1
99:59:59,999 --> 99:59:59,999
f(x)趋向于这个值

2
99:59:59,999 --> 99:59:59,999
下期视频再见

3
99:59:59,999 --> 99:59:59,999
下期视频我们会做一些例题

4
99:59:59,999 --> 99:59:59,999
不会超过给定的数

5
99:59:59,999 --> 99:59:59,999
之间的距离

6
99:59:59,999 --> 99:59:59,999
以及一些极限结论

7
99:59:59,999 --> 99:59:59,999
但这个定义是数学上严格精确的

8
99:59:59,999 --> 99:59:59,999
使之成立

9
99:59:59,999 --> 99:59:59,999
只要选这些x

10
99:59:59,999 --> 99:59:59,999
在我们讲这之前 你们知道的是

11
99:59:59,999 --> 99:59:59,999
在数学上的定义则是

12
99:59:59,999 --> 99:59:59,999
如果你们要学习高等的微积分知识

13
99:59:59,999 --> 99:59:59,999
如果你们觉着这看起来很复杂

14
99:59:59,999 --> 99:59:59,999
它会迷惑很多学生

15
99:59:59,999 --> 99:59:59,999
它是一个很“数学”的问题

16
99:59:59,999 --> 99:59:59,999
它确实是很合乎逻辑的

17
99:59:59,999 --> 99:59:59,999
定义非常的严格

18
99:59:59,999 --> 99:59:59,999
就可以确保函数值和极限值

19
99:59:59,999 --> 99:59:59,999
希望当用到实际的数字时

20
99:59:59,999 --> 99:59:59,999
当x趋向于这点时

21
99:59:59,999 --> 99:59:59,999
当取这些x值时

22
99:59:59,999 --> 99:59:59,999
很难理解

23
99:59:59,999 --> 99:59:59,999
我会用这个定义证明一些极限

24
99:59:59,999 --> 99:59:59,999
我想很多人

25
99:59:59,999 --> 99:59:59,999
我想要f(x)和极限值的距离

26
99:59:59,999 --> 99:59:59,999
我想要非常接近

27
99:59:59,999 --> 99:59:59,999
我有很深的感触

28
99:59:59,999 --> 99:59:59,999
或者说要进修数学 那么这是很重要的

29
99:59:59,999 --> 99:59:59,999
或者这个x 或者这个

30
99:59:59,999 --> 99:59:59,999
是0.000000001

31
99:59:59,999 --> 99:59:59,999
比如第三周你们就会用到

32
99:59:59,999 --> 99:59:59,999
视频时间快到了

33
99:59:59,999 --> 99:59:59,999
计算这些点处的f(x)

34
99:59:59,999 --> 99:59:59,999
这个定义

35
99:59:59,999 --> 99:59:59,999
这个定义会更好理解一些

36
99:59:59,999 --> 99:59:59,999
这个定义会运用在大多数微积分课程

37
99:59:59,999 --> 99:59:59,999
这个距离--

38
99:59:59,999 --> 99:59:59,999
这时我仍可以给出一个x的范围

39
99:59:59,999 --> 99:59:59,999
那么f(x)和极限值的距离

40
99:59:59,999 --> 99:59:59,999
那时甚至还没学导数

41
00:00:00,900 --> 00:00:02,810
我来画一个函数

42
00:00:02,810 --> 00:00:04,490
对这个函数取极限会很有意思

43
00:00:04,490 --> 00:00:06,880
现在我就把它画出来

44
00:00:06,880 --> 00:00:08,390
之后我们会做一些练习

45
00:00:08,390 --> 00:00:11,870
这是y轴 这是x轴

46
00:00:11,870 --> 00:00:14,180
函数应该是这样的--

47
00:00:14,180 --> 00:00:15,950
我尽量想一个简单一些的

48
00:00:15,950 --> 00:00:19,760
在大多数情况下它是条直线

49
00:00:19,760 --> 00:00:23,100
像这样

50
00:00:23,100 --> 00:00:27,080
在某一点有一个洞

51
00:00:27,080 --> 00:00:28,690
比如x=a这一点 没有定义

52
00:00:28,690 --> 00:00:32,030
我把这一点抹黑 这样就知道

53
00:00:32,030 --> 00:00:33,110
此处无定义

54
00:00:33,110 --> 00:00:38,780
这一点是x=a

55
00:00:38,780 --> 00:00:45,180
这是x轴 这是y=f(x)轴

56
00:00:45,180 --> 00:00:47,120
就简单的称之为y轴吧

57
00:00:47,120 --> 00:00:51,030
这条线是函数f(x)

