[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,f(x)趋向于这个值
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,下期视频再见
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,下期视频我们会做一些例题
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,不会超过给定的数
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,之间的距离
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,以及一些极限结论
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,但这个定义是数学上严格精确的
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,使之成立
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,只要选这些x
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,在我们讲这之前 你们知道的是
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,在数学上的定义则是
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,如果你们要学习高等的微积分知识
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,如果你们觉着这看起来很复杂
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,它会迷惑很多学生
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,它是一个很“数学”的问题
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,它确实是很合乎逻辑的
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,定义非常的严格
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,就可以确保函数值和极限值
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,希望当用到实际的数字时
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,当x趋向于这点时
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,当取这些x值时
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,很难理解
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,我会用这个定义证明一些极限
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,我想很多人
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,我想要f(x)和极限值的距离
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,我想要非常接近
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,我有很深的感触
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,或者说要进修数学 那么这是很重要的
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,或者这个x 或者这个
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,是0.000000001
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,比如第三周你们就会用到
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,视频时间快到了
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,计算这些点处的f(x)
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,这个定义
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,这个定义会更好理解一些
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,这个定义会运用在大多数微积分课程
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,这个距离--
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,这时我仍可以给出一个x的范围
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,那么f(x)和极限值的距离
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,那时甚至还没学导数
