0:00:00.900,0:00:02.810
Let me draw a function[br]that would be interesting

0:00:02.810,0:00:04.490
to take a limit of.

0:00:04.490,0:00:06.880
And I'll just draw it visually[br]for now, and we'll do some

0:00:06.880,0:00:08.390
specific examples[br]a little later.

0:00:08.390,0:00:11.870
So that's my y-axis,[br]and that's my x-axis.

0:00:11.870,0:00:14.180
And let;s say the function[br]looks something like--

0:00:14.180,0:00:15.950
I'll make it a fairly[br]straightforward function

0:00:15.950,0:00:19.760
--let's say it's a line,[br]for the most part.

0:00:19.760,0:00:23.100
Let's say it looks just[br]like, accept it has a

0:00:23.100,0:00:27.080
hole at some point.

0:00:27.080,0:00:28.690
x is equal to a, so[br]it's undefined there.

0:00:28.690,0:00:32.030
Let me black that point[br]out so you can see that

0:00:32.030,0:00:33.110
it's not defined there.

0:00:33.110,0:00:38.780
And that point there[br]is x is equal to a.

0:00:38.780,0:00:45.180
This is the x-axis, this is[br]the y is equal f of x-axis.

0:00:45.180,0:00:47.120
Let's just say[br]that's the y-axis.

0:00:47.120,0:00:51.030
And let's say that this[br]is f of x, or this is

0:00:51.030,0:00:53.880
y is equal to f of x.

0:00:53.880,0:00:55.740
Now we've done a bunch[br]of videos on limits.

0:00:55.740,0:00:57.160
I think you have an[br]intuition on this.

0:00:57.160,0:00:59.850
If I were to say what is the[br]limit as x approaches a,

0:00:59.850,0:01:04.020
and let's say that this[br]point right here is l.

0:01:04.020,0:01:06.480
We know from our previous[br]videos that-- well first of all

0:01:06.480,0:01:10.940
I could write it down --the[br]limit as x approaches

0:01:10.940,0:01:13.690
a of f of x.

0:01:13.690,0:01:17.560
What this means intuitively is[br]as we approach a from either

0:01:17.560,0:01:20.980
side, as we approach it from[br]that side, what does

0:01:20.980,0:01:22.290
f of x approach?

0:01:22.290,0:01:27.030
So when x is here,[br]f of x is here.

0:01:27.030,0:01:29.490
When x is here, f[br]of x is there.

0:01:29.490,0:01:33.080
And we see that it's[br]approaching this l right there.

0:01:35.950,0:01:40.320
And when we approach a from[br]that side-- and we've done

0:01:40.320,0:01:42.200
limits where you approach from[br]only the left or right side,

0:01:42.200,0:01:44.750
but to actually have a limit it[br]has to approach the same thing

0:01:44.750,0:01:48.670
from the positive direction and[br]the negative direction --but as

0:01:48.670,0:01:52.380
you go from there, if you pick[br]this x, then this is f of x.

0:01:52.380,0:01:54.440
f of x is right there.

0:01:54.440,0:01:57.460
If x gets here then it goes[br]here, and as we get closer and

0:01:57.460,0:02:03.860
closer to a, f of x approaches[br]this point l, or this value l.

0:02:03.860,0:02:06.600
So we say that the limit[br]of f of x ax x approaches

0:02:06.600,0:02:07.960
a is equal to l.

0:02:07.960,0:02:09.640
I think we have that intuition.

0:02:09.640,0:02:13.360
But this was not very, it's[br]actually not rigorous at all

0:02:13.360,0:02:15.480
in terms of being specific[br]in terms of what we

0:02:15.480,0:02:16.290
mean is a limit.

0:02:16.290,0:02:19.340
All I said so far is as[br]we get closer, what does

0:02:19.340,0:02:21.440
f of x get closer to?

0:02:21.440,0:02:27.360
So in this video I'll attempt[br]to explain to you a definition

0:02:27.360,0:02:29.360
of a limit that has a little[br]bit more, or actually a lot

0:02:29.360,0:02:32.180
more, mathematical rigor than[br]just saying you know, as x gets

0:02:32.180,0:02:36.990
closer to this value, what[br]does f of x get closer to?

0:02:36.990,0:02:39.290
And the way I think about it's:[br]kind of like a little game.

0:02:39.290,0:02:48.640
The definition is, this[br]statement right here means that

0:02:48.640,0:02:55.150
I can always give you a range[br]about this point-- and when I

0:02:55.150,0:02:57.190
talk about range I'm not[br]talking about it in the whole

0:02:57.190,0:03:00.960
domain range aspect, I'm just[br]talking about a range like you

0:03:00.960,0:03:05.980
know, I can give you a distance[br]from a as long as I'm no

0:03:05.980,0:03:12.360
further than that, I can[br]guarantee you that f of x is go

0:03:12.360,0:03:16.160
it not going to be any further[br]than a given distance from l

0:03:16.160,0:03:18.030
--and the way I think about it[br]is, it could be viewed

0:03:18.030,0:03:18.490
as a little game.

