Epsilon-delta limit definition 1 | Limits | Differential Calculus | Khan Academy — Transcript

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1. 0:00Let me draw a function that would be interesting
2. 0:02to take a limit of.
3. 0:04And I'll just draw it visually for now, and we'll do some
4. 0:06specific examples a little later.
5. 0:08So that's my y-axis, and that's my x-axis.
6. 0:11And let;s say the function looks something like--
7. 0:14I'll make it a fairly straightforward function
8. 0:15--let's say it's a line, for the most part.
9. 0:19Let's say it looks just like, accept it has a
10. 0:23hole at some point.
11. 0:27x is equal to a, so it's undefined there.
12. 0:28Let me black that point out so you can see that
13. 0:32it's not defined there.
14. 0:33And that point there is x is equal to a.
15. 0:38This is the x-axis, this is the y is equal f of x-axis.
16. 0:45Let's just say that's the y-axis.
17. 0:47And let's say that this is f of x, or this is
18. 0:51y is equal to f of x.
19. 0:53Now we've done a bunch of videos on limits.
20. 0:55I think you have an intuition on this.
21. 0:57If I were to say what is the limit as x approaches a,
22. 0:59and let's say that this point right here is l.
23. 1:04We know from our previous videos that-- well first of all
24. 1:06I could write it down --the limit as x approaches
25. 1:10a of f of x.
26. 1:13What this means intuitively is as we approach a from either
27. 1:17side, as we approach it from that side, what does
28. 1:20f of x approach?
29. 1:22So when x is here, f of x is here.
30. 1:27When x is here, f of x is there.
31. 1:29And we see that it's approaching this l right there.
32. 1:35And when we approach a from that side-- and we've done
33. 1:40limits where you approach from only the left or right side,
34. 1:42but to actually have a limit it has to approach the same thing
35. 1:44from the positive direction and the negative direction --but as
36. 1:48you go from there, if you pick this x, then this is f of x.
37. 1:52f of x is right there.
38. 1:54If x gets here then it goes here, and as we get closer and
39. 1:57closer to a, f of x approaches this point l, or this value l.
40. 2:03So we say that the limit of f of x ax x approaches
41. 2:06a is equal to l.
42. 2:07I think we have that intuition.
43. 2:09But this was not very, it's actually not rigorous at all
44. 2:13in terms of being specific in terms of what we
45. 2:15mean is a limit.
46. 2:16All I said so far is as we get closer, what does
47. 2:19f of x get closer to?
48. 2:21So in this video I'll attempt to explain to you a definition
49. 2:27of a limit that has a little bit more, or actually a lot
50. 2:29more, mathematical rigor than just saying you know, as x gets
51. 2:32closer to this value, what does f of x get closer to?
52. 2:36And the way I think about it's: kind of like a little game.
53. 2:39The definition is, this statement right here means that
54. 2:48I can always give you a range about this point-- and when I
55. 2:55talk about range I'm not talking about it in the whole
56. 2:57domain range aspect, I'm just talking about a range like you
57. 3:00know, I can give you a distance from a as long as I'm no
58. 3:05further than that, I can guarantee you that f of x is go
59. 3:12it not going to be any further than a given distance from l
60. 3:16--and the way I think about it is, it could be viewed
61. 3:18as a little game.
62. 3:18Let's say you say, OK Sal, I don't believe you.
63. 3:21I want to see you know, whether f of x can get within 0.5 of l.
64. 3:29So let's say you give me 0.5 and you say Sal, by this
65. 3:37definition you should always be able to give me a range
66. 3:39around a that will get f of x within 0.5 of l, right?
67. 3:46So the values of f of x are always going to be right in
68. 3:49this range, right there.
69. 3:51And as long as I'm in that range around a, as long as I'm
70. 3:54the range around you give me, f of x will always be at least
71. 3:57that close to our limit point.
72. 4:02Let me draw it a little bit bigger, just because I think
73. 4:07I'm just overriding the same diagram over and over again.
74. 4:10So let's say that this is f of x, this is the hole point.
