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Limits.

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[MUSIC]

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Taylor series can provide some useful
intuition for thinking about limits.

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For example you might remember thinking
about this limit.

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The limit as x approaches zero of sine x
over x.

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And one way to think about this is via
Taylor series.

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And what do Taylor series tell us?

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Well Taylor series tells us that sine of
x, its equal to x minus x cubed over

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6 plus x to 5th over 5 factorial minus,
and the series

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would keep on going.
But at least I know that

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sine of x is approximately x plus,

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I'll just write higher order terms.
So

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if you believe this, you might try to
evaluate this limit by making use

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of this, admittedly at this point, very
vague fact.

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So what's the limit then as x approaches 0

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of x plus some higher order terms divided
by x?

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Well that's the sort of limit that I
could, you know, really approach.

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Right?

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I mean it's not a transitional function
anymore, it's just it

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looks like a polynomial as far as I'm
concerned or imagining here.

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So I could think about how do I calculate

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the limit as x approaches 0 of x plus
higher order terms over x.

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Well what I would probably want to do is
multiply

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the numerator and the denominator by 1
over x.

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And I've got these higher order terms and
the denominator's got x.

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So this looks like the limit, as x
approaches 0

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of the numerator now is 1 plus these
higher order terms.

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And the denominator is just 1.

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And if

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I'm taking the limit then, as x approaches
0, be

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very tempted to say that the limit is just
1.

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And that is not even close to a proof.

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Well, what's wrong with this argument?

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Well, basically, this is a circular
argument.

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I mean, how so?

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So I'm trying to, trying to calculate the
limit of sine

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x over x by thinking about a Taylor series
for sine.

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And to find a Taylor series for sine,
right, to find

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this Taylor series, I need to be able to
differentiate sine.

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So what do you have to do to be able to
differentiate sine?

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Well, if you're trying to differentiate
sine, you in

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particular need to be able to
differentiate sine at 0.

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So let me just write down the limit of the

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difference quotient that calculates the
derivative of sine at 0.

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That's the limit as h approaches 0 of sine
of 0 plus h minus sine of 0 divided by h.

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Right,

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this limit is calculating the derivative
of sine at 0, cause its the ratio

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of how the output changes when you go from
0 to 0 plus h.

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That's how the input changes when you go
from 0 to h.

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Okay.
But now, how do I calculate this limit?

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Well, what is this limit?

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This is this limit, as h goes to 0, of
what sine of h minus

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sine of 0 oh, sine of 0's 0, so the whole
numerator's just sine of h, divided by h.

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[LAUGH]

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So that's exactly the limit that we've got
here, just with x replaced by h.

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So what happened here, right?

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I'm imagining that I'm trying to calculate

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this limit by thinking about Taylor
series.

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But to calculate the Taylor series for
sine,

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you have to be able to differentiate sine.

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But to be able to differentiate sine, I

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really need to know, or calculate, this
limit.

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So, you know, if you think you're really

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proving anything by this method, you're
really not.

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I mean, this is just a circular argument.

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Admittedly, it's a circular argument, in a
really big circle!

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So maybe it looks like a straight line, or
a reasonable

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argument, but there really is something
essentially circular going on here.

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But the fact that the argument's circular,
shouldn't stop us

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from making use of that kind of thinking
where it's appropriate.

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Let's find the limit as x approaches 0 of
cosine of x minus 1 over the numerator,

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divided by sine of x times log of 1 minus
x.

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That's a very complicated looking limit
question.

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Let me try to approach that more

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complicated limit question by using Taylor
series.

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So let's think about cosine.

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So the Taylor series for cosine looks like
this.

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Cosine of x is 1 minus x squared over 2
plus, and those higher order terms.

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So I'm going to write big O, x to the 4th.
And to be a little

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more pedantic, I'll say as x approaches 0.
And what does this mean?

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[LAUGH]

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Well, I don't want to talk about this too
precisely yet, but

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morally, or intuitively at least, you
should think about it like this.

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Cosine of x is 1 minus x squared over 2
plus, you can just think of

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these as being the higher order terms in

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the Taylor series expansion of cosine
around 0.

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Now the next term is an x to the 4th term

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but there's also an x to the 6th term and
so on.

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So what about cosine x minus 1?
So that means that cosine

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of x minus 1 is negative x squared over 2
plus

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more terms of degree at least 4.
We can write sine of x in the same way.

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Sine of x, now sine also has a Taylor
series expansion, it's x.

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And then there's higher order terms,
right?

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So I'm just going to write plus higher
order terms starting with an x

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cubed term.
Let's do the log term.

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Well, log of 1 minus x, as a Taylor

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series expansion that starts like this,
minus x minus x

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squared over 2 minus x cubed over 3 minus
x

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to the 4th over 4 and that keeps on going.