Dialogue: 0,0:00:00.90,0:00:02.81,Default,,0000,0000,0000,,我来画一个函数
Dialogue: 0,0:00:02.81,0:00:04.49,Default,,0000,0000,0000,,对这个函数取极限会很有意思
Dialogue: 0,0:00:04.49,0:00:06.88,Default,,0000,0000,0000,,现在我就把它画出来
Dialogue: 0,0:00:06.88,0:00:08.39,Default,,0000,0000,0000,,之后我们会做一些练习
Dialogue: 0,0:00:08.39,0:00:11.87,Default,,0000,0000,0000,,这是y轴 这是x轴
Dialogue: 0,0:00:11.87,0:00:14.18,Default,,0000,0000,0000,,函数应该是这样的--
Dialogue: 0,0:00:14.18,0:00:15.95,Default,,0000,0000,0000,,我尽量想一个简单一些的
Dialogue: 0,0:00:15.95,0:00:19.76,Default,,0000,0000,0000,,在大多数情况下它是条直线
Dialogue: 0,0:00:19.76,0:00:23.10,Default,,0000,0000,0000,,像这样
Dialogue: 0,0:00:23.10,0:00:27.08,Default,,0000,0000,0000,,在某一点有一个洞
Dialogue: 0,0:00:27.08,0:00:28.69,Default,,0000,0000,0000,,比如x=a这一点 没有定义
Dialogue: 0,0:00:28.69,0:00:32.03,Default,,0000,0000,0000,,我把这一点抹黑 这样就知道
Dialogue: 0,0:00:32.03,0:00:33.11,Default,,0000,0000,0000,,此处无定义
Dialogue: 0,0:00:33.11,0:00:38.78,Default,,0000,0000,0000,,这一点是x=a
Dialogue: 0,0:00:38.78,0:00:45.18,Default,,0000,0000,0000,,这是x轴 这是y=f(x)轴
Dialogue: 0,0:00:45.18,0:00:47.12,Default,,0000,0000,0000,,就简单的称之为y轴吧
Dialogue: 0,0:00:47.12,0:00:51.03,Default,,0000,0000,0000,,这条线是函数f(x)
Dialogue: 0,0:00:51.03,0:00:53.88,Default,,0000,0000,0000,,或者说是y=f(x)
Dialogue: 0,0:00:53.88,0:00:55.74,Default,,0000,0000,0000,,我们已经学习了一些极限的视频
Dialogue: 0,0:00:55.74,0:00:57.16,Default,,0000,0000,0000,,我想你们有一个大致的理解了
Dialogue: 0,0:00:57.16,0:00:59.85,Default,,0000,0000,0000,,这里要求求出x趋向于a时函数的极限
Dialogue: 0,0:00:59.85,0:01:04.02,Default,,0000,0000,0000,,假设这一点是L
Dialogue: 0,0:01:04.02,0:01:06.48,Default,,0000,0000,0000,,从前面的视频可以知道--
Dialogue: 0,0:01:06.48,0:01:10.94,Default,,0000,0000,0000,,我先把它写出来--
Dialogue: 0,0:01:10.94,0:01:13.69,Default,,0000,0000,0000,,f(x)当x趋向于a时的极限
Dialogue: 0,0:01:13.69,0:01:17.56,Default,,0000,0000,0000,,直观地说 就是指
Dialogue: 0,0:01:17.56,0:01:20.98,Default,,0000,0000,0000,,x从任一边趋向a时
Dialogue: 0,0:01:20.98,0:01:22.29,Default,,0000,0000,0000,,比如从这边
Dialogue: 0,0:01:22.29,0:01:27.03,Default,,0000,0000,0000,,这时f(x)趋向于什么？
Dialogue: 0,0:01:27.03,0:01:29.49,Default,,0000,0000,0000,,当x在这的时候 f(x)在这
Dialogue: 0,0:01:29.49,0:01:33.08,Default,,0000,0000,0000,,x在这的时候 f(x)在这
Dialogue: 0,0:01:35.95,0:01:40.32,Default,,0000,0000,0000,,可以看到 它是趋向于L的
Dialogue: 0,0:01:40.32,0:01:42.20,Default,,0000,0000,0000,,从另一边接近a时--
Dialogue: 0,0:01:42.20,0:01:44.75,Default,,0000,0000,0000,,我们求过只从左边或者右边
Dialogue: 0,0:01:44.75,0:01:48.67,Default,,0000,0000,0000,,趋向于某个数时的极限
Dialogue: 0,0:01:48.67,0:01:52.38,Default,,0000,0000,0000,,但为了极限存在
Dialogue: 0,0:01:52.38,0:01:54.44,Default,,0000,0000,0000,,必须从正负方向同时趋向
