证明:(1) 有限个无穷小的和与差都是无穷小;(2) 有界函数与无穷小的乘积是无穷小，有限个无穷小的积是无穷小.
证明 (1) [tex=2.857x1.071]ghJXJtBd3IXQSpXoMpDSwes+EpetTKeX+WUqMmM0piI=[/tex] , 设[tex=4.429x2.286]MhC0sa4kP8ihnFHLNuEHS1yBudEkNnRRlhrTnGq/Od8aghdkB0XyilPP/g8LqRJDS8FLZihMrIiRjCPZOBq73DSzjeAy0E9Whj1PNccoplg=[/tex] 则 [tex=7.357x1.357]zlKeabRmX2Dxj8dDdgdiwM2BMwjCRwJqlfA/Lho5ho6+Ha6YniCxdxaP+twJqRQJ4efnfBRh3zrLgkM3MwHQAg==[/tex], 当 [tex=11.571x1.357]FPVoynMy8FsiHnXLyIw1Dsun1QT5qv5Ph/gukxHCKyG3TQS1agb6iv1RkMI2hsR9o4jSwId0n8Hiw2Kj8LGE0g==[/tex]时, 恒有 [tex=3.714x2.143]a12nMGe9ciABiLXyiLLmo9rODOs67f9sGlNitQMJZJ/a5z4sqPOvhpjou5ICAfih[/tex] 设[tex=4.357x2.286]MhC0sa4kP8ihnFHLNuEHS1yBudEkNnRRlhrTnGq/Od8aghdkB0XyilPP/g8LqRJDPpQzh/HfSzcKYwM3OvvSXR0q/8TDc8g3U60Jl4QSRBo=[/tex] 则 [tex=7.786x1.357]zlKeabRmX2Dxj8dDdgdiwOMpzE4MLWFAX0sy70pneonvKGb1yvLRqu80lyPIpsPiS/fR78IGcRdc9jY1jO2Gqw==[/tex] 当[tex=11.571x1.357]FPVoynMy8FsiHnXLyIw1DvW6mAPOXvwZFujKWmJ8WIjMY3hJYHhabem+SLahF85kAROieEb30cD5yj4rwGoRyQ==[/tex]时, 恒有[tex=4.5x2.571]TKTp5DuEu6S2cjGD0XOvpUmJHNB6IiRaObzASvKbo/xvAVS+zT8cyluP8ldO12yqTj/RaZwtRxELHXQMO5hMaA==[/tex]故[tex=3.357x1.214]ghJXJtBd3IXQSpXoMpDSwecsBerDOl5Mk6AoraSLy40=[/tex]取 [tex=6.5x1.357]Iz4IgXBK6au7bnbY0iQUIcrEC/mPPysMpuP9C1tAmiLLcHwoOa1OW0whUoyQrCeL[/tex]或 [tex=8.5x1.357]7Y8QmVkryA+bQ0ZZJ2xGBR8Zzn0FD9NWhz2q8BdiMothf4f8X0pps2teQCb8ftSi[/tex] 当[tex=10.571x1.357]FPVoynMy8FsiHnXLyIw1DhacZbYrnf01xvUVhyFhTnvgWCLSA/loNXYAYQIrbZNA[/tex]时 [tex=4.5x1.786]QcyfprOGQzcua+5W4aS/uRbhM4kweVu8HTWD1N7t51tzD1wDReCaVPj0znPu+bDE[/tex][tex=8.929x2.143]MWKrx9mccJUJFfVBnlDXwbcKjvKnlq292fzWL3kagMbEuF+AFVDxLUH9ztqSoUrUb18o8KHgdk1MvxVmBdDGREwkUHCtK3zeVM6IP69IV/4yVG6SmYutblyZDPh9HJRp[/tex]即[tex=6.5x2.357]MhC0sa4kP8ihnFHLNuEHS1yBudEkNnRRlhrTnGq/Od8aghdkB0XyilPP/g8LqRJDMEWqf84nDcVq7brCv20OvOq+eASvV4t8cxe4LxCs3/E=[/tex](2) 设 [tex=1.929x1.357]8Qeytu4ESRuQNId5g8rDJA==[/tex]在 [tex=4.0x1.357]YIa0r+4Y2+Xa83rrlMCwflF55GzhAssp7lSaCA8Jf+jt12u1R7TQIxNbs+TXrAfW[/tex]内(或[tex=3.643x1.357]HAkKti07tHFmHS1FGf0o8tmndEvnFX+KR8NiOF9HxCw=[/tex]时)有界, 即[tex=3.857x1.214]MpZSkO31U46tyl0MCjUrD2S370U7qMiLWgWZT1mq4PU=[/tex] 使[tex=5.071x1.357]sDda1haMrJGOg7vXOx07nEpC7u/2x9XFGsIsY/Yfh2E=[/tex] 