0:03:18.490,0:03:21.840
Let's say you say, OK Sal,[br]I don't believe you.

0:03:21.840,0:03:29.900
I want to see you know, whether[br]f of x can get within 0.5 of l.

0:03:29.900,0:03:37.460
So let's say you give me 0.5[br]and you say Sal, by this

0:03:37.460,0:03:39.760
definition you should always[br]be able to give me a range

0:03:39.760,0:03:46.330
around a that will get f of[br]x within 0.5 of l, right?

0:03:46.330,0:03:49.980
So the values of f of x are[br]always going to be right in

0:03:49.980,0:03:51.160
this range, right there.

0:03:51.160,0:03:54.300
And as long as I'm in that[br]range around a, as long as I'm

0:03:54.300,0:03:57.890
the range around you give me, f[br]of x will always be at least

0:03:57.890,0:04:00.030
that close to our limit point.

0:04:02.820,0:04:07.830
Let me draw it a little bit[br]bigger, just because I think

0:04:07.830,0:04:10.870
I'm just overwriting the same[br]diagram over and over again.

0:04:10.870,0:04:16.770
So let's say that this is f of[br]x, this is the hole point.

0:04:16.770,0:04:19.340
There doesn't have to be a hole[br]there; the limit could equal

0:04:19.340,0:04:21.020
actually a value of the[br]function, but the limit is more

0:04:21.020,0:04:22.560
interesting when the function[br]isn't defined there

0:04:22.560,0:04:23.910
but the limit is.

0:04:23.910,0:04:28.770
So this point right here-- that[br]is, let me draw the axes again.

0:04:31.530,0:04:44.010
So that's x-axis, y-axis x,[br]y, this is the limit point

0:04:44.010,0:04:47.310
l, this is the point a.

0:04:47.310,0:04:49.630
So the definition of the limit,[br]and I'll go back to this in

0:04:49.630,0:04:52.690
second because now that it's[br]bigger I want explain it again.

0:04:52.690,0:04:58.090
It says this means-- and this[br]is the epsilon delta definition

0:04:58.090,0:05:01.260
of limits, and we'll touch on[br]epsilon and delta in a second,

0:05:01.260,0:05:05.790
is I can guarantee you that[br]f of x, you give me any

0:05:05.790,0:05:08.860
distance from l you want.

0:05:08.860,0:05:10.450
And actually let's[br]call that epsilon.

0:05:10.450,0:05:12.590
And let's just hit on[br]the definition right

0:05:12.590,0:05:13.050
from the get go.

0:05:13.050,0:05:17.090
So you say I want to be no more[br]than epsilon away from l.

0:05:17.090,0:05:19.510
And epsilon can just be any[br]number greater, any real

0:05:19.510,0:05:20.960
number, greater than 0.

0:05:20.960,0:05:24.320
So that would be, this distance[br]right here is epsilon.

0:05:24.320,0:05:27.810
This distance there is epsilon.

0:05:27.810,0:05:30.480
And for any epsilon you give[br]me, any real number-- so this

0:05:30.480,0:05:36.810
is, this would be l plus[br]epsilon right here, this would

0:05:36.810,0:05:43.030
be l minus epsilon right here[br]--the epsilon delta definition

0:05:43.030,0:05:48.030
of this says that no matter[br]what epsilon one you give me, I

0:05:48.030,0:05:51.650
can always specify a[br]distance around a.

0:05:51.650,0:05:54.000
And I'll call that delta.

0:05:54.000,0:05:57.710
I can always specify[br]a distance around a.

0:05:57.710,0:06:02.320
So let's say this is delta[br]less than a, and this

0:06:02.320,0:06:04.440
is delta more than a.

0:06:04.440,0:06:05.365
This is the letter delta.

0:06:09.970,0:06:15.680
Where as long as you pick an x[br]that's within a plus delta and

0:06:15.680,0:06:19.440
a minus delta, as long as the x[br]is within here, I can guarantee

0:06:19.440,0:06:23.160
you that the f of x, the[br]corresponding f of x is going

0:06:23.160,0:06:24.350
to be within your range.

0:06:24.350,0:06:26.060
And if you think about it[br]this makes sense right?

