高等数学——高阶、同阶、低阶无穷小的性质与区别

红太狼 2023-07-20 12:17 51阅读 0赞

已知： ![\\alpha \\rightarrow 0 , \\beta \\rightarrow 0][alpha _rightarrow 0 _ _beta _rightarrow 0]

case1：![lim\\frac\{\\beta \}\{\\alpha \}=0][lim_frac_beta _alpha _0] , ![\\beta][beta] 为 ![\\alpha][alpha] 的高阶无穷小，记作：**![\\beta = o \\left ( \\alpha \\right )][beta _ o _left _ _alpha _right]**

case2：![lim\\tfrac\{\\beta \}\{\\alpha \} = k,(k\\neq 0,\\infty )][lim_tfrac_beta _alpha _ _ k_k_neq 0_infty] , ![\\beta][beta] 为 ![\\alpha][alpha] 的同阶无穷小，记作：![\\beta =O \\left ( \\alpha \\right )][beta _O _left _ _alpha _right]

case3：![lim\\frac\{\\beta \}\{\\alpha \}= 1][lim_frac_beta _alpha _ 1] , ![\\beta][beta] 为 ![\\alpha][alpha] 的等价无穷小，记作：![\\alpha \\sim \\beta][alpha _sim _beta]

[alpha _rightarrow 0 _ _beta _rightarrow 0]: https://private.codecogs.com/gif.latex?%5Calpha%20%5Crightarrow%200%20%2C%20%5Cbeta%20%5Crightarrow%200
[lim_frac_beta _alpha _0]: https://private.codecogs.com/gif.latex?lim%5Cfrac%7B%5Cbeta%20%7D%7B%5Calpha%20%7D%3D0
[beta]: https://private.codecogs.com/gif.latex?%5Cbeta
[alpha]: https://private.codecogs.com/gif.latex?%5Calpha
[beta _ o _left _ _alpha _right]: https://private.codecogs.com/gif.latex?%5Cbeta%20%3D%20o%20%5Cleft%20%28%20%5Calpha%20%5Cright%20%29
[lim_tfrac_beta _alpha _ _ k_k_neq 0_infty]: https://private.codecogs.com/gif.latex?lim%5Ctfrac%7B%5Cbeta%20%7D%7B%5Calpha%20%7D%20%3D%20k%2C%28k%5Cneq%200%2C%5Cinfty%20%29
[beta _O _left _ _alpha _right]: https://private.codecogs.com/gif.latex?%5Cbeta%20%3DO%20%5Cleft%20%28%20%5Calpha%20%5Cright%20%29
[lim_frac_beta _alpha _ 1]: https://private.codecogs.com/gif.latex?lim%5Cfrac%7B%5Cbeta%20%7D%7B%5Calpha%20%7D%3D%201
[alpha _sim _beta]: https://private.codecogs.com/gif.latex?%5Calpha%20%5Csim%20%5Cbeta