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PROFESSOR: Hi.
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Welcome, once again, to another
lecture on limits.
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Actually, from a certain point
of view, today's lecture will
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00:00:44,880 --> 00:00:47,290
be the same as the last
lecture, only
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00:00:47,290 --> 00:00:49,130
from a different viewpoint.
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Our lecture today is
called 'Limits: a
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00:00:51,850 --> 00:00:53,440
More Rigorous Approach'.
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00:00:53,440 --> 00:00:57,690
And what our objective for the
day is, aside from helping you
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00:00:57,690 --> 00:01:01,660
gain experience and facility
with using limit expressions
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00:01:01,660 --> 00:01:05,630
and absolute values and the
like, is to have you see how
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we can use the power of
objective, well-defined
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mathematical definitions to
find rather easy ways of
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solving certain types
of problems.
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00:01:18,210 --> 00:01:23,620
Now to this end, let's very
briefly review our fundamental
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definition of last time.
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00:01:26,730 --> 00:01:31,540
Namely, the limit of 'f of x'
as 'x' approaches 'a' equals
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00:01:31,540 --> 00:01:36,820
'l' means that for each epsilon
greater than 0, we can
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00:01:36,820 --> 00:01:40,770
find delta greater than 0,
such that whenever the
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00:01:40,770 --> 00:01:45,120
absolute value of 'x minus a' is
less than delta but greater
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00:01:45,120 --> 00:01:48,580
than 0, then the absolute value
of ''f of x' minus 'l''
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will be less than epsilon.
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00:01:50,800 --> 00:01:56,270
To state that once again, but
in more intuitive terms, for
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any tolerance limit epsilon at
all, we can suitably find
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00:02:02,400 --> 00:02:06,630
another tolerance limit, delta,
such that whenever we
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00:02:06,630 --> 00:02:12,130
make 'x' within delta of 'a', 'f
of x' will automatically be
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00:02:12,130 --> 00:02:14,800
within epsilon of 'l'.
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00:02:14,800 --> 00:02:19,290
And again, the very, very
important emphasis here, we do
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00:02:19,290 --> 00:02:21,980
not allow 'x' to equal 'a'.
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00:02:21,980 --> 00:02:25,680
Again, in terms of a diagram,
if this is the curve 'y'
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00:02:25,680 --> 00:02:28,650
equals 'f of x', this
is 'x' equals 'a'.
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00:02:28,650 --> 00:02:29,890
This is 'l'.
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00:02:29,890 --> 00:02:34,710
If we call this 'l' plus
epsilon, if we call this 'l'
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minus epsilon-- in other words,
epsilon is this width
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00:02:36,950 --> 00:02:38,230
over here--
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00:02:38,230 --> 00:02:44,520
then the way we find delta is to
reflect back to the curve,
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00:02:44,520 --> 00:02:47,100
emphasizing, again, in
the neighborhood
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00:02:47,100 --> 00:02:48,360
of the point 'a'--
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00:02:48,360 --> 00:02:51,230
I can't stress that point
strongly enough, that if this
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00:02:51,230 --> 00:02:55,510
curve were not 1:1, there are
going to be other places where
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00:02:55,510 --> 00:02:57,690
this line meets the curve.
49
00:02:57,690 --> 00:03:01,320
So if that happens, you must
make sure that you pick the
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00:03:01,320 --> 00:03:06,280
neighborhood of 'a', not some
other point over here.
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00:03:06,280 --> 00:03:08,530
We're interested in what
happens near 'a'.
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But at any rate, again notice
that the fact that these two
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00:03:11,660 --> 00:03:15,670
intervals here were equal does
not guarantee that when you
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00:03:15,670 --> 00:03:18,090
project down here, they
will be equal.
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00:03:18,090 --> 00:03:20,480
In fact, the only time that
these two widths would be
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00:03:20,480 --> 00:03:24,210
equal is if the curve happened
to be a straight line.
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00:03:24,210 --> 00:03:27,120
And what it means, again, is
that the delta that we're
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00:03:27,120 --> 00:03:30,170
talking about, for example,
is the minimum
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00:03:30,170 --> 00:03:32,140
of these two widths.
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00:03:32,140 --> 00:03:38,310
In other words, as I've drawn
this diagram, delta would be
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00:03:38,310 --> 00:03:41,900
the distance from 'a'
to this end point.
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And all we're saying is that
once 'x' is in this open
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00:03:46,560 --> 00:03:51,090
interval but not including 'a'
itself, 'f of x' will be in
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00:03:51,090 --> 00:03:52,960
this open interval.
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00:03:52,960 --> 00:03:56,140
And again, notice, as we were
emphasizing last time, once
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00:03:56,140 --> 00:04:01,970
this delta happens to work,
automatically any smaller
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00:04:01,970 --> 00:04:03,700
delta will also work.
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00:04:03,700 --> 00:04:06,640
In other words, if something
is true for everything in
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00:04:06,640 --> 00:04:10,250
here, it's certainly true
for everything in some
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00:04:10,250 --> 00:04:11,370
subinterval of this.
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00:04:11,370 --> 00:04:13,440
What you must be careful
about is not to
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00:04:13,440 --> 00:04:14,520
reverse this process.
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Don't get outside and pick
bigger deltas, then you might
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00:04:18,100 --> 00:04:20,320
very well be in a little
bit of trouble.
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00:04:20,320 --> 00:04:23,200
At any rate, once we've reviewed
what the basic
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definition is, it seems about
the only way to show what
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00:04:27,900 --> 00:04:33,290
mathematics is all about is to
actually do a few theorems,
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00:04:33,290 --> 00:04:37,050
that is, derive a few
inescapable consequences of
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the definition that show how
our theorems coincide with
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00:04:42,830 --> 00:04:45,150
what we believe to be
intuitively true
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00:04:45,150 --> 00:04:46,470
in the first place.
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00:04:46,470 --> 00:04:48,840
And for obvious reasons, we
should start with what
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hopefully would be the simplest
possible theorems and
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00:04:51,750 --> 00:04:54,630
then proceed to tougher
ones as we go along.
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So as my first one, I've
chosen the following.
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00:04:58,130 --> 00:05:02,170
The limit of 'c' as 'x'
approaches 'a' is 'c', where
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00:05:02,170 --> 00:05:04,080
'c' is any constant.
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00:05:04,080 --> 00:05:09,520
Now again, that may look a
little bit strange to you.
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00:05:09,520 --> 00:05:10,560
Let's look at it this way.
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What I'm saying is let
'f of x' equal 'c',
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00:05:16,250 --> 00:05:18,120
where 'c' is a constant.
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00:05:18,120 --> 00:05:22,600
Then what we're saying is for
this choice of 'f of x', the
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00:05:22,600 --> 00:05:27,410
limit of 'f of x' as 'x'
approaches 'a' is 'c' itself.
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00:05:27,410 --> 00:05:29,280
That's what this thing says.
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00:05:29,280 --> 00:05:30,450
How do we prove this?
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00:05:30,450 --> 00:05:33,680
Well, you see, we have
a criteria given.
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00:05:33,680 --> 00:05:36,190
Namely, what is our
basic definition?
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00:05:36,190 --> 00:05:39,750
Let's just juxtaposition our
basic definition with this
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00:05:39,750 --> 00:05:40,940
particular result.
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00:05:40,940 --> 00:05:44,800
To prove that the limit here
is 'c', notice that in this
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00:05:44,800 --> 00:05:51,150
particular problem, if we come
back and compare this with our
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00:05:51,150 --> 00:05:54,180
basic definition, notice that
we have the same basic
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00:05:54,180 --> 00:06:00,630
definition as before, only the
role of 'f of x' is played by
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00:06:00,630 --> 00:06:03,390
'c' and 'l' is also
played by 'c'.
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00:06:03,390 --> 00:06:05,710
In other words, in this
particular problem 'f of x' is
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00:06:05,710 --> 00:06:09,230
'c' and 'l' is also 'c'.
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00:06:09,230 --> 00:06:10,410
Now what must we do?
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00:06:10,410 --> 00:06:14,430
We must show that for each
epsilon greater than 0, or for
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00:06:14,430 --> 00:06:16,910
an arbitrary epsilon
greater than 0--
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now what does it mean to say
an arbitrary epsilon?
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00:06:19,530 --> 00:06:22,590
In a way, think of it as being
a game, a battle of wits
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between you and your worst
enemy, and you're out to win
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00:06:25,320 --> 00:06:27,160
and your worst enemy
is out to beat you.
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00:06:27,160 --> 00:06:31,180
So to make this as difficult a
game as possible, you allow
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00:06:31,180 --> 00:06:33,790
your worst enemy to choose the
epsilon, provided only that
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00:06:33,790 --> 00:06:36,570
it's positive.
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00:06:36,570 --> 00:06:39,690
For whichever one he picks, you
must be able to find the
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00:06:39,690 --> 00:06:43,340
delta that matches that epsilon,
such that what?
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00:06:43,340 --> 00:06:47,750
Whenever 'x' is within delta of
'a' but not equal to 'a',
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00:06:47,750 --> 00:06:51,170
'f of x' will be within
epsilon of 'l'.
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00:06:51,170 --> 00:06:52,900
So let's write that
down over here.
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Let's write what that says.
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00:06:54,330 --> 00:07:05,480
Given epsilon greater than 0,
we must find a delta greater
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00:07:05,480 --> 00:07:11,040
than 0, such that what?
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00:07:11,040 --> 00:07:12,590
Well, by definition.
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00:07:12,590 --> 00:07:19,330
Such that when the absolute
value of 'x minus a' is less
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00:07:19,330 --> 00:07:23,160
than delta but greater
than 0, the absolute
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00:07:23,160 --> 00:07:24,620
value of 'f of x'--
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00:07:24,620 --> 00:07:26,660
well, that of course, is
'c' in this case--
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00:07:26,660 --> 00:07:30,430
minus 'l', which is also 'c' in
this case, the 'l' stands
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00:07:30,430 --> 00:07:33,640
for the limit, has to be
less than epsilon.
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00:07:33,640 --> 00:07:37,740
And lo and behold, we find that
this is a rather simple
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00:07:37,740 --> 00:07:41,700
procedure because what
is 'c' minus 'c'?
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00:07:41,700 --> 00:07:45,090
'c' minus 'c' is 0, and
automatically that will be
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00:07:45,090 --> 00:07:47,710
smaller than any positive
epsilon.