0:06:26.060,0:06:29.630
It's essentially saying, I can[br]get you as close as you want to

0:06:29.630,0:06:32.980
this limit point just by-- and[br]when I say as close as you

0:06:32.980,0:06:36.430
want, you define what you want[br]by giving me an epsilon; on

0:06:36.430,0:06:38.940
it's a little bit of a game[br]--and I can get you as close as

0:06:38.940,0:06:43.000
you want to that limit point by[br]giving you a range around the

0:06:43.000,0:06:44.680
point that x is approaching.

0:06:44.680,0:06:49.420
And as long as you pick an x[br]value that's within this range

0:06:49.420,0:06:52.570
around a, long as you pick an x[br]value around there, I can

0:06:52.570,0:06:55.440
guarantee you that f of x will[br]be within the range

0:06:55.440,0:06:57.290
you specify.

0:06:57.290,0:07:01.270
Just make this a little bit[br]more concrete, let's say you

0:07:01.270,0:07:04.490
say, I want f of x to be within[br]0.5-- let's just you know, make

0:07:04.490,0:07:05.380
everything concrete numbers.

0:07:05.380,0:07:11.750
Let's say this is the number 2[br]and let's say this is number 1.

0:07:11.750,0:07:16.575
So we're saying that the limit[br]as x approaches 1 of f of x-- I

0:07:16.575,0:07:18.880
haven't defined f of x, but it[br]looks like a line with the hole

0:07:18.880,0:07:21.480
right there, is equal to 2.

0:07:21.480,0:07:23.820
This means that you can[br]give me any number.

0:07:23.820,0:07:27.380
Let's say you want to try it[br]out for a couple of examples.

0:07:27.380,0:07:30.220
Let's say you say I want f of x[br]to be within point-- let me do

0:07:30.220,0:07:35.680
a different color --I want f[br]of x to be within 0.5 of 2.

0:07:35.680,0:07:39.970
I want f of x to be[br]between 2.5 and 1.5.

0:07:39.970,0:07:45.650
Then I could say, OK, as long[br]as you pick an x within-- I

0:07:45.650,0:07:48.190
don't know, it could be[br]arbitrarily close but as long

0:07:48.190,0:07:50.920
as you pick an x that's --let's[br]say it works for this function

0:07:50.920,0:07:57.790
that's between, I don't[br]know, 0.9 and 1.1.

0:07:57.790,0:08:02.980
So in this case the delta from[br]our limit point is only 0.1.

0:08:02.980,0:08:09.320
As long as you pick an x that's[br]within 0.1 of this point, or 1,

0:08:09.320,0:08:13.640
I can guarantee you that your[br]f of x is going to

0:08:13.640,0:08:15.740
lie in that range.

0:08:15.740,0:08:17.220
So hopefully you get a little[br]bit of a sense of that.

0:08:17.220,0:08:19.750
Let me define that with the[br]actual epsilon delta, and this

0:08:19.750,0:08:22.580
is what you'll actually see in[br]your mat textbook, and then

0:08:22.580,0:08:24.110
we'll do a couple of examples.

0:08:24.110,0:08:26.730
And just to be clear, that[br]was just a specific example.

0:08:26.730,0:08:29.870
You gave me one epsilon and I[br]gave you a delta that worked.

0:08:29.870,0:08:36.270
But by definition if this is[br]true, or if someone writes

0:08:36.270,0:08:40.290
this, they're saying it doesn't[br]just work for one specific

0:08:40.290,0:08:42.900
instance, it works for[br]any number you give me.

0:08:42.900,0:08:48.800
You can say I want to be within[br]one millionth of, you know, or

0:08:48.800,0:08:52.180
ten to the negative hundredth[br]power of 2, you know, super

0:08:52.180,0:08:55.590
close to 2, and I can always[br]give you a range around this

0:08:55.590,0:09:00.270
point where as long as you pick[br]an x in that range, f of x will

0:09:00.270,0:09:03.540
always be within this range[br]that you specify, within that

0:09:03.540,0:09:08.240
were you know, one trillionth[br]of a unit away from

0:09:08.240,0:09:09.470
the limit point.

0:09:09.470,0:09:11.270
And of course, the one thing[br]I can't guarantee is what

0:09:11.270,0:09:12.760
happens when x is equal to a.

0:09:12.760,0:09:15.580
I'm just saying as long as you[br]pick an x that's within my

0:09:15.580,0:09:17.950
range but not on a, it'll work.

0:09:17.950,0:09:21.720
Your f of x will show up to be[br]within the range you specify.

0:09:21.720,0:09:23.680
And just to make the math[br]clear-- because I've been

0:09:23.680,0:09:26.250
speaking only in words so far[br]--and this is what we see the

0:09:26.250,0:09:33.460
textbook: it says look, you[br]give me any epsilon

0:09:33.460,0:09:35.810
greater than 0.

0:09:35.810,0:09:37.390
Anyway, this is a[br]definition, right?

