注:本文所有的讨论都基于经典微积分,非标准分析不在讨论范围内。
我们都知道,无穷大有特定的数学符号,但无穷小有吗?
其实无穷小也有个写法,而且 和无穷大有关系:
发现了吗?其实无穷小是“0”旋转
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,而无穷大是0/0(或者说无穷小/无穷小)的简写(在作出这个命名的年代,把0/0作为无穷大看也是正常的,那时候的数学没有现在这么严格)。
既然无穷小和0写法都这么像,它们有什么关系吗?
1.1 无穷小的由来
无穷小最早指的是比零大,但绝对值小于任意正实数的“数”。即:
若
,则
为无穷小。
这是很符合我们的直觉的。你看我找到一个绝对值比任意正实数都小的“数”,它就是无穷小。
0在当时是不被当作无穷小看待的。
不过有没有“数”,它的绝对值小于任意正实数?这个问题一直没有得到解答(这里就有一个逻辑上的悖论,如果无穷小是数,那么它绝对值就不小于任意非零实数--因为它不小于自身,除非它是零)。
因此人们给了它一些直观形象的解释,如微不足道,九牛一毛。曾经数学家也用山上的一粒尘埃来解释什么是无穷小。
在这个逻辑下,伟大的数学家莱布尼茨和牛顿都独立发展出了微积分学。
1.2 无穷小的问题
我们知道,“小”是相对于“大”存在的。
当改变参照物时,一些“小”的东西也可以变的很“大”。
如那根毛和牛比,很小,但它和尘埃比也许很大。
而那粒尘埃和山比很小,但在显微镜下,却比许多微生物大。
但我们希望的是绝对的小,这个小不因参照物的改变而改变。因为它要小于任何数。
正是由于无穷小没有严格的定义。所以引发了数学史上的第二次数学危机。
1.3 无穷小的精确定义
数学领域容不得沙子,混在这里是行不通的。经过长时间的努力,终于在极限被定义的情况下(极限定义参看
请问如何理解极限的精确定义?),无穷小的精确定义终于出现了(要理解无穷小,一定要先理解极限)。
在经典的微积分或数学分析中,无穷小量通常它以函数、序列等形式出现,例如:
无穷小有了精确定义。那它都包含哪些成员呢?
2.1 0和无穷小
首先进入无穷小这个大家庭的是之前被排除在外的0。
还记得这个可怜的家伙吗?一开始莱布尼茨在说大于零小于任意正实数的时候可是没算上它的。但此时,咸鱼翻身了。
0,你可以看成是常数,也可以是常数函数
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或者常数数列
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,都符合无穷小的定义,绝对值小于任意正实数。
如果看作常数函数
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的话,有:
不仅如此,甚至可以是:
也就是说,此函数在定义域范围内处处极限为0,处处都是无穷小。
平反了,平反了,彻底平反了,不止是无穷小!还是处处无穷小!!!
不仅如此,零还是实数内唯一一个无穷小。在实数范围内再也找不到绝对值小于任意正实数的数了。想一想这不仅是千里挑一,万里挑一,是无穷里挑一。零能够作为实数入选无穷小家族,是多么荣幸啊。
0孤零零的站在无穷小的房子,看看四周。心想难道我没有其他伙伴了吗?
