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==Definition for finite limit for function of one variable== | ==Definition for finite limit for function of one variable== | ||
Revision as of 23:00, 7 September 2012
ORIGINAL FULL PAGE: Limit
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Definition for finite limit for function of one variable
Two-sided limit
Left hand limit
Right hand limit
Full timed transcript: [SHOW MORE]
0:00:15.940,0:00:20.740 Vipul: In this talk, I'm going to give definitions of one-sided limits.
0:00:20.740,0:00:25.650 So it is going to be the left hand limit and the right hand limit, and I'm going to basically
0:00:25.650,0:00:42.650 compare it with the definition of two-sided limit which was in a previous video. Let's just write this down--left-hand limit.
0:00:44.110,0:00:48.679 Let me first remind you what the definition of two-sided limit says.
0:00:48.679,0:00:57.679 So here's what it says. It says limit as x approaches c, f(x) = L
0:00:58.469,0:01:03.140 so f has to be defined on the immediate left and the immediate right of c.
0:01:03.140,0:01:07.960 It says that this is true if the following holds so for every epsilon greater than zero
0:01:07.960,0:01:13.960 there exists a delta > 0 such that for all x which are within delta of c
0:01:14.000,0:01:22.771 either delta on the left of c or within a delta on the right of c we have that f(x) is within an epsilon
0:01:23.650,0:01:30.530 distance of L. Okay. Now with the left and right hand limit
0:01:30.530,0:01:37.460 what we are trying to do we are trying to consider only one-sided approaches on the, on the x
0:01:39.000,0:01:41.510 What will change when we do the left-hand limit
0:01:42.001,0:01:44.641 what will be different from this definition? [ANSWER!]
0:01:45.710,0:01:48.330 Rui: We approach c from the left.
0:01:48.330,0:01:52.790 Vipul: We'll approach c from the left so what part of this definition will change? [ANSWER!]
0:01:52.790,0:01:54.880 Rui: From the fourth line?
0:01:54.880,0:01:56.890 Vipul: You mean this line?
0:01:56.890,0:02:06.810 Rui: Oh for all x within c distance, within delta distance of c
0:02:06.810,0:02:08.700 Vipul: So what will change?
0:02:08.700,0:02:14.020 Rui: We will not have (c, c + delta).
0:02:14.020,0:02:18.390 Vipul: This part won’t be there. We will just be concerned about whether when x is
0:02:18.390,0:02:28.000 delta close on the left side of c, f(x) is here [in (L - epsilon, L + epsilon)]. Will we change this one also to only include the left?
0:02:28.000,0:02:33.460 Or this one will remain as it is. Rui: I think it will remain. Vipul: It will remain as it is because we
0:02:33.460,0:02:36.340 are just saying as x approaches c from the left f(x) approaches L.
0:02:36.340,0:02:43.340 We are not claiming that f(x) approaches L from the left. Let me make a number line picture.
0:02:51.750,0:02:56.130 We will do a full geometric understanding of the thing later. Right now it's just very [formal].
0:02:56.130,0:03:00.850 So the function is defined on the immediate left of c, maybe not defined at c. It is defined
0:03:00.850,0:03:01.920 on the immediate left of c.
0:03:01.920,0:03:06.410 We don’t even know if the function is defined on the right of c and what we are
0:03:06.410,0:03:13.410 saying is that for any epsilon, so any epsilon around L you can find a delta such that if you restrict
0:03:13.800,0:03:20.800 attention to the interval from c minus delta to c [i.e., (c- delta, c) in math notation]
0:03:21.450,0:03:23.130 then the f value there is within the epsilon distance of L.
0:03:24.130,0:03:28.959 Now the f value could be epsilon to the left or the right so we take left hand limit on
0:03:28.959,0:03:33.840 the domain side it doesn’t have to approach from the left on the other side.
0:03:33.840,0:03:40.690 Let me just write down the full definition. We want to keep this on the side.
0:03:40.690,0:04:03.690 What it says that for every epsilon > 0 there exists
0:04:05.180,0:04:16.680 by the way, the understanding of the what this definition really means will come in another video you may have seen before this or after this
0:04:16.680,0:04:21.209 ... for all x ... [continuing definition]
0:04:21.209,0:04:26.500 Now we should also change it if we are writing in this form so how will it read now?
0:04:26.500,0:04:28.030 Rui: For all x ...
0:04:35.000,0:04:38.000 Vipul: So will you put x – c or c – x? [ANSWER!]
0:04:38.330,0:04:40.990 Rui: It will be x – c, oh c – x.
0:04:41.000,0:04:46.760 Vipul: c – x. Because you want c to be bigger than x. You want x to be on the left of c.
0:04:46.850,0:05:01.850 What would this read, i.e. x is in (c – delta,c). Okay.
0:05:05.000,0:05:11.460 What do we have? We have the same thing. This part doesn’t change.
0:05:13.000,0:05:19.000 Rui: f(x) is within epsilon distance of L.
0:05:34.400,0:05:40.400 Vipul: Why do I keep saying this thing about the L approach doesn’t have to be from the left.
0:05:41.000,0:05:44.350 What’s the significance of that? Why is that important? [ANSWER!]