0:09:37.390,0:09:41.730
If someone writes this they[br]mean that you can give them any

0:09:41.730,0:09:52.800
epsilon greater than 0, and[br]then they'll give you a delta--

0:09:52.800,0:09:56.590
remember your epsilon is how[br]close you want f of x to be

0:09:56.590,0:09:57.760
to your limit point, right?

0:09:57.760,0:10:00.530
It's a range around f of x[br]--they'll give you a delta

0:10:00.530,0:10:04.860
which is a range[br]around a, right?

0:10:04.860,0:10:05.520
Let me write this.

0:10:05.520,0:10:11.830
So limit as approaches a[br]of f of x is equal to l.

0:10:11.830,0:10:15.210
So they'll give you a delta[br]where as long as x is no more

0:10:15.210,0:10:23.025
than delta-- So the distance[br]between x and a, so if we pick

0:10:23.025,0:10:27.950
an x here-- let me do another[br]color --if we pick an x here,

0:10:27.950,0:10:31.340
the distance between that value[br]and a, as long as one, that's

0:10:31.340,0:10:34.840
greater than 0 so that x[br]doesn't show up on top of a,

0:10:34.840,0:10:37.980
because its function might be[br]undefined at that point.

0:10:37.980,0:10:40.750
But as long as the distance[br]between x and a is greater

0:10:40.750,0:10:45.400
than 0 and less than this x[br]range that they gave you,

0:10:45.400,0:10:46.450
it's less than delta.

0:10:46.450,0:10:49.930
So as long as you take an x,[br]you know if I were to zoom the

0:10:49.930,0:10:55.680
x-axis right here-- this is a[br]and so this distance right here

0:10:55.680,0:10:59.240
would be delta, and this[br]distance right here would be

0:10:59.240,0:11:03.920
delta --as long as you pick an[br]x value that falls here-- so as

0:11:03.920,0:11:07.520
long as you pick that x value[br]or this x value or this x value

0:11:07.520,0:11:10.560
--as long as you pick one of[br]those x values, I can guarantee

0:11:10.560,0:11:17.010
you that the distance between[br]your function and the limit

0:11:17.010,0:11:19.670
point, so the distance between[br]you know, when you take one of

0:11:19.670,0:11:23.460
these x values and you evaluate[br]f of x at that point, that the

0:11:23.460,0:11:27.170
distance between that f of x[br]and the limit point is

0:11:27.170,0:11:31.560
going to be less than the[br]number you gave them.

0:11:31.560,0:11:36.470
And if you think of, it seems[br]very complicated, and I have

0:11:36.470,0:11:38.690
mixed feelings about where[br]this is included in most

0:11:38.690,0:11:39.640
calculus curriculums.

0:11:39.640,0:11:42.345
It's included in like the, you[br]know, the third week before you

0:11:42.345,0:11:44.670
even learn derivatives, and[br]it's kind of this very mathy

0:11:44.670,0:11:47.560
and rigorous thing to think[br]about, and you know, it tends

0:11:47.560,0:11:49.720
to derail a lot of students and[br]a lot of people I don't think

0:11:49.720,0:11:53.010
get a lot of the intuition[br]behind it, but it is

0:11:53.010,0:11:54.050
mathematically rigorous.

0:11:54.050,0:11:56.910
And I think it is very valuable[br]once you study you know, more

0:11:56.910,0:11:58.910
advanced calculus or[br]become a math major.

0:11:58.910,0:12:01.330
But with that said, this[br]does make a lot of sense

0:12:01.330,0:12:02.160
intuitively, right?

0:12:02.160,0:12:05.550
Because before we were talking[br]about, look you know, I can get

0:12:05.550,0:12:12.945
you as close as x approaches[br]this value f of x is going

0:12:12.945,0:12:13.960
to approach this value.

0:12:13.960,0:12:17.620
And the way we mathematically[br]define it is, you say Sal,

0:12:17.620,0:12:19.970
I want to be super close.

0:12:19.970,0:12:22.180
I want the distance to be[br]f of x [UNINTELLIGIBLE].

0:12:22.180,0:12:25.640
And I want it to be[br]0.000000001, then I can always

0:12:25.640,0:12:29.540
give you a distance around x[br]where this will be true.

0:12:29.540,0:12:31.320
And I'm all out of[br]time in this video.

0:12:31.320,0:12:34.260
In the next video I'll do some[br]examples where I prove the

0:12:34.260,0:12:38.120
limits, where I prove some[br]limit statements using

0:12:38.120,0:12:39.330
this definition.

0:12:39.330,0:12:43.370
And hopefully you know, when we[br]use some tangible numbers, this

0:12:43.370,0:12:45.440
definition will make a[br]little bit more sense.

0:12:45.440,0:12:47.270
See you in the next video.