75. 4:16There doesn't have to be a hole there; the limit could equal
76. 4:19actually a value of the function, but the limit is more
77. 4:21interesting when the function isn't defined there
78. 4:22but the limit is.
79. 4:23So this point right here-- that is, let me draw the axes again.
80. 4:31So that's x-axis, y-axis x, y, this is the limit point
81. 4:44l, this is the point a.
82. 4:47So the definition of the limit, and I'll go back to this in
83. 4:49second because now that it's bigger I want explain it again.
84. 4:52It says this means-- and this is the epsilon delta definition
85. 4:58of limits, and we'll touch on epsilon and delta in a second,
86. 5:01is I can guarantee you that f of x, you give me any
87. 5:05distance from l you want.
88. 5:08And actually let's call that epsilon.
89. 5:10And let's just hit on the definition right
90. 5:12from the get go.
91. 5:13So you say I want to be no more than epsilon away from l.
92. 5:17And epsilon can just be any number greater, any real
93. 5:19number, greater than 0.
94. 5:20So that would be, this distance right here is epsilon.
95. 5:24This distance there is epsilon.
96. 5:27And for any epsilon you give me, any real number-- so this
97. 5:30is, this would be l plus epsilon right here, this would
98. 5:36be l minus epsilon right here --the epsilon delta definition
99. 5:43of this says that no matter what epsilon one you give me, I
100. 5:48can always specify a distance around a.
101. 5:51And I'll call that delta.
102. 5:54I can always specify a distance around a.
103. 5:57So let's say this is delta less than a, and this
104. 6:02is delta more than a.
105. 6:04This is the letter delta.
106. 6:09Where as long as you pick an x that's within a plus delta and
107. 6:15a minus delta, as long as the x is within here, I can guarantee
108. 6:19you that the f of x, the corresponding f of x is going
109. 6:23to be within your range.
110. 6:24And if you think about it this makes sense right?
111. 6:26It's essentially saying, I can get you as close as you want to
112. 6:29this limit point just by-- and when I say as close as you
113. 6:32want, you define what you want by giving me an epsilon; on
114. 6:36it's a little bit of a game --and I can get you as close as
115. 6:38you want to that limit point by giving you a range around the
116. 6:43point that x is approaching.
117. 6:44And as long as you pick an x value that's within this range
118. 6:49around a, long as you pick an x value around there, I can
119. 6:52guarantee you that f of x will be within the range
120. 6:55you specify.
121. 6:57Just make this a little bit more concrete, let's say you
122. 7:01say, I want f of x to be within 0.5-- let's just you know, make
123. 7:04everything concrete numbers.
124. 7:05Let's say this is the number 2 and let's say this is number 1.
125. 7:11So we're saying that the limit as x approaches 1 of f of x-- I
126. 7:16haven't defined f of x, but it looks like a line with the hole
127. 7:18right there, is equal to 2.
128. 7:21This means that you can give me any number.
129. 7:23Let's say you want to try it out for a couple of examples.
130. 7:27Let's say you say I want f of x to be within point-- let me do
131. 7:30a different color --I want f of x to be within 0.5 of 2.
132. 7:35I want f of x to be between 2.5 and 1.5.
133. 7:39Then I could say, OK, as long as you pick an x within-- I
134. 7:45don't know, it could be arbitrarily close but as long
135. 7:48as you pick an x that's --let's say it works for this function
136. 7:50that's between, I don't know, 0.9 and 1.1.
137. 7:57So in this case the delta from our limit point is only 0.1.
138. 8:02As long as you pick an x that's within 0.1 of this point, or 1,
139. 8:09I can guarantee you that your f of x is going to
140. 8:13lie in that range.
141. 8:15So hopefully you get a little bit of a sense of that.
142. 8:17Let me define that with the actual epsilon delta, and this
143. 8:19is what you'll actually see in your mat textbook, and then
144. 8:22we'll do a couple of examples.
145. 8:24And just to be clear, that was just a specific example.
146. 8:26You gave me one epsilon and I gave you a delta that worked.