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00:07:47,710 --> 00:07:50,520
In other words, what this says
is even your worst enemy can't
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00:07:50,520 --> 00:07:52,720
give you a tough time
with this problem.
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00:07:52,720 --> 00:07:56,340
Namely, no matter what epsilon
he prescribes, no matter how
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00:07:56,340 --> 00:08:01,020
sensitive, you say, well, for
delta I'll pick anything.
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00:08:01,020 --> 00:08:02,670
And it works.
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00:08:02,670 --> 00:08:03,700
And why does that work?
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00:08:03,700 --> 00:08:06,350
Well, here again we
can emphasize
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00:08:06,350 --> 00:08:07,920
the geometric approach.
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00:08:07,920 --> 00:08:14,980
Namely, if we plot the graph 'f
of x' equals 'c', observe
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00:08:14,980 --> 00:08:18,180
that that plot equals a straight
line, 'y' equals 'c'.
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00:08:18,180 --> 00:08:21,150
Now take 'x' equals
'a' over here.
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00:08:21,150 --> 00:08:23,290
What is 'f of a'?
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00:08:23,290 --> 00:08:25,860
'f of a' is 'c'.
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00:08:25,860 --> 00:08:28,860
Now pick an epsilon, which is
this half-width over here,
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00:08:28,860 --> 00:08:34,220
knock that off on either
side of 'c'.
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00:08:34,220 --> 00:08:36,780
And all you're saying in this
particular case is no matter
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00:08:36,780 --> 00:08:41,679
how far away 'x' is from 'a', 'f
of x' will be within these
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00:08:41,679 --> 00:08:43,860
tolerance limits of 'c'.
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00:08:43,860 --> 00:08:45,050
And why is that?
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00:08:45,050 --> 00:08:50,040
Because 'f' is defined in such
a way that the output for any
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00:08:50,040 --> 00:08:52,360
element in its domain
is 'c' itself.
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00:08:52,360 --> 00:08:55,930
In other words, every element
maps up here.
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00:08:55,930 --> 00:08:58,390
This is what you mean by saying
that the graph is the
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00:08:58,390 --> 00:09:00,160
straight line 'y' equals 'c'.
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00:09:00,160 --> 00:09:02,110
By the way, another
word of caution.
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00:09:02,110 --> 00:09:09,930
We must always make sure that
your solution does not depend
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00:09:09,930 --> 00:09:10,930
on the diagram.
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00:09:10,930 --> 00:09:13,160
You see, if a person were to
look at this diagram very
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00:09:13,160 --> 00:09:16,330
quickly, he would assume that
'c' had to be a positive
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00:09:16,330 --> 00:09:19,470
constant here, because look at
how I've drawn the line 'y'
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00:09:19,470 --> 00:09:19,870
equals 'c'.
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00:09:19,870 --> 00:09:21,430
It's above the x-axis.
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00:09:21,430 --> 00:09:23,900
Well, 'c' could just as easily
be a negative constant.
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00:09:23,900 --> 00:09:25,990
And if I drew the diagram
that way, 'c' would
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00:09:25,990 --> 00:09:27,690
be below the x-axis.
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00:09:27,690 --> 00:09:29,290
The important thing
to check is this.
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00:09:29,290 --> 00:09:32,750
When you draw a diagram, you
can't have a certain number
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00:09:32,750 --> 00:09:35,780
being positive and negative at
the same time, so you choose
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00:09:35,780 --> 00:09:37,650
it one way or the other.
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00:09:37,650 --> 00:09:41,960
Always make sure when you do
this that your formal proof,
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00:09:41,960 --> 00:09:45,150
your analytic proof, goes
through word for word if you
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00:09:45,150 --> 00:09:47,800
reverse the signature of the
sign of the number that you're
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00:09:47,800 --> 00:09:48,920
working with.
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00:09:48,920 --> 00:09:53,010
Make sure that your answer does
not depend on the picture
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00:09:53,010 --> 00:09:54,240
that you've drawn.
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00:09:54,240 --> 00:09:57,040
Make sure that your answer
follows, inescapably, from
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00:09:57,040 --> 00:09:58,410
your basic definition.
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00:09:58,410 --> 00:10:00,000
And notice that this is
what we did here.
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00:10:00,000 --> 00:10:01,200
We showed what?
185
00:10:01,200 --> 00:10:04,210
That no matter what epsilon we
were given, we could find a
186
00:10:04,210 --> 00:10:06,420
delta-- in fact, in this
case, any delta--
187
00:10:06,420 --> 00:10:09,950
such that when 'x' was within
delta of a as long as 'x'
188
00:10:09,950 --> 00:10:10,790
wasn't equal to 'a'.
189
00:10:10,790 --> 00:10:12,940
Well in this case, even
if 'x' equaled 'a',
190
00:10:12,940 --> 00:10:14,160
there was no harm done.
191
00:10:14,160 --> 00:10:17,780
But the point is we want to keep
away from a 0 over 0 form
192
00:10:17,780 --> 00:10:19,760
so we always impose
this condition.
193
00:10:19,760 --> 00:10:23,940
In this case, once 'x' was
within delta of 'a', 'f of x'
194
00:10:23,940 --> 00:10:27,410
was automatically within epsilon
of 'c', as long as
195
00:10:27,410 --> 00:10:29,920
epsilon was positive,
because 'f of x' was
196
00:10:29,920 --> 00:10:31,570
already equal to 'c'.
197
00:10:31,570 --> 00:10:34,010
The difference was already 0.
198
00:10:34,010 --> 00:10:36,620
Now again, notice what
happens here.
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00:10:36,620 --> 00:10:41,850
The more pragmatic student will
say, why did you use this
200
00:10:41,850 --> 00:10:46,660
long math when it was obvious
from either the diagram or
201
00:10:46,660 --> 00:10:49,820
from intuition that this
is the correct answer?
202
00:10:49,820 --> 00:10:52,380
And the reason, is as we have
already seen and as we will
203
00:10:52,380 --> 00:10:55,640
see many, many times during
our course, the intuitive
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00:10:55,640 --> 00:10:58,270
answer and the correct
answer will not
205
00:10:58,270 --> 00:11:00,200
necessarily be the same.
206
00:11:00,200 --> 00:11:03,650
The point is we always want to
make sure that when we prove
207
00:11:03,650 --> 00:11:09,110
something, the proof follows
from the assumed definitions
208
00:11:09,110 --> 00:11:11,770
in a logically rigorous way.
209
00:11:11,770 --> 00:11:15,080
If it happens once you prove
this that you can intuitively
210
00:11:15,080 --> 00:11:18,340
recognize the same result,
that's like a double reward
211
00:11:18,340 --> 00:11:21,240
because now you won't have to
memorize what the result was,
212
00:11:21,240 --> 00:11:22,980
you'll just use this thing
automatically.
213
00:11:22,980 --> 00:11:25,910
But the beauty is that for
somebody who doesn't have the
214
00:11:25,910 --> 00:11:29,290
same intuitive insights that you
have, if he says it's not
215
00:11:29,290 --> 00:11:32,240
self-evident to me, explain
to me what's happening.
216
00:11:32,240 --> 00:11:35,110
Then, you see, once he's
accepted the basic definition
217
00:11:35,110 --> 00:11:38,210
and assuming that he knows the
rules of mathematics, he has
218
00:11:38,210 --> 00:11:40,770
to come up with the same
answer that you did.
219
00:11:40,770 --> 00:11:43,850
And this is what we mean by an
objective criterion for doing
220
00:11:43,850 --> 00:11:45,560
mathematics, okay.
221
00:11:45,560 --> 00:11:47,550
Well, this was kind
of an easy one.
222
00:11:47,550 --> 00:11:50,425
Let's do another kind of
easy one that's harder.
223
00:11:50,425 --> 00:11:54,370
Let's make some gradual
transitions, here.
224
00:11:54,370 --> 00:11:57,230
Let's pick another theorem.
225
00:11:57,230 --> 00:11:59,520
Here's another one that sounds
pretty self-evident.
226
00:11:59,520 --> 00:12:04,410
The limit as 'x' approaches
'a' is 'a'.
227
00:12:04,410 --> 00:12:07,310
In fact, what that seems to say
in a self-evident way is
228
00:12:07,310 --> 00:12:11,620
that as 'x' gets arbitrarily
close to 'a', 'x' gets
229
00:12:11,620 --> 00:12:14,450
arbitrarily close
to equal to 'a'.
230
00:12:14,450 --> 00:12:17,490
And if anything is a truism,
I guess that's it.
231
00:12:17,490 --> 00:12:20,890
So we certainly suspect that
this is a true statement.
232
00:12:20,890 --> 00:12:23,530
All I'm trying to get you used
to is not to make a mountain
233
00:12:23,530 --> 00:12:25,780
out of a mole hill, not to have
to think that mathematics
234
00:12:25,780 --> 00:12:30,050
becomes a severe thing where
we try to find hard ways of
235
00:12:30,050 --> 00:12:33,375
doing easy things, but rather
that we can find an
236
00:12:33,375 --> 00:12:37,660
unambiguous, logically
constructed language from
237
00:12:37,660 --> 00:12:41,020
which all of our results can be
proven without recourse to
238
00:12:41,020 --> 00:12:44,260
intuition once our basic
definitions are chosen.
239
00:12:44,260 --> 00:12:46,010
So let's see how this
would work.
240
00:12:46,010 --> 00:12:48,820
Let's again go back to
our basic definition.
241
00:12:48,820 --> 00:12:51,850
You see, again, what we're
saying here is that this is
242
00:12:51,850 --> 00:12:55,780
just a special case now where
'f of x' equals 'x'.
243
00:12:55,780 --> 00:12:57,740
You see the function
is 'f of x'.
244
00:12:57,740 --> 00:13:00,720
In this case, it's 'x', and this
is a special case where
245
00:13:00,720 --> 00:13:02,550
'a' is 'l'.
246
00:13:02,550 --> 00:13:04,650
Now what are you saying here?
247
00:13:04,650 --> 00:13:08,540
You're saying given epsilon
greater than 0, we must find a
248
00:13:08,540 --> 00:13:11,380
delta greater than 0,
such that what?
249
00:13:11,380 --> 00:13:14,510
Such that whenever 'x' is within
delta of 'a' but not
250
00:13:14,510 --> 00:13:20,900
equal to 'a', then 'x' will
be within epsilon of 'a'.