Dialogue: 0,0:01:54.44,0:01:57.46,Default,,0000,0000,0000,,一个相同的数
Dialogue: 0,0:01:57.46,0:02:03.86,Default,,0000,0000,0000,,从右边接近a时 选这个点
Dialogue: 0,0:02:03.86,0:02:06.60,Default,,0000,0000,0000,,这是f(x)
Dialogue: 0,0:02:06.60,0:02:07.96,Default,,0000,0000,0000,,在这
Dialogue: 0,0:02:07.96,0:02:09.64,Default,,0000,0000,0000,,x到了这一点时 f(x)在这
Dialogue: 0,0:02:09.64,0:02:13.36,Default,,0000,0000,0000,,随着x越来越接近a点
Dialogue: 0,0:02:13.36,0:02:15.48,Default,,0000,0000,0000,,f(x)趋向于L
Dialogue: 0,0:02:15.48,0:02:16.29,Default,,0000,0000,0000,,所以f(x)当x趋向于a时的极限
Dialogue: 0,0:02:16.29,0:02:19.34,Default,,0000,0000,0000,,为L
Dialogue: 0,0:02:19.34,0:02:21.44,Default,,0000,0000,0000,,我想我们理解这种定义了
Dialogue: 0,0:02:21.44,0:02:27.36,Default,,0000,0000,0000,,但这并不是很--
Dialogue: 0,0:02:27.36,0:02:29.36,Default,,0000,0000,0000,,实际上相对于极限的概念
Dialogue: 0,0:02:29.36,0:02:32.18,Default,,0000,0000,0000,,这样的说法很不精确
Dialogue: 0,0:02:32.18,0:02:36.99,Default,,0000,0000,0000,,我目前所说的仅限于x趋向于某个数时
Dialogue: 0,0:02:36.99,0:02:39.29,Default,,0000,0000,0000,,f(x)趋近什么
Dialogue: 0,0:02:39.29,0:02:48.64,Default,,0000,0000,0000,,所以这个视频中
Dialogue: 0,0:02:48.64,0:02:55.15,Default,,0000,0000,0000,,我会尝试介绍一种新的
Dialogue: 0,0:02:55.15,0:02:57.19,Default,,0000,0000,0000,,极限的定义
Dialogue: 0,0:02:57.19,0:03:00.96,Default,,0000,0000,0000,,这种定义比简单的当x趋向某个值
Dialogue: 0,0:03:00.96,0:03:05.98,Default,,0000,0000,0000,,f(x)趋向于什么的定义方式
Dialogue: 0,0:03:05.98,0:03:12.36,Default,,0000,0000,0000,,要稍微精确一些 或者说要精确很多
Dialogue: 0,0:03:12.36,0:03:16.16,Default,,0000,0000,0000,,我认为可以把它看成一个小游戏
Dialogue: 0,0:03:16.16,0:03:18.03,Default,,0000,0000,0000,,新的定义就是
Dialogue: 0,0:03:18.03,0:03:18.49,Default,,0000,0000,0000,,这个式子意义在于
Dialogue: 0,0:03:18.49,0:03:21.84,Default,,0000,0000,0000,,我总是可以给出这个点的一个范围
Dialogue: 0,0:03:21.84,0:03:29.90,Default,,0000,0000,0000,,这里谈到范围
Dialogue: 0,0:03:29.90,0:03:37.46,Default,,0000,0000,0000,,我并不是针对整个定义域而言
Dialogue: 0,0:03:37.46,0:03:39.76,Default,,0000,0000,0000,,这里说的范围是像 比如说
Dialogue: 0,0:03:39.76,0:03:46.33,Default,,0000,0000,0000,,规定这么一段距离
Dialogue: 0,0:03:46.33,0:03:49.98,Default,,0000,0000,0000,,只要在这范围之内
Dialogue: 0,0:03:49.98,0:03:51.16,Default,,0000,0000,0000,,可以保证f(x)的值和L的距离
Dialogue: 0,0:03:51.16,0:03:54.30,Default,,0000,0000,0000,,始终不会超过某个给定值
Dialogue: 0,0:03:54.30,0:03:57.89,Default,,0000,0000,0000,,我把它看做
Dialogue: 0,0:03:57.89,0:04:00.03,Default,,0000,0000,0000,,一个小游戏
Dialogue: 0,0:04:02.82,0:04:07.83,Default,,0000,0000,0000,,你们可以不相信我
Dialogue: 0,0:04:07.83,0:04:10.87,Default,,0000,0000,0000,,并怀疑f(x)能不能
Dialogue: 0,0:04:10.87,0:04:16.77,Default,,0000,0000,0000,,始终位于和L相距0.5的范围内