58
00:00:51,030 --> 00:00:53,880
或者说是y=f(x)

59
00:00:53,880 --> 00:00:55,740
我们已经学习了一些极限的视频

60
00:00:55,740 --> 00:00:57,160
我想你们有一个大致的理解了

61
00:00:57,160 --> 00:00:59,850
这里要求求出x趋向于a时函数的极限

62
00:00:59,850 --> 00:01:04,020
假设这一点是L

63
00:01:04,020 --> 00:01:06,480
从前面的视频可以知道--

64
00:01:06,480 --> 00:01:10,940
我先把它写出来--

65
00:01:10,940 --> 00:01:13,690
f(x)当x趋向于a时的极限

66
00:01:13,690 --> 00:01:17,560
直观地说 就是指

67
00:01:17,560 --> 00:01:20,980
x从任一边趋向a时

68
00:01:20,980 --> 00:01:22,290
比如从这边

69
00:01:22,290 --> 00:01:27,030
这时f(x)趋向于什么？

70
00:01:27,030 --> 00:01:29,490
当x在这的时候 f(x)在这

71
00:01:29,490 --> 00:01:33,080
x在这的时候 f(x)在这

72
00:01:35,950 --> 00:01:40,320
可以看到 它是趋向于L的

73
00:01:40,320 --> 00:01:42,200
从另一边接近a时--

74
00:01:42,200 --> 00:01:44,750
我们求过只从左边或者右边

75
00:01:44,750 --> 00:01:48,670
趋向于某个数时的极限

76
00:01:48,670 --> 00:01:52,380
但为了极限存在

77
00:01:52,380 --> 00:01:54,440
必须从正负方向同时趋向

78
00:01:54,440 --> 00:01:57,460
一个相同的数

79
00:01:57,460 --> 00:02:03,860
从右边接近a时 选这个点

80
00:02:03,860 --> 00:02:06,600
这是f(x)