251
00:13:20,900 --> 00:13:24,340
And now you just look at this
thing, I hope, and you say
252
00:13:24,340 --> 00:13:26,640
well, there's sort of a
similarity over here.
253
00:13:26,640 --> 00:13:29,690
254
00:13:29,690 --> 00:13:34,850
How close should 'x' be chosen
to 'a' if you want 'x' to be
255
00:13:34,850 --> 00:13:36,920
within epsilon of 'a'?
256
00:13:36,920 --> 00:13:38,680
And the answer, quite obviously
in this case
257
00:13:38,680 --> 00:13:42,490
without, cause to define again,
is to simply say what
258
00:13:42,490 --> 00:13:43,460
in this case?
259
00:13:43,460 --> 00:13:48,520
For the given epsilon,
choose, for example,
260
00:13:48,520 --> 00:13:50,940
delta to equal epsilon.
261
00:13:50,940 --> 00:13:54,590
Because certainly, if the
absolute value of 'x minus a'
262
00:13:54,590 --> 00:13:57,810
is less than epsilon but greater
than 0, then certainly
263
00:13:57,810 --> 00:14:02,920
the absolute value of 'x minus
a' is less than epsilon.
264
00:14:02,920 --> 00:14:05,070
Don't be upset that the proof
happened to be fairly
265
00:14:05,070 --> 00:14:06,280
easy in this case.
266
00:14:06,280 --> 00:14:08,890
More importantly, don't be upset
that there was a more
267
00:14:08,890 --> 00:14:10,330
intuitive way of doing it.
268
00:14:10,330 --> 00:14:13,060
Remember what we want to do is
to get these rigorous ways
269
00:14:13,060 --> 00:14:19,370
down pat, interpret them in
terms of pictures wherever the
270
00:14:19,370 --> 00:14:20,960
pictures are available.
271
00:14:20,960 --> 00:14:23,560
And then, when we get to the
case where pictures aren't
272
00:14:23,560 --> 00:14:28,110
available, to be able to extend
the analytic concepts.
273
00:14:28,110 --> 00:14:30,780
Again, to see what happens here
pictorially, you notice
274
00:14:30,780 --> 00:14:34,090
that if you start with the
function 'f of x' equals 'x,'
275
00:14:34,090 --> 00:14:37,220
its graph is a straight
line 'y' equals 'x'.
276
00:14:37,220 --> 00:14:41,620
And I'll risk some freehand
drawing here.
277
00:14:41,620 --> 00:14:45,600
So in other words, we suspect
that this is 'a'.
278
00:14:45,600 --> 00:14:48,550
It better be, if this says that
all points on this line
279
00:14:48,550 --> 00:14:50,600
have the property that the
y-coordinate is equal to the
280
00:14:50,600 --> 00:14:51,740
x-coordinate.
281
00:14:51,740 --> 00:14:52,770
Now what do we want to do?
282
00:14:52,770 --> 00:14:58,610
We pick an epsilon and
knock off a plus
283
00:14:58,610 --> 00:15:01,150
epsilon and a minus epsilon.
284
00:15:01,150 --> 00:15:05,150
And now what we want to find out
is how close must x be to
285
00:15:05,150 --> 00:15:10,440
a on the x-axis in order that 'f
of x'-- namely 'y' in this
286
00:15:10,440 --> 00:15:11,730
case, or 'x' itself--
287
00:15:11,730 --> 00:15:15,660
be within this prescribed
tolerance limits of 'a'?
288
00:15:15,660 --> 00:15:17,070
You see what happens here?
289
00:15:17,070 --> 00:15:20,230
This is that one case where
it happens that what?
290
00:15:20,230 --> 00:15:27,320
If you draw these lines over
and project down, but by
291
00:15:27,320 --> 00:15:28,800
proportional parts--
292
00:15:28,800 --> 00:15:30,140
and this is crucial here.
293
00:15:30,140 --> 00:15:33,520
Even if this weren't the 45
degree line, the fact that
294
00:15:33,520 --> 00:15:36,580
these two pieces here are equal
would guarantee that
295
00:15:36,580 --> 00:15:39,440
these two pieces here are equal,
in spite of how I've
296
00:15:39,440 --> 00:15:40,410
drawn this.
297
00:15:40,410 --> 00:15:44,280
The fact that this is the 45
degree line not only says that
298
00:15:44,280 --> 00:15:46,920
these two pieces here are
equal but it says what?
299
00:15:46,920 --> 00:15:52,330
That each of these pieces is
equal to each of these pieces.
300
00:15:52,330 --> 00:15:54,930
I wish I had drawn this better
for you, but in fact the worse
301
00:15:54,930 --> 00:15:57,590
I draw it, the more you have
to rely on being able to
302
00:15:57,590 --> 00:16:00,040
visualize abstractly what's
happening here.
303
00:16:00,040 --> 00:16:03,290
What I'm saying is that this
point here would be labeled 'a
304
00:16:03,290 --> 00:16:06,410
plus epsilon', this point here
would be labeled 'a minus
305
00:16:06,410 --> 00:16:09,540
epsilon', and this is the
pictorial version of what it
306
00:16:09,540 --> 00:16:13,660
means to say you could have
chosen delta to equal epsilon,
307
00:16:13,660 --> 00:16:16,180
in this particular case.
308
00:16:16,180 --> 00:16:20,040
Well, that's enough of the
easier ones, so let's pick one
309
00:16:20,040 --> 00:16:22,510
that gets slightly tougher.
310
00:16:22,510 --> 00:16:27,700
And this one gets tougher in one
sense, but insidiously--
311
00:16:27,700 --> 00:16:29,250
meaning it's real sneaky--
312
00:16:29,250 --> 00:16:30,950
simple in another sense.
313
00:16:30,950 --> 00:16:34,280
In other words, it turns out
that one can reason falsely
314
00:16:34,280 --> 00:16:37,690
and get the right answer
just by a quirk.
315
00:16:37,690 --> 00:16:40,020
You see, what I want to prove
next is a very important
316
00:16:40,020 --> 00:16:43,450
theorem that says that the limit
of a sum is equal to the
317
00:16:43,450 --> 00:16:45,740
sum of the limits.
318
00:16:45,740 --> 00:16:49,690
Written out more formally, if I
have two functions 'f of x'
319
00:16:49,690 --> 00:16:54,240
and 'g of x' and formed the sum
''f of x' plus 'g of x''
320
00:16:54,240 --> 00:16:56,610
and I want to take the limit of
that sum as 'x' approaches
321
00:16:56,610 --> 00:17:00,660
'a', what this theorem says is
you can find the limit of each
322
00:17:00,660 --> 00:17:03,320
of the functions separately
first, and
323
00:17:03,320 --> 00:17:05,500
then add the two results.
324
00:17:05,500 --> 00:17:07,980
Now at first glance, you might
be tempted to say, well, what
325
00:17:07,980 --> 00:17:10,140
else would you expect
to happen?
326
00:17:10,140 --> 00:17:11,859
The answer is, I don't know,
again, what else you would
327
00:17:11,859 --> 00:17:12,609
expect to happen.
328
00:17:12,609 --> 00:17:13,970
But this is a luxury.
329
00:17:13,970 --> 00:17:17,010
You see, evidently what's
happened here is that we've
330
00:17:17,010 --> 00:17:19,819
reversed the order
of operations.
331
00:17:19,819 --> 00:17:20,950
You see, this says what?
332
00:17:20,950 --> 00:17:24,329
First add these two, and
then take the limit.
333
00:17:24,329 --> 00:17:26,440
What are we doing down here?
334
00:17:26,440 --> 00:17:30,130
First we're taking the limits
and then we're adding.
335
00:17:30,130 --> 00:17:33,380
Now, is it self-evident that
just by changing the order of
336
00:17:33,380 --> 00:17:35,740
operations you don't
change anything?
337
00:17:35,740 --> 00:17:39,290
We've seen many examples already
in the short time that
338
00:17:39,290 --> 00:17:42,660
this course has been in
existence where changing the
339
00:17:42,660 --> 00:17:46,720
order, changing the voice
inflection, what have you,
340
00:17:46,720 --> 00:17:48,930
changes the answer.
341
00:17:48,930 --> 00:17:50,830
And in fact, we're going
to see more drastic
342
00:17:50,830 --> 00:17:52,510
examples later on.
343
00:17:52,510 --> 00:17:54,470
I guess this is one of
the tragedies of
344
00:17:54,470 --> 00:17:56,710
a course like this.
345
00:17:56,710 --> 00:17:58,740
I guess it's typical of
problems every place.
346
00:17:58,740 --> 00:18:01,320
If the place that the person is
going to get into trouble
347
00:18:01,320 --> 00:18:05,010
comes far beyond the time at
which you're lecturing to him,
348
00:18:05,010 --> 00:18:08,030
it's kind of empty to threaten
him with the trouble he's
349
00:18:08,030 --> 00:18:09,220
going to get into.
350
00:18:09,220 --> 00:18:11,560
So I'm not going to threaten
you with the trouble you're
351
00:18:11,560 --> 00:18:13,510
going to get into until
we get into it.
352
00:18:13,510 --> 00:18:16,760
All I'm going to say is be
careful when you say that it's
353
00:18:16,760 --> 00:18:19,960
self-evident that we can first
add and then take the limit or
354
00:18:19,960 --> 00:18:22,620
whether we first take the
limits and then add.
355
00:18:22,620 --> 00:18:25,230
In general, it does make a
difference in which order you
356
00:18:25,230 --> 00:18:26,840
perform operations.
357
00:18:26,840 --> 00:18:28,760
Well, let's take this just a
little bit more formally to
358
00:18:28,760 --> 00:18:30,770
see what this thing says.
359
00:18:30,770 --> 00:18:33,460
First of all, when we write
something like this, we assume
360
00:18:33,460 --> 00:18:36,030
that the limit of 'f of x' as
'x' approaches 'a' exists,
361
00:18:36,030 --> 00:18:37,940
otherwise we wouldn't
write this thing.
362
00:18:37,940 --> 00:18:39,770
So let's call that limit 'l1'.
363
00:18:39,770 --> 00:18:42,020
In other words, let the limit of
'f of x' as 'x' approaches
364
00:18:42,020 --> 00:18:43,570
'a' equal 'l1'.
365
00:18:43,570 --> 00:18:47,910
Let the limit of 'g of x' as 'x'
approaches 'a' equal 'l2'.