Dialogue: 0,0:04:16.77,0:04:19.34,Default,,0000,0000,0000,,你们给出的距离是0.5
Dialogue: 0,0:04:19.34,0:04:21.02,Default,,0000,0000,0000,,此时 我必须能找到
Dialogue: 0,0:04:21.02,0:04:22.56,Default,,0000,0000,0000,,点a附近的一段范围
Dialogue: 0,0:04:22.56,0:04:23.91,Default,,0000,0000,0000,,确保f(x)值始终和L相距0.5以内 对吧？
Dialogue: 0,0:04:23.91,0:04:28.77,Default,,0000,0000,0000,,也就是说f(x)始终位于
Dialogue: 0,0:04:31.53,0:04:44.01,Default,,0000,0000,0000,,这段范围内
Dialogue: 0,0:04:44.01,0:04:47.31,Default,,0000,0000,0000,,只要x位于以a为中心的那段范围
Dialogue: 0,0:04:47.31,0:04:49.63,Default,,0000,0000,0000,,只要满足你们所给出的范围
Dialogue: 0,0:04:49.63,0:04:52.69,Default,,0000,0000,0000,,那么f(x)肯定
Dialogue: 0,0:04:52.69,0:04:58.09,Default,,0000,0000,0000,,能达到你们的要求
Dialogue: 0,0:04:58.09,0:05:01.26,Default,,0000,0000,0000,,我把图画大一些
Dialogue: 0,0:05:01.26,0:05:05.79,Default,,0000,0000,0000,,因为我觉着我是在同一个地方上
Dialogue: 0,0:05:05.79,0:05:08.86,Default,,0000,0000,0000,,反复的写
Dialogue: 0,0:05:08.86,0:05:10.45,Default,,0000,0000,0000,,这是曲线f(x) 这点没有定义
Dialogue: 0,0:05:10.45,0:05:12.59,Default,,0000,0000,0000,,这里不一定非得是个洞
Dialogue: 0,0:05:12.59,0:05:13.05,Default,,0000,0000,0000,,其极限值可以等于一个函数值
Dialogue: 0,0:05:13.05,0:05:17.09,Default,,0000,0000,0000,,但当函数在这点无定义却存在极限时
Dialogue: 0,0:05:17.09,0:05:19.51,Default,,0000,0000,0000,,情况会更为有趣
Dialogue: 0,0:05:19.51,0:05:20.96,Default,,0000,0000,0000,,所以这一点是--
Dialogue: 0,0:05:20.96,0:05:24.32,Default,,0000,0000,0000,,重新画出坐标轴
Dialogue: 0,0:05:24.32,0:05:27.81,Default,,0000,0000,0000,,这是x轴 y轴 x y
Dialogue: 0,0:05:27.81,0:05:30.48,Default,,0000,0000,0000,,这是极限值L 这是点a
Dialogue: 0,0:05:30.48,0:05:36.81,Default,,0000,0000,0000,,所以极限的定义是
Dialogue: 0,0:05:36.81,0:05:43.03,Default,,0000,0000,0000,,我一会儿会回到这个问题
Dialogue: 0,0:05:43.03,0:05:48.03,Default,,0000,0000,0000,,因为既然图大了些 我想再解释一遍
Dialogue: 0,0:05:48.03,0:05:51.65,Default,,0000,0000,0000,,这个式子表示--
Dialogue: 0,0:05:51.65,0:05:54.00,Default,,0000,0000,0000,,这是极限的ε-δ定义
Dialogue: 0,0:05:54.00,0:05:57.71,Default,,0000,0000,0000,,稍后我们会接触到ε和δ
Dialogue: 0,0:05:57.71,0:06:02.32,Default,,0000,0000,0000,,这式子的意义就是 你给我
Dialogue: 0,0:06:02.32,0:06:04.44,Default,,0000,0000,0000,,距L的任何一个距离
Dialogue: 0,0:06:04.44,0:06:05.36,Default,,0000,0000,0000,,实际上我们称这段距离为ε
Dialogue: 0,0:06:09.97,0:06:15.68,Default,,0000,0000,0000,,与最开始提出的定义相对应
Dialogue: 0,0:06:15.68,0:06:19.44,Default,,0000,0000,0000,,f(x)与L的距离不超过ε
Dialogue: 0,0:06:19.44,0:06:23.16,Default,,0000,0000,0000,,ε可以是任何
Dialogue: 0,0:06:23.16,0:06:24.35,Default,,0000,0000,0000,,大于0的实数
Dialogue: 0,0:06:24.35,0:06:26.06,Default,,0000,0000,0000,,所以这里的这段距离就是ε
Dialogue: 0,0:06:26.06,0:06:29.63,Default,,0000,0000,0000,,这一段也是ε