2.2 数列和无穷小
实数范围内是没有了,不过无穷数列过来了。
无穷数列可是个大家族。其中有两个重要派系:
一派叫发散派系:
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mstyle%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-texatom%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-7C%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-munderover%22%20transform%3D%22translate(445%2C0)%22%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(224%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-6C%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-69%22%20x%3D%22278%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-6D%22%20x%3D%22557%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-texatom%22%20transform%3D%22translate(0%2C-601)%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(424%2C0)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-2192%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(1132%2C0)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-221E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-msubsup%22%20transform%3D%22translate(2451%2C0)%22%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(529%2C-150)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-texatom%22%20transform%3D%22translate(3505%2C0)%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-7C%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(4061%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-3D%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(5118%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-221E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
。
另一派叫收敛派系,
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mstyle%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-munderover%22%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(224%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-6C%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-69%22%20x%3D%22278%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-6D%22%20x%3D%22557%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-texatom%22%20transform%3D%22translate(0%2C-601)%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(424%2C0)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-2192%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(1132%2C0)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-221E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-msubsup%22%20transform%3D%22translate(2006%2C0)%22%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(529%2C-150)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(3338%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-3D%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(4394%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-4C%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(5075%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2C%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(5521%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-4C%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(6480%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2208%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(7425%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-52%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
。
而能够入住无穷小的,就是来自收敛派系中,
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-4C%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(959%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-3D%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mn%22%20transform%3D%22translate(2015%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-30%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
的这一支。
别看它们已经是某一派系里的一小支,但成员仍然十分众多。如(说明一下,为了严格,我们用
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-7B%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-msubsup%22%20transform%3D%22translate(500%2C0)%22%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(529%2C-150)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(1554%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-7D%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
表示通项为
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-msubsup%22%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(529%2C-150)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
的数列):
还有:
更有:
它们的数量甚至是无穷多的。
它们在无穷处的极限都为0,即:
它们都住进了无穷小的房子里。
特别强调一下,
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-7B%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-msubsup%22%20transform%3D%22translate(500%2C0)%22%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(529%2C-150)%22%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(1554%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-7D%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
是数列,所以我们可以说
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-7B%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mfrac%22%20transform%3D%22translate(500%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(120%2C0)%22%3E%0A%3Crect%20stroke%3D%22none%22%20width%3D%22720%22%20height%3D%2260%22%20x%3D%220%22%20y%3D%22220%22%3E%3C%2Frect%3E%0A%3Cg%20class%3D%22mjx-svg-mn%22%20transform%3D%22translate(110%2C676)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-31%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(60%2C-686)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(1461%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-7D%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
是无穷小,而不能说
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mfrac%22%3E%0A%3Cg%20transform%3D%22translate(120%2C0)%22%3E%0A%3Crect%20stroke%3D%22none%22%20width%3D%22720%22%20height%3D%2260%22%20x%3D%220%22%20y%3D%22220%22%3E%3C%2Frect%3E%0A%3Cg%20class%3D%22mjx-svg-mn%22%20transform%3D%22translate(110%2C676)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-31%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(60%2C-686)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
是无穷小。
2.3 函数和无穷小
不能只有
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2192%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(1278%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-221E%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
时的无穷小啊,而