366
00:18:47,910 --> 00:18:51,790
Now define a new function 'h of
x' to be equal to ''f of x'
367
00:18:51,790 --> 00:18:53,070
plus 'g of x''.
368
00:18:53,070 --> 00:18:54,510
And by the way, this is
something I didn't say
369
00:18:54,510 --> 00:18:57,050
strongly enough in one of our
early lectures, and I want to
370
00:18:57,050 --> 00:19:00,950
make sure that it's very clear
that this is understood.
371
00:19:00,950 --> 00:19:06,420
And that is, notice that when
you define 'h' to be the sum
372
00:19:06,420 --> 00:19:11,000
of 'f' and 'g', you had better
make sure that 'f' and 'g'
373
00:19:11,000 --> 00:19:13,810
have the same domain.
374
00:19:13,810 --> 00:19:17,540
You see, if some number 'x'
belongs to the domain of 'f'
375
00:19:17,540 --> 00:19:21,590
but it doesn't belong to the
domain of 'g', then how can
376
00:19:21,590 --> 00:19:24,280
you form ''f of x' plus
'g of x''? 'g'
377
00:19:24,280 --> 00:19:26,320
doesn't operate on 'x'.
378
00:19:26,320 --> 00:19:29,250
By the way, this is not quite
as serious a problem as it
379
00:19:29,250 --> 00:19:33,600
seems if you understand the
language of our new
380
00:19:33,600 --> 00:19:35,670
mathematics and sets
and the like.
381
00:19:35,670 --> 00:19:42,350
Namely, if the domain of 'f'
happens to look like this and
382
00:19:42,350 --> 00:19:49,900
the domain of 'g' happens to
look like this, what we do is
383
00:19:49,900 --> 00:19:51,730
we restrict the sum to the
384
00:19:51,730 --> 00:19:53,560
intersection of the two domains.
385
00:19:53,560 --> 00:19:56,170
386
00:19:56,170 --> 00:20:00,150
In other words, referring back
to our function 'h', we define
387
00:20:00,150 --> 00:20:01,980
the domain of 'h' to
be the intersection
388
00:20:01,980 --> 00:20:03,290
of these two domains.
389
00:20:03,290 --> 00:20:06,580
And that way, for any 'x', which
is in the domain of 'h',
390
00:20:06,580 --> 00:20:10,910
it automatically belongs to both
the domain of 'f' and the
391
00:20:10,910 --> 00:20:11,950
domain of 'g'.
392
00:20:11,950 --> 00:20:14,730
And so this becomes
well-defined.
393
00:20:14,730 --> 00:20:17,220
Another way of looking at this
is what you're really saying
394
00:20:17,220 --> 00:20:21,820
is that 'f' and 'g' must
include in their domain
395
00:20:21,820 --> 00:20:25,340
intervals surrounding
'x' equals 'a'.
396
00:20:25,340 --> 00:20:28,100
And since you're only interested
in what's happening
397
00:20:28,100 --> 00:20:31,750
near 'x' equals 'a', you don't
really care whether these have
398
00:20:31,750 --> 00:20:35,150
the same domains or not,
provided they have what?
399
00:20:35,150 --> 00:20:38,520
An intersection that can serve
as a common domain.
400
00:20:38,520 --> 00:20:41,600
But that, I think, is more
clear from the context.
401
00:20:41,600 --> 00:20:44,280
It's a very, very important
fine point.
402
00:20:44,280 --> 00:20:47,690
It's a tragedy to try to add
two numbers, one of which
403
00:20:47,690 --> 00:20:48,890
doesn't exist.
404
00:20:48,890 --> 00:20:50,240
I don't know if it's a tragedy,
405
00:20:50,240 --> 00:20:52,760
it's certainly futile.
406
00:20:52,760 --> 00:20:55,710
At any rate, though, let's see
what this thing then says.
407
00:20:55,710 --> 00:21:00,260
If we now define 'h' to be 'f
plus g', what we want to prove
408
00:21:00,260 --> 00:21:04,250
is that the limit of 'h of x'
as 'x' approaches 'a' equals
409
00:21:04,250 --> 00:21:06,700
'l1 plus l2'.
410
00:21:06,700 --> 00:21:08,310
Now here's the point again.
411
00:21:08,310 --> 00:21:11,420
What does this mean
by definition?
412
00:21:11,420 --> 00:21:17,300
It means that given epsilon
greater than 0, we must be
413
00:21:17,300 --> 00:21:21,170
able to find a delta such that
when 'x' is within delta of
414
00:21:21,170 --> 00:21:25,540
'a' but not equal to 'a',
'h of x' is within
415
00:21:25,540 --> 00:21:28,380
epsilon of 'l1 plus l2'.
416
00:21:28,380 --> 00:21:31,000
That's probably kind of hard
to keep track of, so I've
417
00:21:31,000 --> 00:21:32,860
taken the liberty of writing
this down for
418
00:21:32,860 --> 00:21:34,740
you right over here.
419
00:21:34,740 --> 00:21:38,310
See, given epsilon greater than
0-- so that's given, we
420
00:21:38,310 --> 00:21:39,800
have no control over that--
421
00:21:39,800 --> 00:21:45,180
what we must do is find delta
greater than 0, such that 0
422
00:21:45,180 --> 00:21:48,280
less than the absolute value
of 'x minus a' less than
423
00:21:48,280 --> 00:21:49,930
delta, implies--
424
00:21:49,930 --> 00:21:51,140
now, what is 'h of x'?
425
00:21:51,140 --> 00:21:53,860
It's ''f of x' plus 'g of x'',
and our limit that we're
426
00:21:53,860 --> 00:21:56,640
looking for in this case
is 'l1 plus l2'.
427
00:21:56,640 --> 00:22:00,640
So mathematically, we replace 'h
of x' by ''f of x' plus 'g
428
00:22:00,640 --> 00:22:03,430
of x'', the limit
is 'l1 plus l2'.
429
00:22:03,430 --> 00:22:06,935
So what must we show that the
absolute value of the quantity
430
00:22:06,935 --> 00:22:10,830
of ''f of x' plus 'g of x''
minus the quantity 'l1 plus
431
00:22:10,830 --> 00:22:12,690
l2' is less than epsilon.
432
00:22:12,690 --> 00:22:15,560
And now we start to play
detective again.
433
00:22:15,560 --> 00:22:17,980
This is the expression
that we want to
434
00:22:17,980 --> 00:22:19,570
make less than epsilon.
435
00:22:19,570 --> 00:22:23,970
So what we do is we look at this
particular expression and
436
00:22:23,970 --> 00:22:27,500
we try to see what kind of cute
things we can do with it.
437
00:22:27,500 --> 00:22:29,220
Now, what do I mean
by a cute thing?
438
00:22:29,220 --> 00:22:32,490
Well, we're assuming that the
limit of 'f of x' as 'x'
439
00:22:32,490 --> 00:22:36,150
approaches 'a' equals 'l1'.
440
00:22:36,150 --> 00:22:39,230
Let me write this as
an aside over here.
441
00:22:39,230 --> 00:22:42,730
Among other things, what this
tells us is that we have a
442
00:22:42,730 --> 00:22:47,480
hold on expressions like this.
443
00:22:47,480 --> 00:22:50,080
In other words, the fact that
the limit of 'f of x' as 'x'
444
00:22:50,080 --> 00:22:53,310
approaches 'a' is equal to 'l1'
tells us that we can make
445
00:22:53,310 --> 00:22:55,970
this as small as we want.
446
00:22:55,970 --> 00:22:57,940
I'll use a subscript over here
for epsilon 1 because it
447
00:22:57,940 --> 00:23:00,050
doesn't have to be the same
epsilon that was given here.
448
00:23:00,050 --> 00:23:03,540
For any positive number, say
epsilon 1, the point is I can
449
00:23:03,540 --> 00:23:08,980
make 'f of x' within epsilon 1
of 'l1' just by choosing 'x'
450
00:23:08,980 --> 00:23:11,330
sufficiently close to
'a' by definition
451
00:23:11,330 --> 00:23:12,610
of what limit means.
452
00:23:12,610 --> 00:23:15,390
So in other words, I like
expressions of this form.
453
00:23:15,390 --> 00:23:20,140
And similarly, I like
expressions of this form.
454
00:23:20,140 --> 00:23:22,530
And again, the reason is that
since the limit of 'g of x' as
455
00:23:22,530 --> 00:23:26,260
'x' approaches 'a' is 'l sub 2',
it gives me a hold on the
456
00:23:26,260 --> 00:23:29,100
difference between 'g
of x' and 'l2'.
457
00:23:29,100 --> 00:23:33,420
So again, using the old adage
that hindsight is better than
458
00:23:33,420 --> 00:23:36,860
foresight by a darn site,
knowing exactly what it is I
459
00:23:36,860 --> 00:23:40,260
have to do, I come back here
and try to doctor things up
460
00:23:40,260 --> 00:23:41,360
for myself.
461
00:23:41,360 --> 00:23:45,360
The first thing I observe is
that this can be rewritten.
462
00:23:45,360 --> 00:23:46,800
There's no calculus
in this, notice.
463
00:23:46,800 --> 00:23:49,090
Just plain ordinary algebra,
arithmetic.
464
00:23:49,090 --> 00:23:53,830
This can be rewritten so that
I can group the 'f sub 'f of
465
00:23:53,830 --> 00:23:58,160
x' and 'l1'' together and 'g
of x' and 'l2' together.
466
00:23:58,160 --> 00:24:01,940
In other words, this is indeed
nothing more than 1.
467
00:24:01,940 --> 00:24:04,700
The absolute value of the
quantity of ''f of x' minus
468
00:24:04,700 --> 00:24:07,370
l1' plus the absolute value
of the quantity
469
00:24:07,370 --> 00:24:09,290
''g of x' minus l2'.
470
00:24:09,290 --> 00:24:12,980
Now, the point is since we
already know that the absolute
471
00:24:12,980 --> 00:24:15,810
value of a sum is less than
or equal to the sum of the
472
00:24:15,810 --> 00:24:18,760
absolute values, that
tells me that--
473
00:24:18,760 --> 00:24:21,480
treating this is one number and
this is another number,
474
00:24:21,480 --> 00:24:26,831
the absolute value of a sum is
less than or equal to the sum
475
00:24:26,831 --> 00:24:29,010
of the absolute values.