Dialogue: 0,0:06:29.63,0:06:32.98,Default,,0000,0000,0000,,对于给出的任何ε 任何实数--
Dialogue: 0,0:06:32.98,0:06:36.43,Default,,0000,0000,0000,,这点是L+ε
Dialogue: 0,0:06:36.43,0:06:38.94,Default,,0000,0000,0000,,这点是L-ε
Dialogue: 0,0:06:38.94,0:06:43.00,Default,,0000,0000,0000,,极限的ε-δ定义是说
Dialogue: 0,0:06:43.00,0:06:44.68,Default,,0000,0000,0000,,不论ε是多少
Dialogue: 0,0:06:44.68,0:06:49.42,Default,,0000,0000,0000,,总可以在a的附近确定一段范围
Dialogue: 0,0:06:49.42,0:06:52.57,Default,,0000,0000,0000,,并称之为δ
Dialogue: 0,0:06:52.57,0:06:55.44,Default,,0000,0000,0000,,总是可以确定一个δ
Dialogue: 0,0:06:55.44,0:06:57.29,Default,,0000,0000,0000,,这段是比a小δ
Dialogue: 0,0:06:57.29,0:07:01.27,Default,,0000,0000,0000,,这段是比a大δ
Dialogue: 0,0:07:01.27,0:07:04.49,Default,,0000,0000,0000,,这是字母δ
Dialogue: 0,0:07:04.49,0:07:05.38,Default,,0000,0000,0000,,只要在a+δ和a-δ之间
Dialogue: 0,0:07:05.38,0:07:11.75,Default,,0000,0000,0000,,选取一个x值
Dialogue: 0,0:07:11.75,0:07:16.58,Default,,0000,0000,0000,,只要x位于这段范围
Dialogue: 0,0:07:16.58,0:07:18.88,Default,,0000,0000,0000,,就可以保证与x相对应的f(x)
Dialogue: 0,0:07:18.88,0:07:21.48,Default,,0000,0000,0000,,位于你所给出的范围
Dialogue: 0,0:07:21.48,0:07:23.82,Default,,0000,0000,0000,,思考一下 你们会觉着有道理 对吧？
Dialogue: 0,0:07:23.82,0:07:27.38,Default,,0000,0000,0000,,实质上是说
Dialogue: 0,0:07:27.38,0:07:30.22,Default,,0000,0000,0000,,我可以无限接近极限值
Dialogue: 0,0:07:30.22,0:07:35.68,Default,,0000,0000,0000,,只要--
Dialogue: 0,0:07:35.68,0:07:39.97,Default,,0000,0000,0000,,我所说的无限接近
Dialogue: 0,0:07:39.97,0:07:45.65,Default,,0000,0000,0000,,是指你们可以任意给出一个ε值
Dialogue: 0,0:07:45.65,0:07:48.19,Default,,0000,0000,0000,,因为这有点像一个小游戏
Dialogue: 0,0:07:48.19,0:07:50.92,Default,,0000,0000,0000,,通过给出一个需要趋近的点
Dialogue: 0,0:07:50.92,0:07:57.79,Default,,0000,0000,0000,,附近的一段范围
Dialogue: 0,0:07:57.79,0:08:02.98,Default,,0000,0000,0000,,f(x)就可以无限接近极限值
Dialogue: 0,0:08:02.98,0:08:09.32,Default,,0000,0000,0000,,只要是在a附近的这段范围内
Dialogue: 0,0:08:09.32,0:08:13.64,Default,,0000,0000,0000,,选取x的值
Dialogue: 0,0:08:13.64,0:08:15.74,Default,,0000,0000,0000,,只要是在这里取一个x值
Dialogue: 0,0:08:15.74,0:08:17.22,Default,,0000,0000,0000,,我就可以保证f(x)
Dialogue: 0,0:08:17.22,0:08:19.75,Default,,0000,0000,0000,,位于你们所指定的范围
Dialogue: 0,0:08:19.75,0:08:22.58,Default,,0000,0000,0000,,为了更具体点
Dialogue: 0,0:08:22.58,0:08:24.11,Default,,0000,0000,0000,,假设 x位于--
Dialogue: 0,0:08:24.11,0:08:26.73,Default,,0000,0000,0000,,我们一律换成具体的数字来表示
Dialogue: 0,0:08:26.73,0:08:29.87,Default,,0000,0000,0000,,假设这个是2
Dialogue: 0,0:08:29.87,0:08:36.27,Default,,0000,0000,0000,,这是1
Dialogue: 0,0:08:36.27,0:08:40.29,Default,,0000,0000,0000,,也就是求x趋向1时f(x)的极限