147. 8:29But by definition if this is true, or if someone writes
148. 8:36this, they're saying it doesn't just work for one specific
149. 8:40instance, it works for any number you give me.
150. 8:42You can say I want to be within one millionth of, you know, or
151. 8:48ten to the negative hundredth power of 2, you know, super
152. 8:52close to 2, and I can always give you a range around this
153. 8:55point where as long as you pick an x in that range, f of x will
154. 9:00always be within this range that you specify, within that
155. 9:03were you know, one trillionth of a unit away from
156. 9:08the limit point.
157. 9:09And of course, the one thing I can't guarantee is what
158. 9:11happens when x is equal to a.
159. 9:12I'm just saying as long as you pick an x that's within my
160. 9:15range but not on a, it'll work.
161. 9:17Your f of x will show up to be within the range you specify.
162. 9:21And just to make the math clear-- because I've been
163. 9:23speaking only in words so far --and this is what we see the
164. 9:26textbook: it says look, you give me any epsilon
165. 9:33greater than 0.
166. 9:35Anyway, this is a definition, right?
167. 9:37If someone writes this they mean that you can give them any
168. 9:41epsilon greater than 0, and then they'll give you a delta--
169. 9:52remember your epsilon is how close you want f of x to be
170. 9:56to your limit point, right?
171. 9:57It's a range around f of x --they'll give you a delta
172. 10:00which is a range around a, right?
173. 10:04Let me write this.
174. 10:05So limit as approaches a of f of x is equal to l.
175. 10:11So they'll give you a delta where as long as x is no more
176. 10:15than delta-- So the distance between x and a, so if we pick
177. 10:23an x here-- let me do another color --if we pick an x here,
178. 10:27the distance between that value and a, as long as one, that's
179. 10:31greater than 0 so that x doesn't show up on top of a,
180. 10:34because its function might be undefined at that point.
181. 10:37But as long as the distance between x and a is greater
182. 10:40than 0 and less than this x range that they gave you,
183. 10:45it's less than delta.
184. 10:46So as long as you take an x, you know if I were to zoom the
185. 10:49x-axis right here-- this is a and so this distance right here
186. 10:55would be delta, and this distance right here would be
187. 10:59delta --as long as you pick an x value that falls here-- so as
188. 11:03long as you pick that x value or this x value or this x value
189. 11:07--as long as you pick one of those x values, I can guarantee
190. 11:10you that the distance between your function and the limit
191. 11:17point, so the distance between you know, when you take one of
192. 11:19these x values and you evaluate f of x at that point, that the
193. 11:23distance between that f of x and the limit point is
194. 11:27going to be less than the number you gave them.
195. 11:31And if you think of, it seems very complicated, and I have
196. 11:36mixed feelings about where this is included in most
197. 11:38calculus curriculums.
198. 11:39It's included in like the, you know, the third week before you
199. 11:42even learn derivatives, and it's kind of this very mathy
200. 11:44and rigorous thing to think about, and you know, it tends
201. 11:47to derail a lot of students and a lot of people I don't think
202. 11:49get a lot of the intuition behind it, but it is
203. 11:53mathematically rigorous.
204. 11:54And I think it is very valuable once you study you know, more
205. 11:56advanced calculus or become a math major.
206. 11:58But with that said, this does make a lot of sense
207. 12:01intuitively, right?
208. 12:02Because before we were talking about, look you know, I can get
209. 12:05you as close as x approaches this value f of x is going
210. 12:12to approach this value.
211. 12:13And the way we mathematically define it is, you say Sal,
212. 12:17I want to be super close.
213. 12:19I want the distance to be f of x [UNINTELLIGIBLE].
214. 12:22And I want it to be 0.000000001, then I can always
215. 12:25give you a distance around x where this will be true.
216. 12:29And I'm all out of time in this video.
217. 12:31In the next video I'll do some examples where I prove the
218. 12:34limits, where I prove some limit statements using
219. 12:38this definition.
220. 12:39And hopefully you know, when we use some tangible numbers, this
221. 12:43definition will make a little bit more sense.
222. 12:45See you in the next video.