321affa3b4b6dc45999b02bf3960ba)
解答:分子与分母同阶,故极限为最高次项系数之比,即![{\displaystyle \mathbf {\frac {3}{2}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc439bf9abe6861856c1679399e7a455ac8151fb)
![{\displaystyle \lim _{x\to -\infty }{\frac {x^{3}-3x^{2}+1}{3x^{2}+x+5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3a11e1d0c7b46a6e20e3a806a9079a1445dfdc)
解答:分子比分母高阶,故极限为![{\displaystyle \mathbf {-\infty } }](https://wikimedia.org/api/rest_v1/media/math/render/svg/41ce82a60cdea80429fa702027b1b6a77de05b10)
![{\displaystyle \lim _{x\to \infty }{\frac {x^{2}+2}{x^{3}-2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4288844f8839c314db76ece8e2af49515904c6d3)
解答:分母比分子高阶,故极限为![{\displaystyle \mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d)
![{\displaystyle \lim _{x\to 2^{-}}f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4af40c31548d01d27995d477816edd4a48c717bd)
解答:![{\displaystyle (2-2)^{2}=\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ade295ce30899f7084fc7b4bd0e017b867f4c9ef)
![{\displaystyle \lim _{x\to 2^{+}}f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2511dfa778173de8f0851c336631d209185efe1f)
解答:![{\displaystyle 2-3=\mathbf {-1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c64ddc1ee5bffe554322aaadffa610e3630a9d1)
![{\displaystyle \lim _{x\to 2}f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/663bed7e3cc2b22135e8d17ad2e7227f2c0faa9d)
解答:左右两侧极限不相等,故极限不存在
![{\displaystyle \lim _{x\to 4^{+}}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d24b1c736ad975571fa5fec405bfef3dd7d1b1e)
解答:![{\displaystyle 4^{2}+2=16+2=\mathbf {18} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f36a5280fc6f49faa236cf479cc4cfd9d3a25921)
![{\displaystyle \lim _{x\to 4^{-}}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d3bac7a3ae5d43020f55c8232a7ddca16d06071)
解答:![{\displaystyle 4+1=\mathbf {5} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f025eb27da779985875beca5d3d1800f0dbf0a)
![{\displaystyle \lim _{x\to 0^{+}}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4036702fee1e50eae95b4c5b4c59c3b43a90a3)
解答:![{\displaystyle 0+1=\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/158f3c88a430accfc46632b71a6b5b18ad3ecf7e)
![{\displaystyle \lim _{x\to 0^{-}}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6b26bbcfbe8395807a337066bc7eb9e6a2a325)
解答:![{\displaystyle -2(0)+1=\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/91feb7fe902162944bbebf6b812b99aeb4af7c88)
![{\displaystyle \lim _{x\to 0}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc67622028d074c9b6281a5f8ad2d4d1d078a67)
解答:左右两侧极限相等,故极限为![{\displaystyle \mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e)
![{\displaystyle \lim _{x\to 1}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e415bb6b0cc015b443f5ed7681d04355f1f3b7d3)
解答:![{\displaystyle 1+1=\mathbf {2} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f23732ba3bc848ab836e249b7f20efc36c273d3f)
![{\displaystyle \lim _{x\to 0}h(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1fc23b9be87d5e6373d2b64b5c4763be6b1951)
解答:![{\displaystyle 2(0)-3=\mathbf {-3} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/42a0cc45385e79d285118c85798714264f875cde)
![{\displaystyle \lim _{x\to 2^{-}}h(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a206d69c2e80a565280ce77b0be913ec3679975f)
解答:![{\displaystyle 2(2)-3=4-3=\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/017ba4475e5105972f5d52a9bfa352bd0671d341)
![{\displaystyle \lim _{x\to 2^{+}}h(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/503442e28a4f3df38727d507b3f4f9f55751d8d6)
解答:![{\displaystyle -(2)+3=\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/03cb9958366b5e8503880c4157d01625eb1de727)
![{\displaystyle \lim _{x\to 2}h(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31b65463035e82e46efea1478e884501d165a514)
解答:左右两侧极限相等,故极限为![{\displaystyle \mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e)