Dialogue: 0,0:08:40.29,0:08:42.90,Default,,0000,0000,0000,,还没定义f(x)
Dialogue: 0,0:08:42.90,0:08:48.80,Default,,0000,0000,0000,,但函数曲线是条有一个洞的直线
Dialogue: 0,0:08:48.80,0:08:52.18,Default,,0000,0000,0000,,那里的f(x)值是2
Dialogue: 0,0:08:52.18,0:08:55.59,Default,,0000,0000,0000,,你们可以给我任何一个数字
Dialogue: 0,0:08:55.59,0:09:00.27,Default,,0000,0000,0000,,假定你们想用几个具体例子来验证一下
Dialogue: 0,0:09:00.27,0:09:03.54,Default,,0000,0000,0000,,比如想要f(x)位于--
Dialogue: 0,0:09:03.54,0:09:08.24,Default,,0000,0000,0000,,换种颜色--
Dialogue: 0,0:09:08.24,0:09:09.47,Default,,0000,0000,0000,,想要使f(x)位于距2这点0.5的范围内
Dialogue: 0,0:09:09.47,0:09:11.27,Default,,0000,0000,0000,,也就是在1.5和2.5之间
Dialogue: 0,0:09:11.27,0:09:12.76,Default,,0000,0000,0000,,那么只要x选在--
Dialogue: 0,0:09:12.76,0:09:15.58,Default,,0000,0000,0000,,x可以任意接近
Dialogue: 0,0:09:15.58,0:09:17.95,Default,,0000,0000,0000,,但只要x选在--
Dialogue: 0,0:09:17.95,0:09:21.72,Default,,0000,0000,0000,,对于这个函数
Dialogue: 0,0:09:21.72,0:09:23.68,Default,,0000,0000,0000,,假定是在0.9和1.1之间
Dialogue: 0,0:09:23.68,0:09:26.25,Default,,0000,0000,0000,,那么在这个例子中 δ和极限点的距离只有0.1
Dialogue: 0,0:09:26.25,0:09:33.46,Default,,0000,0000,0000,,只要在和1相距0.1的范围内
Dialogue: 0,0:09:33.46,0:09:35.81,Default,,0000,0000,0000,,选取x
Dialogue: 0,0:09:35.81,0:09:37.39,Default,,0000,0000,0000,,就可以确保f(x)
Dialogue: 0,0:09:37.39,0:09:41.73,Default,,0000,0000,0000,,位于要求的范围
Dialogue: 0,0:09:41.73,0:09:52.80,Default,,0000,0000,0000,,希望你们有初步的理解了
Dialogue: 0,0:09:52.80,0:09:56.59,Default,,0000,0000,0000,,现在我用ε和δ来给出定义
Dialogue: 0,0:09:56.59,0:09:57.76,Default,,0000,0000,0000,,实际上在课本中见到的就是这种
Dialogue: 0,0:09:57.76,0:10:00.53,Default,,0000,0000,0000,,接下来再做几个练习
Dialogue: 0,0:10:00.53,0:10:04.86,Default,,0000,0000,0000,,要搞清楚上面只是个特例
Dialogue: 0,0:10:04.86,0:10:05.52,Default,,0000,0000,0000,,你们给出一个ε 我再给出一个δ
Dialogue: 0,0:10:05.52,0:10:11.83,Default,,0000,0000,0000,,但如果它在定义里成立
Dialogue: 0,0:10:11.83,0:10:15.21,Default,,0000,0000,0000,,或者说要把它写出来
Dialogue: 0,0:10:15.21,0:10:23.02,Default,,0000,0000,0000,,那么就是说这并不仅仅适用于
Dialogue: 0,0:10:23.02,0:10:27.95,Default,,0000,0000,0000,,一个特定的例子
Dialogue: 0,0:10:27.95,0:10:31.34,Default,,0000,0000,0000,,而是适用于任何给定的数
Dialogue: 0,0:10:31.34,0:10:34.84,Default,,0000,0000,0000,,ε可以是百万分之一
Dialogue: 0,0:10:34.84,0:10:37.98,Default,,0000,0000,0000,,或者10的负一百次方
Dialogue: 0,0:10:37.98,0:10:40.75,Default,,0000,0000,0000,,也就是说非常接近于2
Dialogue: 0,0:10:40.75,0:10:45.40,Default,,0000,0000,0000,,我都可以给出一个a点附近的范围
Dialogue: 0,0:10:45.40,0:10:46.45,Default,,0000,0000,0000,,只要在里面选x值
Dialogue: 0,0:10:46.45,0:10:49.93,Default,,0000,0000,0000,,那么f(x)就位于所给出的范围
Dialogue: 0,0:10:49.93,0:10:55.68,Default,,0000,0000,0000,,也就是距极限值