476
00:24:29,010 --> 00:24:32,700
What this now tells me is I can
say that this, that I'm
477
00:24:32,700 --> 00:24:35,580
trying to get a hold on,
is less than this.
478
00:24:35,580 --> 00:24:37,670
But look at this expression.
479
00:24:37,670 --> 00:24:40,560
This expression is the
absolute value of
480
00:24:40,560 --> 00:24:42,240
''f of x' minus l1'.
481
00:24:42,240 --> 00:24:44,680
And this expression is
the absolute value of
482
00:24:44,680 --> 00:24:47,140
''g of x' minus l2'.
483
00:24:47,140 --> 00:24:50,360
In other words, then, since
I can make 'f of x' as
484
00:24:50,360 --> 00:24:54,300
arbitrarily nearly equal to 'l1'
as I want and 'g of x' as
485
00:24:54,300 --> 00:24:57,290
close to 'l2' as I want just by
choosing 'x' sufficiently
486
00:24:57,290 --> 00:25:02,150
close to 'a', why don't I choose
'x' close enough to 'a'
487
00:25:02,150 --> 00:25:05,140
so each of these will be less
than epsilon over 2?
488
00:25:05,140 --> 00:25:07,700
Now again, this calls
for a little aside.
489
00:25:07,700 --> 00:25:10,830
When one talks about epsilon,
that is 'a',
490
00:25:10,830 --> 00:25:12,040
what shall we say?
491
00:25:12,040 --> 00:25:19,325
A generic name for any number
which exceeds 0.
492
00:25:19,325 --> 00:25:22,590
Or I could have written that
less mystically by just saying
493
00:25:22,590 --> 00:25:24,530
any positive number.
494
00:25:24,530 --> 00:25:27,020
Well, if epsilon is positive,
what can you say
495
00:25:27,020 --> 00:25:28,860
about half of epsilon?
496
00:25:28,860 --> 00:25:30,620
It's also positive.
497
00:25:30,620 --> 00:25:32,700
In other words, if I had chosen
a different epsilon,
498
00:25:32,700 --> 00:25:37,300
say 'epsilon sub 1', equal to
the original epsilon divided
499
00:25:37,300 --> 00:25:41,310
by 2, then I'm guaranteed
what?
500
00:25:41,310 --> 00:25:45,170
That I can get 'f of x' with an
epsilon 2 of 'l1', 'g of x'
501
00:25:45,170 --> 00:25:46,860
with an epsilon 2 of 'l2'.
502
00:25:46,860 --> 00:25:49,560
And now adding these two
together, if this term is less
503
00:25:49,560 --> 00:25:52,420
than epsilon over 2 and this
term is less than epsilon over
504
00:25:52,420 --> 00:25:56,110
2, the whole sum is
less than epsilon.
505
00:25:56,110 --> 00:25:58,770
And it seems now
semi-intuitively--
506
00:25:58,770 --> 00:26:00,460
what do I mean by
semi-intuitively?
507
00:26:00,460 --> 00:26:03,100
Well, this is far from an
intuitive job over here.
508
00:26:03,100 --> 00:26:05,150
It's quite mathematical.
509
00:26:05,150 --> 00:26:06,460
It's rigorous in that sense.
510
00:26:06,460 --> 00:26:08,790
It's intuitive in the sense that
I'm not playing around
511
00:26:08,790 --> 00:26:09,620
with the deltas here.
512
00:26:09,620 --> 00:26:12,560
All I'm saying is look, I can
make this as small is I want,
513
00:26:12,560 --> 00:26:15,230
I could make this as small as
I want, therefore I can make
514
00:26:15,230 --> 00:26:16,810
the sum as small as I want.
515
00:26:16,810 --> 00:26:18,700
And the fancy way of saying
that is I can make it less
516
00:26:18,700 --> 00:26:21,730
than any given epsilon, and
therefore it appears that this
517
00:26:21,730 --> 00:26:24,390
will be the limit.
518
00:26:24,390 --> 00:26:27,960
Using our old adage again of
being able, knowing what we
519
00:26:27,960 --> 00:26:31,460
want, to be able to doctor
things up rigorously, once
520
00:26:31,460 --> 00:26:35,090
we've gone through this it's now
relatively easy to clean
521
00:26:35,090 --> 00:26:36,310
up the details.
522
00:26:36,310 --> 00:26:39,350
In other words, for those
of us who are
523
00:26:39,350 --> 00:26:43,130
mathematically-oriented enough
to say, the way you've proven
524
00:26:43,130 --> 00:26:46,220
this last result is the same
sloppiness that I was used to
525
00:26:46,220 --> 00:26:49,400
seeing in certain types of
engineering proofs where
526
00:26:49,400 --> 00:26:51,860
people were more interested in
the result than with the
527
00:26:51,860 --> 00:26:54,600
answer, I still don't see how
you used the epsilons and the
528
00:26:54,600 --> 00:26:55,430
deltas here.
529
00:26:55,430 --> 00:26:58,790
Let me show you what a simple
step it is to now go from the
530
00:26:58,790 --> 00:27:02,960
semi-rigorous approach to the
completely rigorous approach.
531
00:27:02,960 --> 00:27:05,940
All we do is reword what
we've done before.
532
00:27:05,940 --> 00:27:08,590
In fact, this is true
in most mathematics.
533
00:27:08,590 --> 00:27:10,970
You take a geometry book and
there's a theorem that says
534
00:27:10,970 --> 00:27:14,000
something like if 'a', 'b', 'c',
and 'd' are true, then
535
00:27:14,000 --> 00:27:14,680
'e' is true.
536
00:27:14,680 --> 00:27:16,490
And you learn this proof
quite mechanically.
537
00:27:16,490 --> 00:27:18,060
You sort of memorize it.
538
00:27:18,060 --> 00:27:20,820
Well you know, the man who
proved that theorem didn't, in
539
00:27:20,820 --> 00:27:24,020
general, start out by saying, I
wonder what happens if 'a',
540
00:27:24,020 --> 00:27:26,190
'b', 'c', and 'd' are true.
541
00:27:26,190 --> 00:27:28,890
In general, what he tries to
do is to prove that some
542
00:27:28,890 --> 00:27:30,740
result, like 'e', is true.
543
00:27:30,740 --> 00:27:33,360
And as he's proving it,
he hits pitfalls.
544
00:27:33,360 --> 00:27:35,650
And he says, you know, if I
could only be sure 'a' was
545
00:27:35,650 --> 00:27:37,670
true, I could get over
this pitfall.
546
00:27:37,670 --> 00:27:40,440
And if I could be sure 'b' was
true, I could get over the
547
00:27:40,440 --> 00:27:42,310
second pitfall, et cetera.
548
00:27:42,310 --> 00:27:44,710
And when he makes enough
assumptions to get over all
549
00:27:44,710 --> 00:27:48,940
the pitfalls and he has his
answer, he then we writes down
550
00:27:48,940 --> 00:27:51,310
the answer in the reverse
order from
551
00:27:51,310 --> 00:27:52,600
which he invented it.
552
00:27:52,600 --> 00:27:53,620
Namely, he says what?
553
00:27:53,620 --> 00:27:55,580
Suppose 'a', 'b', 'c',
and 'd' are true.
554
00:27:55,580 --> 00:27:57,210
Let's prove that 'e' is true.
555
00:27:57,210 --> 00:28:00,620
And the student is then robbed
of any attempt to see
556
00:28:00,620 --> 00:28:03,320
intuitively how this whole
thing came about.
557
00:28:03,320 --> 00:28:05,450
As a case in point, let
me show you what
558
00:28:05,450 --> 00:28:06,490
I'm driving at here.
559
00:28:06,490 --> 00:28:09,240
My first exposure to formal
limit proofs was
560
00:28:09,240 --> 00:28:10,240
something like this.
561
00:28:10,240 --> 00:28:13,700
When somebody said prove the
limit of a sum equals the sum
562
00:28:13,700 --> 00:28:16,590
of the limits, something
like this would happen.
563
00:28:16,590 --> 00:28:19,410
The first statement in the book
would say, given epsilon
564
00:28:19,410 --> 00:28:24,440
greater than 0, let epsilon
1 equal epsilon over 2.
565
00:28:24,440 --> 00:28:26,870
Now, I respected my teacher,
I respected the
566
00:28:26,870 --> 00:28:27,630
author of the book.
567
00:28:27,630 --> 00:28:30,000
If he says let epsilon
1 equal epsilon over
568
00:28:30,000 --> 00:28:31,150
2, all right, fine.
569
00:28:31,150 --> 00:28:32,250
We can do that.
570
00:28:32,250 --> 00:28:34,730
And more to the point, it turned
out that the problem
571
00:28:34,730 --> 00:28:37,220
worked if you did that.
572
00:28:37,220 --> 00:28:40,880
The part that bothered me is why
did he say epsilon over 2?
573
00:28:40,880 --> 00:28:43,230
Why not 2 epsilon over 3?
574
00:28:43,230 --> 00:28:44,820
Or epsilon over 5?
575
00:28:44,820 --> 00:28:48,090
Or epsilon over 6,872?
576
00:28:48,090 --> 00:28:49,920
Why epsilon over 2?
577
00:28:49,920 --> 00:28:52,220
And the point was that
he had cheated.
578
00:28:52,220 --> 00:28:55,720
He had already done the problem
that we had over here,
579
00:28:55,720 --> 00:28:59,290
and knowing what he needed, then
came back here and said
580
00:28:59,290 --> 00:29:02,120
let epsilon 1 equal
epsilon over 2.
581
00:29:02,120 --> 00:29:03,670
And notice how this
is going to mimic
582
00:29:03,670 --> 00:29:06,240
everything we said before.
583
00:29:06,240 --> 00:29:10,230
For this choice of epsilon 1, we
can find a delta 1 greater
584
00:29:10,230 --> 00:29:14,910
than 0 such that if the absolute
value of 'x minus a'
585
00:29:14,910 --> 00:29:18,200
is less than delta 1 but greater
than 0, then the
586
00:29:18,200 --> 00:29:20,910
absolute value of ''f of
x' minus l1' is less
587
00:29:20,910 --> 00:29:21,800
than epsilon 1.
588
00:29:21,800 --> 00:29:23,260
Why do we know that?