81
00:02:06,600 --> 00:02:07,960
在这

82
00:02:07,960 --> 00:02:09,640
x到了这一点时 f(x)在这

83
00:02:09,640 --> 00:02:13,360
随着x越来越接近a点

84
00:02:13,360 --> 00:02:15,480
f(x)趋向于L

85
00:02:15,480 --> 00:02:16,290
所以f(x)当x趋向于a时的极限

86
00:02:16,290 --> 00:02:19,340
为L

87
00:02:19,340 --> 00:02:21,440
我想我们理解这种定义了

88
00:02:21,440 --> 00:02:27,360
但这并不是很--

89
00:02:27,360 --> 00:02:29,360
实际上相对于极限的概念

90
00:02:29,360 --> 00:02:32,180
这样的说法很不精确

91
00:02:32,180 --> 00:02:36,990
我目前所说的仅限于x趋向于某个数时

92
00:02:36,990 --> 00:02:39,290
f(x)趋近什么

93
00:02:39,290 --> 00:02:48,640
所以这个视频中

94
00:02:48,640 --> 00:02:55,150
我会尝试介绍一种新的

95
00:02:55,150 --> 00:02:57,190
极限的定义

96
00:02:57,190 --> 00:03:00,960
这种定义比简单的当x趋向某个值

97
00:03:00,960 --> 00:03:05,980
f(x)趋向于什么的定义方式

98
00:03:05,980 --> 00:03:12,360
要稍微精确一些 或者说要精确很多

99
00:03:12,360 --> 00:03:16,160
我认为可以把它看成一个小游戏

100
00:03:16,160 --> 00:03:18,030
新的定义就是

101
00:03:18,030 --> 00:03:18,490
这个式子意义在于

102
00:03:18,490 --> 00:03:21,840
我总是可以给出这个点的一个范围

103
00:03:21,840 --> 00:03:29,900
这里谈到范围

104
00:03:29,900 --> 00:03:37,460
我并不是针对整个定义域而言

105
00:03:37,460 --> 00:03:39,760
这里说的范围是像 比如说

106
00:03:39,760 --> 00:03:46,330
规定这么一段距离

107
00:03:46,330 --> 00:03:49,980
只要在这范围之内

108
00:03:49,980 --> 00:03:51,160
可以保证f(x)的值和L的距离

109
00:03:51,160 --> 00:03:54,300
始终不会超过某个给定值

110
00:03:54,300 --> 00:03:57,890
我把它看做

111
00:03:57,890 --> 00:04:00,030
一个小游戏

112
00:04:02,820 --> 00:04:07,830
你们可以不相信我

113
00:04:07,830 --> 00:04:10,870
并怀疑f(x)能不能

114
00:04:10,870 --> 00:04:16,770
始终位于和L相距0.5的范围内

115
00:04:16,770 --> 00:04:19,340
你们给出的距离是0.5

116
00:04:19,340 --> 00:04:21,020
此时 我必须能找到

117
00:04:21,020 --> 00:04:22,560
点a附近的一段范围

118
00:04:22,560 --> 00:04:23,910
确保f(x)值始终和L相距0.5以内 对吧？

119
00:04:23,910 --> 00:04:28,770
也就是说f(x)始终位于

120
00:04:31,530 --> 00:04:44,010
这段范围内

121
00:04:44,010 --> 00:04:47,310
只要x位于以a为中心的那段范围

122
00:04:47,310 --> 00:04:49,630
只要满足你们所给出的范围

123
00:04:49,630 --> 00:04:52,690
那么f(x)肯定

124
00:04:52,690 --> 00:04:58,090
能达到你们的要求

125
00:04:58,090 --> 00:05:01,260
我把图画大一些

126
00:05:01,260 --> 00:05:05,790
因为我觉着我是在同一个地方上

127
00:05:05,790 --> 00:05:08,860
反复的写

128
00:05:08,860 --> 00:05:10,450
这是曲线f(x) 这点没有定义

129
00:05:10,450 --> 00:05:12,590
这里不一定非得是个洞

130
00:05:12,590 --> 00:05:13,050
其极限值可以等于一个函数值

131
00:05:13,050 --> 00:05:17,090
但当函数在这点无定义却存在极限时

132
00:05:17,090 --> 00:05:19,510
情况会更为有趣

133
00:05:19,510 --> 00:05:20,960
所以这一点是--

134
00:05:20,960 --> 00:05:24,320
重新画出坐标轴

135
00:05:24,320 --> 00:05:27,810
这是x轴 y轴 x y

136
00:05:27,810 --> 00:05:30,480
这是极限值L 这是点a

137
00:05:30,480 --> 00:05:36,810
所以极限的定义是

138
00:05:36,810 --> 00:05:43,030
我一会儿会回到这个问题

139
00:05:43,030 --> 00:05:48,030
因为既然图大了些 我想再解释一遍

140
00:05:48,030 --> 00:05:51,650
这个式子表示--

141
00:05:51,650 --> 00:05:54,000
这是极限的ε-δ定义

142
00:05:54,000 --> 00:05:57,710
稍后我们会接触到ε和δ

143
00:05:57,710 --> 00:06:02,320
这式子的意义就是 你给我