0:05:45.000,0:05:51.000 Rui: It’s important because we don’t know whether the function is decreasing or increasing
0:05:51.620,0:05:52.370 at that point.
0:05:52.370,0:05:55.750 Vipul: Yeah, so if your function is actually increasing than L will also be approached
0:05:55.750,0:06:01.590 from the left, and if it’s decreasing it will be approached from the right, but sometimes
0:06:01.590,0:06:07.590 it’s neither increasing nor decreasing, but it's still true it approaches from one side, so that’s a little complicated but the way
0:06:07.590,0:06:12.150 this comes up is that when you are dealing with composition of functions, so when you
0:06:12.150,0:06:16.710 are doing one function and then applying another function to that and you have some results
0:06:16.710,0:06:18.440 with one-sided limits.
0:06:18.440,0:06:30.440 Let me just write this down. If you have one-sided limits and you have composition,
0:06:31.610,0:06:39.550 so you are doing one function and then doing another you have to be very careful.
0:06:45.050,0:06:48.350 You need to be very careful when you are doing one-sided limits and composition.
0:06:48.360,0:06:57.360 Why? Because if you have g of f(x) and x approaches to c from the left, f(x) approaches L but
0:06:57.850,0:06:59.280 not necessarily from the left.
0:06:59.280,0:07:03.560 You then you have another thing which is as f(x) approaches L from the left, g of that
0:07:03.560,0:07:09.280 approaches something you just need to be careful that when you compose things the sidedness
0:07:09.280,0:07:10.930 could change each time you compose.
0:07:10.930,0:07:14.590 Rui: Can you write a composition of the function out?
0:07:14.590,0:07:17.870 Vipul: Not in this video. We will do that in another video.
0:07:17.870,0:07:23.800 That’s something we will see in a subsequent video but this is just something to keep in
0:07:23.800,0:07:27.770 mind so when you see that it will ring a bell.
0:07:30.770,0:07:31.880 Let us do... what other side is left? Rui: Right? Vipul: Right!
0:07:31.880,0:07:36.690 Vipul: By the way, you probably already know this if you have seen this intuitively so
0:07:36.690,0:07:42.300 I'm not stressing this too much but left hand limit is really the limit as you approach
0:07:42.300,0:07:49.300 from the left. You are not moving toward the left you are moving from the left to the point.
0:07:50.160,0:07:55.940 Right hand limit will be approach from the right to the point so it is right, moving from
0:07:55.940,0:07:59.330 the right, so the words left and right are describing where the limit is coming *from*,
0:07:59.330,0:08:06.330 not the direction which it is going to.
0:08:12.569,0:08:17.650 Now you can just tell me what will be the corresponding thing. To make sense of this
0:08:17.650,0:08:19.819 notion we need f to be defined where? [ANSWER!]
0:08:19.819,0:08:21.699 Rui: On its right.
0:08:21.699,0:08:29.199 Vipul: On the immediate right of c. If it is not defined on the immediate right it doesn’t
0:08:29.389,0:08:36.389 even make sense to ask this question what the right hand limit is.
0:08:37.280,0:08:38.550 How will that be defined?
0:08:38.550,0:08:44.240 Rui: For every epsilon greater than zero
0:08:44.240,0:08:51.240 Vipul: The epsilon is the interval on which you are trying to trap the function value.
0:08:51.500,0:08:54.279 Rui: There exists epsilon
0:08:54.279,0:08:55.890 Vipul: No, delta
0:08:55.890,0:09:14.890 Rui: delta> 0 such that for all x with x – c > 0
0:09:15.040,0:09:22.040 Vipul: The general one is for all x with 0<|x-c|<delta because you want to capture both the intervals.
0:09:23.170,0:09:29.270 In this one, the left hand limit one, we just captured the left side interval.
0:09:29.270,0:09:39.270 Now in the right one we just want to capture the right side interval, so as you said 0< x- c < delta.
0:09:44.180,0:09:51.480 In the picture, the function is defined, say c to c + t and you are really saying you can
0:09:52.290,0:10:00.290 find delta if x is in here [between c and c + delta] which actually... this is not including c, it is all the points
0:10:00.390,0:10:05.390 in the immediate right of c. We have? [ANSWER!]
0:10:06.000,0:10:13.000 Rui: The absolute value of f(x) – L is less than epsilon.
0:10:20.010,0:10:22.010 Vipul: So f(x) is? Are we here? We have everything?
0:10:23.010,0:10:23.260 Rui: Yes.
0:10:26.190,0:10:30.890 Vipul: We have both of these here? So do you see what’s the main difference between these
0:10:30.890,0:10:37.430 two and the actual definition.
0:10:37.430,0:10:42.930 For every epsilon there exists delta, the first second and fourth line remain the same.
0:10:42.930,0:10:47.440 It is this line where you are specifying where the x are that’s different.
0:10:47.440,0:10:54.940 In the two-sided thing the x could be either place. For the left hand limit the x,
0:10:55.720,0:11:05.220 you just want x here and for the right hand limit
want x in c to c + delta. Okay? [END!]Relation between the limit notions
Definition of finite limit for function of one variable in terms of a game
Two-sided limit
Non-existence of limit
Conceptual definition and various cases
Formulation of conceptual definition
Functions of one variable case
This covers limits at and to infinity.