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2192%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mn%22%20transform%3D%22translate(1278%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-30%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(1778%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2C%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(2223%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2192%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mn%22%20transform%3D%22translate(3502%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-31%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2E%22%20x%3D%22500%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2E%22%20x%3D%22779%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2E%22%20x%3D%221057%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(5116%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2192%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(6394%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-63%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
时的无穷小有没有啊
"有!!"。函数登场。
形如:
比如:
再如:
等等..等等..
2.4 无穷小家族
OK,如大家所见,无穷小家族的成员至此全部到齐。
它们有唯一的实数0或
%3D0%3C%2Ftitle%3E%0A%3Cdefs%20aria-hidden%3D%22true%22%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-66%22%20d%3D%22M118%20-162Q120%20-162%20124%20-164T135%20-167T147%20-168Q160%20-168%20171%20-155T187%20-126Q197%20-99%20221%2027T267%20267T289%20382V385H242Q195%20385%20192%20387Q188%20390%20188%20397L195%20425Q197%20430%20203%20430T250%20431Q298%20431%20298%20432Q298%20434%20307%20482T319%20540Q356%20705%20465%20705Q502%20703%20526%20683T550%20630Q550%20594%20529%20578T487%20561Q443%20561%20443%20603Q443%20622%20454%20636T478%20657L487%20662Q471%20668%20457%20668Q445%20668%20434%20658T419%20630Q412%20601%20403%20552T387%20469T380%20433Q380%20431%20435%20431Q480%20431%20487%20430T498%20424Q499%20420%20496%20407T491%20391Q489%20386%20482%20386T428%20385H372L349%20263Q301%2015%20282%20-47Q255%20-132%20212%20-173Q175%20-205%20139%20-205Q107%20-205%2081%20-186T55%20-132Q55%20-95%2076%20-78T118%20-61Q162%20-61%20162%20-103Q162%20-122%20151%20-136T127%20-157L118%20-162Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-28%22%20d%3D%22M94%20250Q94%20319%20104%20381T127%20488T164%20576T202%20643T244%20695T277%20729T302%20750H315H319Q333%20750%20333%20741Q333%20738%20316%20720T275%20667T226%20581T184%20443T167%20250T184%2058T225%20-81T274%20-167T316%20-220T333%20-241Q333%20-250%20318%20-250H315H302L274%20-226Q180%20-141%20137%20-14T94%20250Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-78%22%20d%3D%22M52%20289Q59%20331%20106%20386T222%20442Q257%20442%20286%20424T329%20379Q371%20442%20430%20442Q467%20442%20494%20420T522%20361Q522%20332%20508%20314T481%20292T458%20288Q439%20288%20427%20299T415%20328Q415%20374%20465%20391Q454%20404%20425%20404Q412%20404%20406%20402Q368%20386%20350%20336Q290%20115%20290%2078Q290%2050%20306%2038T341%2026Q378%2026%20414%2059T463%20140Q466%20150%20469%20151T485%20153H489Q504%20153%20504%20145Q504%20144%20502%20134Q486%2077%20440%2033T333%20-11Q263%20-11%20227%2052Q186%20-10%20133%20-10H127Q78%20-10%2057%2016T35%2071Q35%20103%2054%20123T99%20143Q142%20143%20142%20101Q142%2081%20130%2066T107%2046T94%2041L91%2040Q91%2039%2097%2036T113%2029T132%2026Q168%2026%20194%2071Q203%2087%20217%20139T245%20247T261%20313Q266%20340%20266%20352Q266%20380%20251%20392T217%20404Q177%20404%20142%20372T93%20290Q91%20281%2088%20280T72%20278H58Q52%20284%2052%20289Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-29%22%20d%3D%22M60%20749L64%20750Q69%20750%2074%20750H86L114%20726Q208%20641%20251%20514T294%20250Q294%20182%20284%20119T261%2012T224%20-76T186%20-143T145%20-194T113%20-227T90%20-246Q87%20-249%2086%20-250H74Q66%20-250%2063%20-250T58%20-247T55%20-238Q56%20-237%2066%20-225Q221%20-64%20221%20250T66%20725Q56%20737%2055%20738Q55%20746%2060%20749Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-3D%22%20d%3D%22M56%20347Q56%20360%2070%20367H707Q722%20359%20722%20347Q722%20336%20708%20328L390%20327H72Q56%20332%2056%20347ZM56%20153Q56%20168%2072%20173H708Q722%20163%20722%20153Q722%20140%20707%20133H70Q56%20140%2056%20153Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-30%22%20d%3D%22M96%20585Q152%20666%20249%20666Q297%20666%20345%20640T423%20548Q460%20465%20460%20320Q460%20165%20417%2083Q397%2041%20362%2016T301%20-15T250%20-22Q224%20-22%20198%20-16T137%2016T82%2083Q39%20165%2039%20320Q39%20494%2096%20585ZM321%20597Q291%20629%20250%20629Q208%20629%20178%20597Q153%20571%20145%20525T137%20333Q137%20175%20145%20125T181%2046Q209%2016%20250%2016Q290%2016%20318%2046Q347%2076%20354%20130T362%20333Q362%20478%20354%20524T321%20597Z%22%3E%3C%2Fpath%3E%0A%3C%2Fdefs%3E%0A%3Cg%20stroke%3D%22currentColor%22%20fill%3D%22currentColor%22%20stroke-width%3D%220%22%20transform%3D%22matrix(1%200%200%20-1%200%200)%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-66%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(550%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mi%22%20transform%3D%22translate(940%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(1512%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(2179%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-3D%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mn%22%20transform%3D%22translate(3236%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-30%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
或
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20class%3D%22mjx-svg-mrow%22%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-7B%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mn%22%20transform%3D%22translate(500%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-30%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(1001%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2C%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mn%22%20transform%3D%22translate(1446%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-30%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(1946%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2C%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(2391%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-22EF%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20class%3D%22mjx-svg-mo%22%20transform%3D%22translate(3564%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-7D%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E)
。
收敛到0的无穷数列。
极限值为0的函数。
来张合影吧:
或许有人会觉得为什么无穷小不是负无穷?我的理解是,比如说物理里面的摩擦力,一般都把摩擦力看成是负的力,只有0才是表示没有力作用的意思,这才符合无穷小的直觉。
无穷小不是数,虽然里面有常数0,它是指代一堆“东西”。
无穷小里面包含有:常数0、函数、数列,我们也将全部统称为无穷小量,注意,不是数哦。
比如数列无穷小,有无数多个数列都是无穷小,而不是只有一个,函数也是一个道理。
无穷小和0相关,而非负无穷大。