589
00:29:23,260 --> 00:29:28,720
That's the definition of what it
means to say that the limit
590
00:29:28,720 --> 00:29:32,530
of 'f of x' as 'x' approaches
'a' equals 'l1'.
591
00:29:32,530 --> 00:29:35,820
In a similar way, he says we can
find delta 2 greater than
592
00:29:35,820 --> 00:29:40,040
0, such that whenever the
absolute value of 'x minus a'
593
00:29:40,040 --> 00:29:43,610
is greater than 0 but less than
delta over 2, we can make
594
00:29:43,610 --> 00:29:49,020
the absolute value of ''g of
x' minus 'l sub 2'' and be
595
00:29:49,020 --> 00:29:51,100
less than epsilon 1.
596
00:29:51,100 --> 00:29:53,610
And now comes the
beautiful step.
597
00:29:53,610 --> 00:29:57,590
He says, now that these delta
1 and delta 2 exist
598
00:29:57,590 --> 00:30:01,210
separately, pick delta to be
the minimum of these two.
599
00:30:01,210 --> 00:30:04,020
In other words, if I let delta
equal the minimum of these
600
00:30:04,020 --> 00:30:06,230
two, what does that
guarantee me?
601
00:30:06,230 --> 00:30:09,500
If delta is the minimum of these
two, it guarantees me
602
00:30:09,500 --> 00:30:14,180
that both of these conditions
are met at the same time.
603
00:30:14,180 --> 00:30:15,120
What does that tell me?
604
00:30:15,120 --> 00:30:20,160
It tells me that as soon as the
absolute value of 'x minus
605
00:30:20,160 --> 00:30:23,430
a' is less than delta but
greater than 0, automatically
606
00:30:23,430 --> 00:30:24,970
these two conditions hold.
607
00:30:24,970 --> 00:30:26,850
And that, in turn,
tells me what?
608
00:30:26,850 --> 00:30:32,600
That the absolute value of ''f
of x' minus l1' is less than
609
00:30:32,600 --> 00:30:37,380
epsilon 1 and the absolute value
of ''g of x' minus l2'
610
00:30:37,380 --> 00:30:39,940
is less than epsilon 1.
611
00:30:39,940 --> 00:30:44,620
And now by adding unequals to
equals, that tells me what?
612
00:30:44,620 --> 00:30:48,910
That this plus this is less
than 2 epsilon 1.
613
00:30:48,910 --> 00:30:52,410
But see, using my hindsight, I
picked epsilon to be what?
614
00:30:52,410 --> 00:30:53,980
Epsilon over 2.
615
00:30:53,980 --> 00:30:59,000
So 2 epsilon 1 is just another
way of saying epsilon.
616
00:30:59,000 --> 00:31:02,100
In other words, what this now
implies is that the absolute
617
00:31:02,100 --> 00:31:07,860
value of ''f of x' minus l1'
plus the absolute value of ''g
618
00:31:07,860 --> 00:31:11,740
of x' minus l2' is less
than 2 epsilon 1,
619
00:31:11,740 --> 00:31:14,140
which equals epsilon.
620
00:31:14,140 --> 00:31:19,110
But you see, this
in turn is what?
621
00:31:19,110 --> 00:31:23,880
This is greater than the
absolute value of ''f of x'
622
00:31:23,880 --> 00:31:28,105
minus l1' plus ''g
of x' minus l2'.
623
00:31:28,105 --> 00:31:31,380
624
00:31:31,380 --> 00:31:33,350
And so this was the thing
we wanted to make
625
00:31:33,350 --> 00:31:35,010
smaller than epsilon.
626
00:31:35,010 --> 00:31:38,250
Since this is smaller than
this and this is already
627
00:31:38,250 --> 00:31:40,910
smaller than epsilon, then
this must be smaller than
628
00:31:40,910 --> 00:31:42,520
epsilon too.
629
00:31:42,520 --> 00:31:44,480
Again, notice, this
is rigorous.
630
00:31:44,480 --> 00:31:47,740
But if you understand what's
happening here piece by piece,
631
00:31:47,740 --> 00:31:51,010
you never really have
to memorize a thing.
632
00:31:51,010 --> 00:31:54,080
Let's just take a look back
here to reinforce what I'm
633
00:31:54,080 --> 00:31:55,050
saying here.
634
00:31:55,050 --> 00:31:55,860
Notice what we did.
635
00:31:55,860 --> 00:31:58,960
We started with the answer that
we wanted to show, worked
636
00:31:58,960 --> 00:32:02,270
around to get ahold of things
that we wanted.
637
00:32:02,270 --> 00:32:03,430
We were lucky enough--
638
00:32:03,430 --> 00:32:05,320
and this is true in any
game, for example.
639
00:32:05,320 --> 00:32:08,210
One can plot masterful strategy
and still lose,
640
00:32:08,210 --> 00:32:10,170
there's no guarantee we're
going to win with our
641
00:32:10,170 --> 00:32:11,730
masterful strategy.
642
00:32:11,730 --> 00:32:14,670
But we usually do in this
course, usually.
643
00:32:14,670 --> 00:32:18,510
What we do is we masterfully
come back to this, see what
644
00:32:18,510 --> 00:32:19,740
has to be done.
645
00:32:19,740 --> 00:32:22,770
Knowing what the right answer
has got to be, we come back
646
00:32:22,770 --> 00:32:25,590
here and then formalize it.
647
00:32:25,590 --> 00:32:28,350
Essentially, what we've done
is reverse the steps here.
648
00:32:28,350 --> 00:32:30,860
I guess there is one thing that
bothers many people that
649
00:32:30,860 --> 00:32:32,830
I should make an aside about.
650
00:32:32,830 --> 00:32:36,580
Why do you need a different
delta 1 and delta 2 for the
651
00:32:36,580 --> 00:32:38,020
same epsilon 1?
652
00:32:38,020 --> 00:32:40,980
See, if you're memorizing,
there's a danger that you
653
00:32:40,980 --> 00:32:43,390
won't realize this and
why this happens.
654
00:32:43,390 --> 00:32:48,670
Let me show you in terms of
a picture what this means.
655
00:32:48,670 --> 00:32:51,710
Let this, for the sake of
argument, be the curve 'y'
656
00:32:51,710 --> 00:32:53,480
equals 'f of x'.
657
00:32:53,480 --> 00:32:58,550
And let this be the curve
'y' equals 'g of x'.
658
00:32:58,550 --> 00:32:59,800
All we're saying is this.
659
00:32:59,800 --> 00:33:02,750
660
00:33:02,750 --> 00:33:10,220
That when you prescribe an
epsilon, the same epsilon that
661
00:33:10,220 --> 00:33:15,400
surrounds 'l2', if you have
that surround 'l1'.
662
00:33:15,400 --> 00:33:16,910
Because the curves may have
different slopes.
663
00:33:16,910 --> 00:33:18,360
Look what happens over
here, even as badly
664
00:33:18,360 --> 00:33:19,150
as I've drawn this.
665
00:33:19,150 --> 00:33:22,760
Notice that in epsilon,
neighborhood of 'l2', projects
666
00:33:22,760 --> 00:33:26,070
down into this size neighborhood
around 'a'.
667
00:33:26,070 --> 00:33:28,380
On the other hand, an epsilon
neighborhood of 'l1'.
668
00:33:28,380 --> 00:33:31,200
669
00:33:31,200 --> 00:33:33,500
You get what you pay
for, I guess.
670
00:33:33,500 --> 00:33:37,950
An epsilon, neighborhood of
'l1', projects into a much
671
00:33:37,950 --> 00:33:39,220
smaller region.
672
00:33:39,220 --> 00:33:40,810
And by the way, that's exactly
what we meant.
673
00:33:40,810 --> 00:33:43,030
This is a delta 1,
for example.
674
00:33:43,030 --> 00:33:46,470
This is delta 2.
675
00:33:46,470 --> 00:33:50,060
And when we said pick delta to
be the minimum of delta 1 and
676
00:33:50,060 --> 00:33:53,730
delta 2, all we were saying was
listen, if we guarantee
677
00:33:53,730 --> 00:33:57,770
that 'x' stays in here, then
certainly both of these two
678
00:33:57,770 --> 00:34:00,730
things will be true
at the same time.
679
00:34:00,730 --> 00:34:04,530
Well again, we have many
exercises and reading material
680
00:34:04,530 --> 00:34:05,840
to reinforce these points.
681
00:34:05,840 --> 00:34:08,940
All I want to do with the
lecture is to give you an
682
00:34:08,940 --> 00:34:12,030
overview as to what's happening
so that you see
683
00:34:12,030 --> 00:34:12,650
these things.
684
00:34:12,650 --> 00:34:15,690
And I'm afraid I might cure
you with details if I just
685
00:34:15,690 --> 00:34:18,670
keep hammering home these
rigorous little points.
686
00:34:18,670 --> 00:34:20,570
As I say, I hope you
get the main idea
687
00:34:20,570 --> 00:34:21,670
from what we're doing.
688
00:34:21,670 --> 00:34:25,360
Let me just, for the sake of
argument, try to work with
689
00:34:25,360 --> 00:34:31,370
just one more idea and we'll see
how this works out also.
690
00:34:31,370 --> 00:34:36,340
Let's, for example, try to play
around with the idea that
691
00:34:36,340 --> 00:34:39,400
a companion to the limit of a
sum equals the sum of the
692
00:34:39,400 --> 00:34:42,219
limits would be what?
693
00:34:42,219 --> 00:34:45,469
The limit of a product equals
the product of the limits.
694
00:34:45,469 --> 00:34:49,050
In other words, as before, if
the limit of 'f of x' as 'x'
695
00:34:49,050 --> 00:34:52,380
approaches 'a' is 'l1' and the
limit of 'g of x' as 'x'
696
00:34:52,380 --> 00:34:55,659
approaches 'a' is 'l2', let's
form a new function, which we
697
00:34:55,659 --> 00:34:58,170
could call 'k of x' which
is equal to the
698
00:34:58,170 --> 00:34:59,810
product of 'f' and 'g'.
699
00:34:59,810 --> 00:35:03,800
Again, noticing that the 'f' and
'g' have to have a common
700
00:35:03,800 --> 00:35:05,780
domain here.
701
00:35:05,780 --> 00:35:08,330
You want to show that the limit
of 'f of x' times 'g of
702
00:35:08,330 --> 00:35:12,940
x' is 'l1' times 'l2'.