144
00:06:02,320 --> 00:06:04,440
距L的任何一个距离

145
00:06:04,440 --> 00:06:05,365
实际上我们称这段距离为ε

146
00:06:09,970 --> 00:06:15,680
与最开始提出的定义相对应

147
00:06:15,680 --> 00:06:19,440
f(x)与L的距离不超过ε

148
00:06:19,440 --> 00:06:23,160
ε可以是任何

149
00:06:23,160 --> 00:06:24,350
大于0的实数

150
00:06:24,350 --> 00:06:26,060
所以这里的这段距离就是ε

151
00:06:26,060 --> 00:06:29,630
这一段也是ε

152
00:06:29,630 --> 00:06:32,980
对于给出的任何ε 任何实数--

153
00:06:32,980 --> 00:06:36,430
这点是L+ε

154
00:06:36,430 --> 00:06:38,940
这点是L-ε

155
00:06:38,940 --> 00:06:43,000
极限的ε-δ定义是说

156
00:06:43,000 --> 00:06:44,680
不论ε是多少

157
00:06:44,680 --> 00:06:49,420
总可以在a的附近确定一段范围

158
00:06:49,420 --> 00:06:52,570
并称之为δ

159
00:06:52,570 --> 00:06:55,440
总是可以确定一个δ

160
00:06:55,440 --> 00:06:57,290
这段是比a小δ

161
00:06:57,290 --> 00:07:01,270
这段是比a大δ

162
00:07:01,270 --> 00:07:04,490
这是字母δ

163
00:07:04,490 --> 00:07:05,380
只要在a+δ和a-δ之间

164
00:07:05,380 --> 00:07:11,750
选取一个x值

165
00:07:11,750 --> 00:07:16,575
只要x位于这段范围

166
00:07:16,575 --> 00:07:18,880
就可以保证与x相对应的f(x)

167
00:07:18,880 --> 00:07:21,480
位于你所给出的范围

168
00:07:21,480 --> 00:07:23,820
思考一下 你们会觉着有道理 对吧？

169
00:07:23,820 --> 00:07:27,380
实质上是说

170
00:07:27,380 --> 00:07:30,220
我可以无限接近极限值

171
00:07:30,220 --> 00:07:35,680
只要--

172
00:07:35,680 --> 00:07:39,970
我所说的无限接近

173
00:07:39,970 --> 00:07:45,650
是指你们可以任意给出一个ε值

174
00:07:45,650 --> 00:07:48,190
因为这有点像一个小游戏

175
00:07:48,190 --> 00:07:50,920
通过给出一个需要趋近的点

176
00:07:50,920 --> 00:07:57,790
附近的一段范围

177
00:07:57,790 --> 00:08:02,980
f(x)就可以无限接近极限值

178
00:08:02,980 --> 00:08:09,320
只要是在a附近的这段范围内

179
00:08:09,320 --> 00:08:13,640
选取x的值

180
00:08:13,640 --> 00:08:15,740
只要是在这里取一个x值

181
00:08:15,740 --> 00:08:17,220
我就可以保证f(x)

182
00:08:17,220 --> 00:08:19,750
位于你们所指定的范围

183
00:08:19,750 --> 00:08:22,580
为了更具体点

184
00:08:22,580 --> 00:08:24,110
假设 x位于--

185
00:08:24,110 --> 00:08:26,730
我们一律换成具体的数字来表示

186
00:08:26,730 --> 00:08:29,870
假设这个是2

187
00:08:29,870 --> 00:08:36,270
这是1

188
00:08:36,270 --> 00:08:40,290
也就是求x趋向1时f(x)的极限

189
00:08:40,290 --> 00:08:42,900
还没定义f(x)

190
00:08:42,900 --> 00:08:48,800
但函数曲线是条有一个洞的直线

191
00:08:48,800 --> 00:08:52,180
那里的f(x)值是2

192
00:08:52,180 --> 00:08:55,590
你们可以给我任何一个数字

193
00:08:55,590 --> 00:09:00,270
假定你们想用几个具体例子来验证一下

194
00:09:00,270 --> 00:09:03,540
比如想要f(x)位于--

195
00:09:03,540 --> 00:09:08,240
换种颜色--

196
00:09:08,240 --> 00:09:09,470
想要使f(x)位于距2这点0.5的范围内

197
00:09:09,470 --> 00:09:11,270
也就是在1.5和2.5之间

198
00:09:11,270 --> 00:09:12,760
那么只要x选在--

199
00:09:12,760 --> 00:09:15,580
x可以任意接近

200
00:09:15,580 --> 00:09:17,950
但只要x选在--

201
00:09:17,950 --> 00:09:21,720
对于这个函数

202
00:09:21,720 --> 00:09:23,680
假定是在0.9和1.1之间

203
00:09:23,680 --> 00:09:26,250
那么在这个例子中 δ和极限点的距离只有0.1

204
00:09:26,250 --> 00:09:33,460
只要在和1相距0.1的范围内

205
00:09:33,460 --> 00:09:35,810
选取x

206
00:09:35,810 --> 00:09:37,390
就可以确保f(x)

