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Epsilon-delta limit definition 1 | Limits | Differential Calculus | Khan Academy - YouTube
Channel: Khan Academy
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Let me draw a function
that would be interesting
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to take a limit of.
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And I'll just draw it visually
for now, and we'll do some
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specific examples
a little later.
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So that's my y-axis,
and that's my x-axis.
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And let;s say the function
looks something like--
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I'll make it a fairly
straightforward function
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--let's say it's a line,
for the most part.
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Let's say it looks just
like, accept it has a
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hole at some point.
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x is equal to a, so
it's undefined there.
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Let me black that point
out so you can see that
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it's not defined there.
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And that point there
is x is equal to a.
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This is the x-axis, this is
the y is equal f of x-axis.
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Let's just say
that's the y-axis.
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And let's say that this
is f of x, or this is
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y is equal to f of x.
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Now we've done a bunch
of videos on limits.
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I think you have an
intuition on this.
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If I were to say what is the
limit as x approaches a,
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and let's say that this
point right here is l.
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We know from our previous
videos that-- well first of all
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I could write it down --the
limit as x approaches
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a of f of x.
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What this means intuitively is
as we approach a from either
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side, as we approach it from
that side, what does
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f of x approach?
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So when x is here,
f of x is here.
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When x is here, f
of x is there.
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And we see that it's
approaching this l right there.
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And when we approach a from
that side-- and we've done
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limits where you approach from
only the left or right side,
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but to actually have a limit it
has to approach the same thing
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from the positive direction and
the negative direction --but as
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you go from there, if you pick
this x, then this is f of x.
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f of x is right there.
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If x gets here then it goes
here, and as we get closer and
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closer to a, f of x approaches
this point l, or this value l.
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So we say that the limit
of f of x ax x approaches
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a is equal to l.
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I think we have that intuition.
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But this was not very, it's
actually not rigorous at all
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in terms of being specific
in terms of what we
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mean is a limit.
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All I said so far is as
we get closer, what does
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f of x get closer to?
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So in this video I'll attempt
to explain to you a definition
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of a limit that has a little
bit more, or actually a lot
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more, mathematical rigor than
just saying you know, as x gets
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closer to this value, what
does f of x get closer to?
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And the way I think about it's:
kind of like a little game.
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The definition is, this
statement right here means that
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I can always give you a range
about this point-- and when I
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talk about range I'm not
talking about it in the whole
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domain range aspect, I'm just
talking about a range like you
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know, I can give you a distance
from a as long as I'm no
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further than that, I can
guarantee you that f of x is go
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it not going to be any further
than a given distance from l
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--and the way I think about it
is, it could be viewed
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as a little game.
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Let's say you say, OK Sal,
I don't believe you.
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I want to see you know, whether
f of x can get within 0.5 of l.
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So let's say you give me 0.5
and you say Sal, by this
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definition you should always
be able to give me a range
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around a that will get f of
x within 0.5 of l, right?
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So the values of f of x are
always going to be right in
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this range, right there.
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And as long as I'm in that
range around a, as long as I'm
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the range around you give me, f
of x will always be at least
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that close to our limit point.
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Let me draw it a little bit
bigger, just because I think
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I'm just overriding the same
diagram over and over again.
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So let's say that this is f of
x, this is the hole point.
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There doesn't have to be a hole
there; the limit could equal
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actually a value of the
function, but the limit is more
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interesting when the function
isn't defined there
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but the limit is.
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So this point right here-- that
is, let me draw the axes again.
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So that's x-axis, y-axis x,
y, this is the limit point
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l, this is the point a.
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So the definition of the limit,
and I'll go back to this in
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second because now that it's
bigger I want explain it again.
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It says this means-- and this
is the epsilon delta definition
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of limits, and we'll touch on
epsilon and delta in a second,
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is I can guarantee you that
f of x, you give me any
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distance from l you want.
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And actually let's
call that epsilon.
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And let's just hit on
the definition right
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from the get go.
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So you say I want to be no more
than epsilon away from l.
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And epsilon can just be any
number greater, any real
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number, greater than 0.
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So that would be, this distance
right here is epsilon.
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This distance there is epsilon.
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And for any epsilon you give
me, any real number-- so this
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is, this would be l plus
epsilon right here, this would
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be l minus epsilon right here
--the epsilon delta definition
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of this says that no matter
what epsilon one you give me, I
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can always specify a
distance around a.
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And I'll call that delta.
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I can always specify
a distance around a.
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So let's say this is delta
less than a, and this
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is delta more than a.
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This is the letter delta.
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Where as long as you pick an x
that's within a plus delta and
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a minus delta, as long as the x
is within here, I can guarantee
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you that the f of x, the
corresponding f of x is going
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to be within your range.
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And if you think about it
this makes sense right?