又设[tex=4.429x2.286]MhC0sa4kP8ihnFHLNuEHS1yBudEkNnRRlhrTnGq/Od8aghdkB0XyilPP/g8LqRJDS8FLZihMrIiRjCPZOBq73DSzjeAy0E9Whj1PNccoplg=[/tex]则[tex=11.071x1.357]ghJXJtBd3IXQSpXoMpDSwVHkUsT0YlZLtOxA51h3tuc8PMllVaCrKtQ/IGA5TNPYsurw8mY2Qn+DZsWjqCFSDmtOwoi2lXwO5xBpGhsaY+A=[/tex] 当[tex=11.571x1.357]FPVoynMy8FsiHnXLyIw1Dsun1QT5qv5Ph/gukxHCKyG3TQS1agb6iv1RkMI2hsR9o4jSwId0n8Hiw2Kj8LGE0g==[/tex]时, 恒有 [tex=4.214x2.143]a12nMGe9ciABiLXyiLLmo5MRCgTZnV1jPimwA/eqvpS5SZqFArlPkZu3CImpxK1q[/tex] 因此, [tex=3.143x1.643]ghJXJtBd3IXQSpXoMpDSwZbn6EW5ZTYv2vHC0vI/mEVQ51JDaQQ+32zrEn10+pmd[/tex]取[tex=6.5x1.357]Iz4IgXBK6au7bnbY0iQUITanJAx/nzBICWZGYK4BkvjxJp+nEly3LZnzWx2mmRXY[/tex]或[tex=8.286x1.357]+o31NhlP8xvehPbrY2K8u//1OJ9U/Vps0e4lvmGYtVkGvzXSYXlpeFU5n4aWf1xu[/tex] 当[tex=10.571x1.357]FPVoynMy8FsiHnXLyIw1DhacZbYrnf01xvUVhyFhTnvgWCLSA/loNXYAYQIrbZNA[/tex] 时, 恒有[tex=14.214x2.143]z7op6zCENs005qF9Se1eiHxgvBWO9Y3SPCrruAIJnqlHxmlaeeBtIMgiqyi+SGAiVo4C968dwOFMBR3qqYx8qKTCD8ftQRoYVuAcPmMqwGNYAqfbLYjoW5O8rT+Wxbd1[/tex] 即 [tex=7.286x2.714]MhC0sa4kP8ihnFHLNuEHS1yBudEkNnRRlhrTnGq/Od8aghdkB0XyilPP/g8LqRJDVYypK1QzFdJZi5THbTbfplxQVVQZ8OT3slORJTnRMNHkL5BLoE38ARAWczME5hbU[/tex]若 [tex=4.357x2.286]MhC0sa4kP8ihnFHLNuEHS1yBudEkNnRRlhrTnGq/Od8aghdkB0XyilPP/g8LqRJDPpQzh/HfSzcKYwM3OvvSXR0q/8TDc8g3U60Jl4QSRBo=[/tex] 则 [tex=0.571x1.214]Pjn9K91WMLn9RRxgkP8doQ==[/tex] 在 [tex=4.0x1.357]YIa0r+4Y2+Xa83rrlMCwflF55GzhAssp7lSaCA8Jf+jt12u1R7TQIxNbs+TXrAfW[/tex]内[tex=0.429x1.357]WGzIBikYVTPQZi9OI1mRPA==[/tex] 或 [tex=3.643x1.357]HAkKti07tHFmHS1FGf0o8tmndEvnFX+KR8NiOF9HxCw=[/tex]时 ) 有界. 已知 [tex=4.429x2.286]MhC0sa4kP8ihnFHLNuEHS1yBudEkNnRRlhrTnGq/Od8aghdkB0XyilPP/g8LqRJDS8FLZihMrIiRjCPZOBq73DSzjeAy0E9Whj1PNccoplg=[/tex]则[tex=4.786x2.286]MhC0sa4kP8ihnFHLNuEHS1yBudEkNnRRlhrTnGq/Od8aghdkB0XyilPP/g8LqRJDcYhrDpJVtQCL/jXSZm0ommOYFwidwZZub9K/A2EMTaE=[/tex]
举一反三
内容
- 0
【单选题】当 时,将下列函数: , , , , 皆与 进行比较,哪些是高价无穷小量、低价无穷小量、同阶无穷小量、等价无穷小量,下列答案正确的是 () (7.0分) A. 高阶无穷小，无法比较，等价无穷小，同阶无穷小， 无法比较 ； B. 无法比较，高阶无穷小，等价无穷小，同阶无穷小，高阶无穷小； C. 高阶无穷小，无法比较，等价无穷小，同阶无穷小，低阶无穷小； D. 高阶无穷小，无法比较，同阶无穷小，等价无穷小，低阶无穷小；
- 1
当时，无穷小量是无穷小量的（ ）无穷小.bb84dd0dc12cc5a44b1c71587a789894.pngb63fcb41b92bca3304da070aa1bbdebc.pngf51c4bc2f5ba521d1de2c5e808f32be3.png
- 2
无穷个无穷小的和仍然是无穷小。 A: 其余都不对 B: 不一定 C: 不正确 D: 正确
- 3
无穷个无穷小的和仍然是无穷小。 A: 不正确 B: 不一定 C: 以上都不对 D: 正确
- 4
无穷个无穷小的和仍然是无穷小。 A: 不一定 B: 不正确 C: 其余都不对 D: 正确