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And I could write that as just minus x
plus terms of degree at least 2.

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Let's put the sine and

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log term together.

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So let's see, I want to multiply together
things that involve these big O's.

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So how does that work?

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Well I'm trying to calculate sine of x
times log of 1 minus x.

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I can write down a Taylor series for that,
but

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I don't want to actually go through the
bother of calculating

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out what the Taylor series is going to be,
so I'm

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just going try to multiply together these
these, these Taylor series.

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So sine of x is x plus terms of degree at
least

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3, and log of 1 minus x, I mean not
everywhere, but at least around 0, is

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given by negative x plus terms of degree
at least 2.

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So now what happens when I multiply
together these two Taylor series?

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Well I've got an x times a negative x, so
that gives me negative x squared.

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But what

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other terms do I get?

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Well, here I've got terms of degree at
least 3 times x.

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These'll give me terms of degree at least
4.

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Things of degree of at least 3 times
things of degree

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at least 2 will give me thing of degree at
least 5.

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And I've got x times things of degree of
at least 2, so this x

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times possibly some x squared terms,
that'll give

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me somethings of degree of at least 3.

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So all told what I know, is that this

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Taylor Series, when I multiply them
together,

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is going to start out minus x squared.

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And there's going to be some more terms of
degree at least 3.

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I'm not bothering to calculate those,
right?

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They're all sort of hidden in this big O
notation.

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Now we're in a position to consider the
original limit.

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So thinking about Taylor series for the
numerator and denominator, I might

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be tempted or I, this is the limit as x
approaches 0.

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I've got a Taylor series for cosine of x

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minus 1.

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It's negative x squared over 2 plus terms,
degree at least 4.

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And I just found a Taylor series for the
denominator by

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multiplying together the Taylor series for
sine and the Taylor series

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for log of 1 minus x, and that ended up
being

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minus x squared plus some terms of degree
at least 3.

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Now I could multiply this by just the
disguised version

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of 1, so 1 over x squared divided by 1
over x squared, and this'll give me what?

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The limit as x approaches 0 of this
numerator is now minus one

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half plus terms of degree at least 2.
And the denominator

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is now just minus 1 plus terms of degree
at least 1.

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But what is this limit, right?
I'm imagining

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that x is approaching 0, so these terms
and these terms are both approaching 0,

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so the numerator has limit minus one half
and the denominator has limit minus 1.

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And that means that this limit, the limit
of the ratio is just one half.

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So maybe, at this point, you're
unimpressed.

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You're thinking, I could have done that
with L'Hopital's rule.

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And you'd be right.

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I mean, you could have used L'Hopital's
rule.

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So why are we thinking about Taylor
series?

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Well, the point here isn't

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just that Taylor series are useful
computationally.

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My claim is that Taylor series is really

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providing some deeper insight into what's
going on.

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So we're presented with this fact, that
this limit is equal to one

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half and you're probably thinking, who
cares, and you'd be right to think that.

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Who cares if this limit is equal to one
half?

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If a machine told me this is equal to one
half, I wouldn't care either.

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What matters isn't that the machines are
telling us that this

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is equal to one half or that we've got
some horribly

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long algebraic calculation that proves to
us that this limit is equal to one half.

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What really matters here is that this
limit is equal

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to one half for a reason that human beings
can understand.

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We can really comprehend why this limit
should be equal to one half.

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We can think about Taylor series.

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And Taylor series are telling us that this
numerator looks

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like negative X squared over 2 plus higher
order terms.

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We think about Taylor series for sine, and
sine looks

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like x plus x cubed over 6 and so on.
X plus higher order terms.

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The Taylor series for log of 1 minus x,

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starts of negative x minus, and then
there's more terms.

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And when I can multiply together Taylor
series, right?

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And I multiply together these two series,
I'm

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getting a Taylor Series for the whole
denominator.

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And the denominator then looks like

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negative x squared plus higher order
terms.

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And now, I can bring to bear all of the
intuition

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that I have about limits of rational
functions,

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limits of polynomials over polynomials,
and as x approaches

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0, it makes perfect sense then, that if

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this is a polynomial, it's really a power
series.

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But if I'm thinking that this is a
polynomial that starts

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off negative x squared over 2 plus higher
order terms, and

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the denominator looks like a polynomial,
really a power series, but

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it's like a polynomial, negative X squared
plus higher order terms.

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Well then what happens, right?

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The limit is exactly the ratio of these
leading terms.

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It's negative one half over negative 1.

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So the fact that this limit equals one
half, you

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can understand a reason for it by thinking
about Taylor series.

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Yeah, wow and that's always the point,
right?

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The point isn't getting answers, it's
getting insight.

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The point of mathematics, you know, it
isn't truth, it's proof.

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Right?

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It's not so much whether something's true,
but why

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is it true.

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[SOUND]

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[NOISE]