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It's essentially saying, I can
get you as close as you want to
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this limit point just by-- and
when I say as close as you
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want, you define what you want
by giving me an epsilon; on
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it's a little bit of a game
--and I can get you as close as
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you want to that limit point by
giving you a range around the
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point that x is approaching.
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And as long as you pick an x
value that's within this range
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around a, long as you pick an x
value around there, I can
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guarantee you that f of x will
be within the range
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you specify.
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Just make this a little bit
more concrete, let's say you
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say, I want f of x to be within
0.5-- let's just you know, make
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everything concrete numbers.
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Let's say this is the number 2
and let's say this is number 1.
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So we're saying that the limit
as x approaches 1 of f of x-- I
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haven't defined f of x, but it
looks like a line with the hole
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right there, is equal to 2.
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This means that you can
give me any number.
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Let's say you want to try it
out for a couple of examples.
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Let's say you say I want f of x
to be within point-- let me do
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a different color --I want f
of x to be within 0.5 of 2.
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I want f of x to be
between 2.5 and 1.5.
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Then I could say, OK, as long
as you pick an x within-- I
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don't know, it could be
arbitrarily close but as long
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as you pick an x that's --let's
say it works for this function
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that's between, I don't
know, 0.9 and 1.1.
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So in this case the delta from
our limit point is only 0.1.
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As long as you pick an x that's
within 0.1 of this point, or 1,
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I can guarantee you that your
f of x is going to
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lie in that range.
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So hopefully you get a little
bit of a sense of that.
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Let me define that with the
actual epsilon delta, and this
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is what you'll actually see in
your mat textbook, and then
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we'll do a couple of examples.
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And just to be clear, that
was just a specific example.
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You gave me one epsilon and I
gave you a delta that worked.
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But by definition if this is
true, or if someone writes
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this, they're saying it doesn't
just work for one specific
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instance, it works for
any number you give me.
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You can say I want to be within
one millionth of, you know, or
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ten to the negative hundredth
power of 2, you know, super
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close to 2, and I can always
give you a range around this
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point where as long as you pick
an x in that range, f of x will
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always be within this range
that you specify, within that
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were you know, one trillionth
of a unit away from
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the limit point.
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And of course, the one thing
I can't guarantee is what
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happens when x is equal to a.
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I'm just saying as long as you
pick an x that's within my
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range but not on a, it'll work.
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Your f of x will show up to be
within the range you specify.
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And just to make the math
clear-- because I've been
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speaking only in words so far
--and this is what we see the
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textbook: it says look, you
give me any epsilon
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greater than 0.
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Anyway, this is a
definition, right?
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If someone writes this they
mean that you can give them any
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epsilon greater than 0, and
then they'll give you a delta--
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remember your epsilon is how
close you want f of x to be
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to your limit point, right?
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It's a range around f of x
--they'll give you a delta
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which is a range
around a, right?
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Let me write this.
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So limit as approaches a
of f of x is equal to l.
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So they'll give you a delta
where as long as x is no more
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than delta-- So the distance
between x and a, so if we pick
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an x here-- let me do another
color --if we pick an x here,
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the distance between that value
and a, as long as one, that's
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greater than 0 so that x
doesn't show up on top of a,
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because its function might be
undefined at that point.
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But as long as the distance
between x and a is greater
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than 0 and less than this x
range that they gave you,
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it's less than delta.
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So as long as you take an x,
you know if I were to zoom the
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x-axis right here-- this is a
and so this distance right here
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would be delta, and this
distance right here would be
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delta --as long as you pick an
x value that falls here-- so as
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long as you pick that x value
or this x value or this x value
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--as long as you pick one of
those x values, I can guarantee
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you that the distance between
your function and the limit
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point, so the distance between
you know, when you take one of
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these x values and you evaluate
f of x at that point, that the
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distance between that f of x
and the limit point is
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going to be less than the
number you gave them.
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And if you think of, it seems
very complicated, and I have
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mixed feelings about where
this is included in most
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calculus curriculums.
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It's included in like the, you
know, the third week before you
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even learn derivatives, and
it's kind of this very mathy
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and rigorous thing to think
about, and you know, it tends
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to derail a lot of students and
a lot of people I don't think
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get a lot of the intuition
behind it, but it is
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mathematically rigorous.
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And I think it is very valuable
once you study you know, more
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advanced calculus or
become a math major.
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But with that said, this
does make a lot of sense
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intuitively, right?
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Because before we were talking
about, look you know, I can get
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you as close as x approaches
this value f of x is going
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to approach this value.
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And the way we mathematically
define it is, you say Sal,
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I want to be super close.
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I want the distance to be
f of x [UNINTELLIGIBLE].
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And I want it to be
0.000000001, then I can always
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give you a distance around x
where this will be true.
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And I'm all out of
time in this video.
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In the next video I'll do some
examples where I prove the
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limits, where I prove some
limit statements using
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this definition.
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And hopefully you know, when we
use some tangible numbers, this
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definition will make a
little bit more sense.
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See you in the next video.
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