207
00:09:37,390 --> 00:09:41,730
位于要求的范围

208
00:09:41,730 --> 00:09:52,800
希望你们有初步的理解了

209
00:09:52,800 --> 00:09:56,590
现在我用ε和δ来给出定义

210
00:09:56,590 --> 00:09:57,760
实际上在课本中见到的就是这种

211
00:09:57,760 --> 00:10:00,530
接下来再做几个练习

212
00:10:00,530 --> 00:10:04,860
要搞清楚上面只是个特例

213
00:10:04,860 --> 00:10:05,520
你们给出一个ε 我再给出一个δ

214
00:10:05,520 --> 00:10:11,830
但如果它在定义里成立

215
00:10:11,830 --> 00:10:15,210
或者说要把它写出来

216
00:10:15,210 --> 00:10:23,025
那么就是说这并不仅仅适用于

217
00:10:23,025 --> 00:10:27,950
一个特定的例子

218
00:10:27,950 --> 00:10:31,340
而是适用于任何给定的数

219
00:10:31,340 --> 00:10:34,840
ε可以是百万分之一

220
00:10:34,840 --> 00:10:37,980
或者10的负一百次方

221
00:10:37,980 --> 00:10:40,750
也就是说非常接近于2

222
00:10:40,750 --> 00:10:45,400
我都可以给出一个a点附近的范围

223
00:10:45,400 --> 00:10:46,450
只要在里面选x值

224
00:10:46,450 --> 00:10:49,930
那么f(x)就位于所给出的范围

225
00:10:49,930 --> 00:10:55,680
也就是距极限值

226
00:10:55,680 --> 00:10:59,240
万亿分之一的距离

227
00:10:59,240 --> 00:11:03,920
当然 我不能确定

228
00:11:03,920 --> 00:11:07,520
当x=a时的情况

229
00:11:07,520 --> 00:11:10,560
我只是说当x不等于a

230
00:11:10,560 --> 00:11:17,010
且位于给定范围内时 它会成立

231
00:11:17,010 --> 00:11:19,670
f(x)会在给定的范围内

232
00:11:19,670 --> 00:11:23,460
为了在数学上更清晰些

233
00:11:23,460 --> 00:11:27,170
因为目前为止我都只是用单词来讲

234
00:11:27,170 --> 00:11:31,560
这是我们会在课本中看到的

235
00:11:31,560 --> 00:11:36,470
给定任意一个大于0的ε

236
00:11:36,470 --> 00:11:38,690
这是一种定义 对吧？

237
00:11:38,690 --> 00:11:39,640
如果写成这样

238
00:11:39,640 --> 00:11:42,345
那意味着ε可以是大于0的任何数

239
00:11:42,345 --> 00:11:44,670
然后会给出一个δ

240
00:11:44,670 --> 00:11:47,560
记住 ε是f(x)与极限值的距离

241
00:11:47,560 --> 00:11:49,720
对吧？

242
00:11:49,720 --> 00:11:53,010
是f(x)的范围

243
00:11:53,010 --> 00:11:54,050
之后会给出δ 也就是与a的距离 对吧？

244
00:11:54,050 --> 00:11:56,910
写出来

245
00:11:56,910 --> 00:11:58,910
f(x)当x趋向a的极限是1

246
00:11:58,910 --> 00:12:01,330
给出δ 只要x与a相距

247
00:12:01,330 --> 00:12:02,160
不超过δ--

248
00:12:02,160 --> 00:12:05,550
x和a的距离

249
00:12:05,550 --> 00:12:12,945
假设x选在这-- 换种颜色--

250
00:12:12,945 --> 00:12:13,960
如果x选在这

251
00:12:13,960 --> 00:12:17,620
那么只要x和a的绝对值

252
00:12:17,620 --> 00:12:19,970
大于0 这是为了保证x不会位于a

253
00:12:19,970 --> 00:12:22,180
因为a点函数无定义

254
00:12:22,180 --> 00:12:25,640
只要x和a的距离

255
00:12:25,640 --> 00:12:29,540
大于0并小于给定的一个值

256
00:12:29,540 --> 00:12:31,320
也就是δ

257
00:12:31,320 --> 00:12:34,260
只要在这个范围选取x

258
00:12:34,260 --> 00:12:38,120
我在这画一个小点的x轴

259
00:12:38,120 --> 00:12:39,330
这是a 这段距离是δ

260
00:12:39,330 --> 00:12:43,370
这段距离也是δ

261
00:12:43,370 --> 00:12:45,440
只要选取位于这里的x--

262
00:12:45,440 --> 00:12:47,270
这个x