703
00:35:12,940 --> 00:35:16,920
And this is the hard part of the
course, this is the part
704
00:35:16,920 --> 00:35:19,110
of the course that I don't think
anybody in the world can
705
00:35:19,110 --> 00:35:20,150
really teach.
706
00:35:20,150 --> 00:35:24,410
All one can do is try to expose
the student to ideas
707
00:35:24,410 --> 00:35:27,400
and hope that the student has
the knack of putting these
708
00:35:27,400 --> 00:35:30,550
things together to form
his own repertoire.
709
00:35:30,550 --> 00:35:32,030
The idea is something
like this.
710
00:35:32,030 --> 00:35:34,020
I'll show you alternative
methods and the like.
711
00:35:34,020 --> 00:35:36,480
One way is we want to
get ahold of 'f of
712
00:35:36,480 --> 00:35:38,710
x' times 'g of x'.
713
00:35:38,710 --> 00:35:40,380
Now what do we have a hold on?
714
00:35:40,380 --> 00:35:41,930
It's very clever what
we do here.
715
00:35:41,930 --> 00:35:45,360
We have a hold of ''f of x'
minus l1' and we have a hold
716
00:35:45,360 --> 00:35:48,540
of ''g of x' minus l2'.
717
00:35:48,540 --> 00:35:52,620
So as we so often do in
mathematics, we simply add and
718
00:35:52,620 --> 00:35:53,870
subtract the same thing.
719
00:35:53,870 --> 00:35:56,570
We frequently add on zeroes
in this cute way.
720
00:35:56,570 --> 00:35:58,360
We add and subtract
the same thing.
721
00:35:58,360 --> 00:35:59,640
Notice what we did over here.
722
00:35:59,640 --> 00:36:04,520
We wrote 'f of x' as 'l1' plus
''f of x' minus l1'.
723
00:36:04,520 --> 00:36:06,180
Again, why did we do that?
724
00:36:06,180 --> 00:36:09,260
Because from our definition of
the limit of 'f of x' as 'x'
725
00:36:09,260 --> 00:36:12,380
approaches 'a' equaling 'l1',
we know that we can
726
00:36:12,380 --> 00:36:14,250
control this side.
727
00:36:14,250 --> 00:36:15,710
And the same thing is
true over here.
728
00:36:15,710 --> 00:36:17,660
The fact that the limit of 'g
of x' as 'x' approaches 'a'
729
00:36:17,660 --> 00:36:20,710
equals 'l2' means we have
some control over this.
730
00:36:20,710 --> 00:36:23,710
Now let's just multiply
everything out.
731
00:36:23,710 --> 00:36:27,975
This is what? 'f of
x' times 'g of x'.
732
00:36:27,975 --> 00:36:30,650
I'm going to save myself some
space and keep the board
733
00:36:30,650 --> 00:36:31,750
somewhat symmetric.
734
00:36:31,750 --> 00:36:34,360
When I multiply these terms out,
I'm going to get what?
735
00:36:34,360 --> 00:36:37,770
An 'l1' times 'l2' term over
here as one of my four terms?
736
00:36:37,770 --> 00:36:40,790
Let me already transpose
that one so I kill two
737
00:36:40,790 --> 00:36:42,050
birds with one stone.
738
00:36:42,050 --> 00:36:44,440
One is I keep a little bit
of symmetry in what
739
00:36:44,440 --> 00:36:45,510
I'm going to write.
740
00:36:45,510 --> 00:36:50,110
And secondly, notice that later
on, this is what I want
741
00:36:50,110 --> 00:36:51,080
to get a hold on.
742
00:36:51,080 --> 00:36:53,350
In other words, notice that the
proof of the limit of 'f
743
00:36:53,350 --> 00:36:57,320
of x' times 'g of x' as 'x'
approaches 'a' is 'l1' times
744
00:36:57,320 --> 00:37:01,360
'l2', this is precisely the
expression I must make small.
745
00:37:01,360 --> 00:37:04,170
Our show can be made as small is
I wish just by picking 'x'
746
00:37:04,170 --> 00:37:05,550
sufficiently close to 'a'.
747
00:37:05,550 --> 00:37:08,050
And again, I will not go through
all the details here,
748
00:37:08,050 --> 00:37:11,810
I will simply outline
what we do here.
749
00:37:11,810 --> 00:37:14,430
Namely, let's multiply out
the rest of this thing.
750
00:37:14,430 --> 00:37:15,570
We have what here?
751
00:37:15,570 --> 00:37:18,390
This times this,
which is what?
752
00:37:18,390 --> 00:37:23,660
'l1' times ''g of
x' minus l2'.
753
00:37:23,660 --> 00:37:32,160
Then we have this times this,
which is 'l2' times
754
00:37:32,160 --> 00:37:35,880
''f of x' minus l1'.
755
00:37:35,880 --> 00:37:37,530
And now we have what?
756
00:37:37,530 --> 00:37:39,770
This times this.
757
00:37:39,770 --> 00:37:47,140
That's ''f of x' minus l1' times
''g of x' minus l2'.
758
00:37:47,140 --> 00:37:49,970
Now, the point is this is the
thing that we'd like to make
759
00:37:49,970 --> 00:37:52,140
very small in absolute value.
760
00:37:52,140 --> 00:37:54,900
Well again, the absolute value
of this is equal to the
761
00:37:54,900 --> 00:37:57,770
absolute value of this, which is
less than or equal to-- and
762
00:37:57,770 --> 00:38:00,560
I'm going to go through these
details rather rapidly, and
763
00:38:00,560 --> 00:38:02,750
allow you to fill these
in for yourself-- all
764
00:38:02,750 --> 00:38:03,930
I'm using is what?
765
00:38:03,930 --> 00:38:07,200
That the absolute value of a sum
is less than or equal to
766
00:38:07,200 --> 00:38:08,740
the sum of the absolute
values.
767
00:38:08,740 --> 00:38:11,010
The absolute value of a product
is equal to the
768
00:38:11,010 --> 00:38:13,650
product of the absolute values,
so this is going to be
769
00:38:13,650 --> 00:38:14,900
less than or equal to.
770
00:38:14,900 --> 00:38:18,680
771
00:38:18,680 --> 00:38:19,930
Let's break these things up.
772
00:38:19,930 --> 00:38:22,630
773
00:38:22,630 --> 00:38:25,050
And now, you see what
the key idea is?
774
00:38:25,050 --> 00:38:31,720
'l1' and 'l2' are certain fixed
numbers, fixed numbers.
775
00:38:31,720 --> 00:38:34,570
Notice that because 'g of x' can
be made as close to 'l2'
776
00:38:34,570 --> 00:38:37,800
as I want and 'f of x' can be
made as close to 'l1' as I
777
00:38:37,800 --> 00:38:41,130
want, notice that no matter what
epsilon I'm given, I can
778
00:38:41,130 --> 00:38:45,660
certainly make this as small as
I wish, just by picking 'x'
779
00:38:45,660 --> 00:38:47,150
close enough to 'a'.
780
00:38:47,150 --> 00:38:49,040
For example, for a given
epsilon, how many
781
00:38:49,040 --> 00:38:50,220
terms do I have here?
782
00:38:50,220 --> 00:38:55,670
One, two, three.
783
00:38:55,670 --> 00:38:59,280
To make this whole sum less than
epsilon, it's sufficient
784
00:38:59,280 --> 00:39:02,240
to make each of these three
factors less than
785
00:39:02,240 --> 00:39:03,490
epsilon over 3.
786
00:39:03,490 --> 00:39:08,310
787
00:39:08,310 --> 00:39:10,950
I can make these two less than
epsilon over 3 pretty easy.
788
00:39:10,950 --> 00:39:13,880
How do I make this less
than epsilon over 3?
789
00:39:13,880 --> 00:39:16,280
And the answer is, if you want
this times this to be less
790
00:39:16,280 --> 00:39:18,820
than epsilon over 3 where these
are positive numbers,
791
00:39:18,820 --> 00:39:22,070
make each of these less than
the square root of
792
00:39:22,070 --> 00:39:23,690
epsilon over 3.
793
00:39:23,690 --> 00:39:27,200
Then when you multiply these two
together, if this is less
794
00:39:27,200 --> 00:39:30,110
than this and this is less than
this, this times this
795
00:39:30,110 --> 00:39:32,610
will be less than
epsilon over 3.
796
00:39:32,610 --> 00:39:35,680
And now what you see what
we do is very simple.
797
00:39:35,680 --> 00:39:39,760
To finish this proof off, all we
do now is say, let epsilon
798
00:39:39,760 --> 00:39:41,430
greater than 0 be given.
799
00:39:41,430 --> 00:39:45,410
Choose epsilon 1 to equal
epsilon over 3.
800
00:39:45,410 --> 00:39:48,480
Choose epsilon 2 to equal
epsilon over 3.
801
00:39:48,480 --> 00:39:53,630
Choose epsilon sub 3 and epsilon
sub 4 to each be the
802
00:39:53,630 --> 00:39:55,680
square root of epsilon over 3.
803
00:39:55,680 --> 00:39:58,870
Then we can find delta
1, delta 2, delta
804
00:39:58,870 --> 00:40:01,660
3, delta 4, et cetera.
805
00:40:01,660 --> 00:40:06,130
Clean up all of these, you see,
and pick delta to be the
806
00:40:06,130 --> 00:40:08,470
minimum of the four
deltas involved.
807
00:40:08,470 --> 00:40:11,800
Again, this is done much more
explicitly both in the text
808
00:40:11,800 --> 00:40:13,090
and in the exercises.
809
00:40:13,090 --> 00:40:15,160
I just wanted you to get
an overview here.
810
00:40:15,160 --> 00:40:17,920
And by the way, while we're
speaking of this, there's
811
00:40:17,920 --> 00:40:20,390
always the danger that some
of you may respect the
812
00:40:20,390 --> 00:40:21,670
professor too much.
813
00:40:21,670 --> 00:40:25,320
So that danger is one I don't
mind living with.
814
00:40:25,320 --> 00:40:26,580
The problem is this.
815
00:40:26,580 --> 00:40:29,240
You may get the idea out of
respect for me that I have
816
00:40:29,240 --> 00:40:31,280
invented a unique proof here.
817
00:40:31,280 --> 00:40:32,250
Let me tell you this.
818
00:40:32,250 --> 00:40:34,460
One, I did not invent
this proof.
819
00:40:34,460 --> 00:40:37,070
Two, it is not unique.