Dialogue: 0,0:10:55.68,0:10:59.24,Default,,0000,0000,0000,,万亿分之一的距离
Dialogue: 0,0:10:59.24,0:11:03.92,Default,,0000,0000,0000,,当然 我不能确定
Dialogue: 0,0:11:03.92,0:11:07.52,Default,,0000,0000,0000,,当x=a时的情况
Dialogue: 0,0:11:07.52,0:11:10.56,Default,,0000,0000,0000,,我只是说当x不等于a
Dialogue: 0,0:11:10.56,0:11:17.01,Default,,0000,0000,0000,,且位于给定范围内时 它会成立
Dialogue: 0,0:11:17.01,0:11:19.67,Default,,0000,0000,0000,,f(x)会在给定的范围内
Dialogue: 0,0:11:19.67,0:11:23.46,Default,,0000,0000,0000,,为了在数学上更清晰些
Dialogue: 0,0:11:23.46,0:11:27.17,Default,,0000,0000,0000,,因为目前为止我都只是用单词来讲
Dialogue: 0,0:11:27.17,0:11:31.56,Default,,0000,0000,0000,,这是我们会在课本中看到的
Dialogue: 0,0:11:31.56,0:11:36.47,Default,,0000,0000,0000,,给定任意一个大于0的ε
Dialogue: 0,0:11:36.47,0:11:38.69,Default,,0000,0000,0000,,这是一种定义 对吧？
Dialogue: 0,0:11:38.69,0:11:39.64,Default,,0000,0000,0000,,如果写成这样
Dialogue: 0,0:11:39.64,0:11:42.34,Default,,0000,0000,0000,,那意味着ε可以是大于0的任何数
Dialogue: 0,0:11:42.34,0:11:44.67,Default,,0000,0000,0000,,然后会给出一个δ
Dialogue: 0,0:11:44.67,0:11:47.56,Default,,0000,0000,0000,,记住 ε是f(x)与极限值的距离
Dialogue: 0,0:11:47.56,0:11:49.72,Default,,0000,0000,0000,,对吧？
Dialogue: 0,0:11:49.72,0:11:53.01,Default,,0000,0000,0000,,是f(x)的范围
Dialogue: 0,0:11:53.01,0:11:54.05,Default,,0000,0000,0000,,之后会给出δ 也就是与a的距离 对吧？
Dialogue: 0,0:11:54.05,0:11:56.91,Default,,0000,0000,0000,,写出来
Dialogue: 0,0:11:56.91,0:11:58.91,Default,,0000,0000,0000,,f(x)当x趋向a的极限是1
Dialogue: 0,0:11:58.91,0:12:01.33,Default,,0000,0000,0000,,给出δ 只要x与a相距
Dialogue: 0,0:12:01.33,0:12:02.16,Default,,0000,0000,0000,,不超过δ--
Dialogue: 0,0:12:02.16,0:12:05.55,Default,,0000,0000,0000,,x和a的距离
Dialogue: 0,0:12:05.55,0:12:12.94,Default,,0000,0000,0000,,假设x选在这-- 换种颜色--
Dialogue: 0,0:12:12.94,0:12:13.96,Default,,0000,0000,0000,,如果x选在这
Dialogue: 0,0:12:13.96,0:12:17.62,Default,,0000,0000,0000,,那么只要x和a的绝对值
Dialogue: 0,0:12:17.62,0:12:19.97,Default,,0000,0000,0000,,大于0 这是为了保证x不会位于a
Dialogue: 0,0:12:19.97,0:12:22.18,Default,,0000,0000,0000,,因为a点函数无定义
Dialogue: 0,0:12:22.18,0:12:25.64,Default,,0000,0000,0000,,只要x和a的距离
Dialogue: 0,0:12:25.64,0:12:29.54,Default,,0000,0000,0000,,大于0并小于给定的一个值
Dialogue: 0,0:12:29.54,0:12:31.32,Default,,0000,0000,0000,,也就是δ
Dialogue: 0,0:12:31.32,0:12:34.26,Default,,0000,0000,0000,,只要在这个范围选取x
Dialogue: 0,0:12:34.26,0:12:38.12,Default,,0000,0000,0000,,我在这画一个小点的x轴
Dialogue: 0,0:12:38.12,0:12:39.33,Default,,0000,0000,0000,,这是a 这段距离是δ
Dialogue: 0,0:12:39.33,0:12:43.37,Default,,0000,0000,0000,,这段距离也是δ
Dialogue: 0,0:12:43.37,0:12:45.44,Default,,0000,0000,0000,,只要选取位于这里的x--
Dialogue: 0,0:12:45.44,0:12:47.27,Default,,0000,0000,0000,,这个x