820
00:40:37,070 --> 00:40:40,040
There are many different ways
of proving the same result.
821
00:40:40,040 --> 00:40:44,690
For example, a person being
told to work on this--
822
00:40:44,690 --> 00:40:46,880
and I'm not going to carry
the details out here--
823
00:40:46,880 --> 00:40:50,230
but a person being told to work
on something like this
824
00:40:50,230 --> 00:40:52,720
might have decided that instead
of doing the clever
825
00:40:52,720 --> 00:40:55,650
thing that we did before, he
was going to do the clever
826
00:40:55,650 --> 00:40:58,280
thing of adding and subtracting
827
00:40:58,280 --> 00:41:00,250
'l1' times 'g of x'.
828
00:41:00,250 --> 00:41:03,050
Because you see, if he did
that, what would happen?
829
00:41:03,050 --> 00:41:06,450
He could now factor out a 'g
of x' from here and rewrite
830
00:41:06,450 --> 00:41:10,450
this as 'g of x' times
''f of x' minus l1'.
831
00:41:10,450 --> 00:41:13,680
832
00:41:13,680 --> 00:41:16,970
And these two terms could be
combined together, we factor
833
00:41:16,970 --> 00:41:19,430
out an 'l1' times what?
834
00:41:19,430 --> 00:41:22,060
''g of x' minus 'l2'.
835
00:41:22,060 --> 00:41:24,150
And even though the details
would have been considerably
836
00:41:24,150 --> 00:41:26,420
different, the intuitive
approach would have been--
837
00:41:26,420 --> 00:41:28,060
well, look.
838
00:41:28,060 --> 00:41:33,730
This is pretty close to 'l2'
when 'x' is near 'a'.
839
00:41:33,730 --> 00:41:35,820
This I can make as
small as I want.
840
00:41:35,820 --> 00:41:38,580
Similarly, I can do the same
things over here, and pretty
841
00:41:38,580 --> 00:41:42,010
soon you've got the idea that
you can make this sum as small
842
00:41:42,010 --> 00:41:46,800
as you want just by choosing
these sufficiently small.
843
00:41:46,800 --> 00:41:48,840
And there's no unique
way of doing this.
844
00:41:48,840 --> 00:41:52,700
Now, here's a main point.
845
00:41:52,700 --> 00:41:56,900
Once all this work has been
done, for a wide variety of
846
00:41:56,900 --> 00:42:01,730
problems we never again have to
use an epsilon or a delta.
847
00:42:01,730 --> 00:42:04,010
Let me illustrate with
one problem.
848
00:42:04,010 --> 00:42:06,760
Let's suppose we were given
the problem limit of 'x
849
00:42:06,760 --> 00:42:10,010
squared plus 7x' as 'x
approaches 3', and we wanted
850
00:42:10,010 --> 00:42:11,450
to find that limit.
851
00:42:11,450 --> 00:42:14,300
Our intuitive thing would
be to do what?
852
00:42:14,300 --> 00:42:19,880
Let 'x' equal 3, in which case
we get 9 plus 21 is 30.
853
00:42:19,880 --> 00:42:23,190
But we know by now that this
instruction says that 'x'
854
00:42:23,190 --> 00:42:25,850
can't equal 3.
855
00:42:25,850 --> 00:42:26,620
This is the problem.
856
00:42:26,620 --> 00:42:28,960
We get what appears to be a nice
answer that we believe
857
00:42:28,960 --> 00:42:31,090
in, but by an illegal method.
858
00:42:31,090 --> 00:42:34,410
Can we use our legal methods
to gain the same result?
859
00:42:34,410 --> 00:42:36,000
The answer is yes.
860
00:42:36,000 --> 00:42:39,440
Because, you see, what is
'x squared plus 7x'?
861
00:42:39,440 --> 00:42:43,260
It's the sum of two functions,
and we've just proven that the
862
00:42:43,260 --> 00:42:46,140
limit of the sum is the
sum of the limits.
863
00:42:46,140 --> 00:42:48,170
For example, what I
can say is this.
864
00:42:48,170 --> 00:42:49,340
I don't know if this is true.
865
00:42:49,340 --> 00:42:56,880
What I do know is true is that
the limit of 'x squared plus
866
00:42:56,880 --> 00:42:59,930
7x' as 'x' approaches 3 is a
limit of 'x squared' as 'x'
867
00:42:59,930 --> 00:43:03,960
approaches 3 plus the limit of
'7x' as 'x' approaches 3.
868
00:43:03,960 --> 00:43:05,000
How do I know that?
869
00:43:05,000 --> 00:43:06,700
I've proven that the
limit of a sum is
870
00:43:06,700 --> 00:43:07,900
the sum of the limits.
871
00:43:07,900 --> 00:43:09,100
Now what is this?
872
00:43:09,100 --> 00:43:11,330
This is really a product.
873
00:43:11,330 --> 00:43:14,210
This is really the limit
of 'x' times 'x'.
874
00:43:14,210 --> 00:43:17,490
But we already know that the
limit of a product is the
875
00:43:17,490 --> 00:43:18,580
product of the limits.
876
00:43:18,580 --> 00:43:20,130
See, we proved that theorem.
877
00:43:20,130 --> 00:43:23,240
Well, we almost proved it,
certainly close enough so I
878
00:43:23,240 --> 00:43:27,000
think that we can
say that we did.
879
00:43:27,000 --> 00:43:28,250
And this is a product also.
880
00:43:28,250 --> 00:43:33,930
881
00:43:33,930 --> 00:43:37,765
So you see, we can get from here
to here to here just by
882
00:43:37,765 --> 00:43:39,100
our theorem.
883
00:43:39,100 --> 00:43:42,420
Now didn't we also prove as one
of our theorems earlier
884
00:43:42,420 --> 00:43:45,910
that the limit of 'x' as 'x'
approaches 'a' is 'a' itself?
885
00:43:45,910 --> 00:43:46,870
Sure, we did.
886
00:43:46,870 --> 00:43:50,580
In particular then, the limit of
'x' as 'x' approaches 3 has
887
00:43:50,580 --> 00:43:54,390
already been proven to be 3.
888
00:43:54,390 --> 00:43:56,800
So this is 3, this is 3.
889
00:43:56,800 --> 00:43:59,670
What is the limit of 7
as 'x' approaches 3?
890
00:43:59,670 --> 00:44:03,810
Well, 7 is a constant, and we
already proved that the limit
891
00:44:03,810 --> 00:44:05,690
of a constant as 'x'
approaches 'a'
892
00:44:05,690 --> 00:44:07,080
is that same constant.
893
00:44:07,080 --> 00:44:08,740
So what is the constant here?
894
00:44:08,740 --> 00:44:09,830
It's 7.
895
00:44:09,830 --> 00:44:12,340
And what's the limit of 'x'
as 'x' approaches 3?
896
00:44:12,340 --> 00:44:13,600
That's 3.
897
00:44:13,600 --> 00:44:15,810
So by using our theorems,
we get from what?
898
00:44:15,810 --> 00:44:20,050
From here to here
to here to here.
899
00:44:20,050 --> 00:44:22,645
And now hopefully from a theorem
that comes from some
900
00:44:22,645 --> 00:44:26,480
place around the third grade,
3 times 3 is 9, 7 times 3 is
901
00:44:26,480 --> 00:44:29,220
21, the sum is 30.
902
00:44:29,220 --> 00:44:32,280
And now you see this is no
longer a conjecture.
903
00:44:32,280 --> 00:44:35,660
This follows, inescapably, from
the rules of our game,
904
00:44:35,660 --> 00:44:38,140
from the rules and our
basic definition.
905
00:44:38,140 --> 00:44:43,470
By the way, you may have a
tendency to feel when you see
906
00:44:43,470 --> 00:44:47,220
something like this that, why
did we need the epsilons and
907
00:44:47,220 --> 00:44:48,210
deltas in the first place?
908
00:44:48,210 --> 00:44:49,450
Wasn't it a terrible waste?
909
00:44:49,450 --> 00:44:50,670
Well, two things.
910
00:44:50,670 --> 00:44:54,160
First of all, we couldn't prove
our theorems without the
911
00:44:54,160 --> 00:44:55,430
epsilons and deltas.
912
00:44:55,430 --> 00:44:58,860
And secondly, and don't lose
sight of this, in many real
913
00:44:58,860 --> 00:45:02,930
life situations you may very
well be faced with the type of
914
00:45:02,930 --> 00:45:06,420
a problem that doesn't ask you
to prove that the limit of 'x
915
00:45:06,420 --> 00:45:10,050
squared plus 7x' is 30 as 'x'
approaches 3, but rather might
916
00:45:10,050 --> 00:45:15,710
say how close must 'x' be chosen
to 3 if we want 'x
917
00:45:15,710 --> 00:45:21,100
squared plus 7x' to be
less than 30.023.
918
00:45:21,100 --> 00:45:23,430
And then, you see, if that's
the kind of a problem you
919
00:45:23,430 --> 00:45:26,030
have, these new theorems
will not solve
920
00:45:26,030 --> 00:45:27,740
that problem for you.
921
00:45:27,740 --> 00:45:29,030
So we're not making
a choice here.
922
00:45:29,030 --> 00:45:32,130
All we're saying is that the
epsilons and deltas are the
923
00:45:32,130 --> 00:45:36,210
backbone of limits, but that
fortunately through
924
00:45:36,210 --> 00:45:40,920
mathematical theorems, we can
get simpler ways of getting
925
00:45:40,920 --> 00:45:41,980
important results.
926
00:45:41,980 --> 00:45:43,800
And that was our main
purpose of today's
927
00:45:43,800 --> 00:45:44,970
lecture, these two things.
928
00:45:44,970 --> 00:45:47,870
That completes our lecture for
today, and so on until next
929
00:45:47,870 --> 00:45:49,120
time, good bye.
930
00:45:49,120 --> 00:45:51,440
931
00:45:51,440 --> 00:45:54,450
NARRATOR: Funding for the
publication of this video was
932
00:45:54,450 --> 00:45:59,160
provided by the Gabriella and
Paul Rosenbaum foundation.
933
00:45:59,160 --> 00:46:03,340
Help OCW continue to provide
free and open access to MIT
934
00:46:03,340 --> 00:46:07,540
courses by making a donation
at ocw.mit.edu/donate.
935
00:46:07,540 --> 00:46:12,287