[SOUND] So here's a table of values of the function F of X equals sin 1 / X.
F of one is really sin of one, it's like 8..
F of 1. which is really sin of one over 1. sin of 10 -.5.
F of 01.. This will be sine of 100, it's also about
-0.5. F of 0.001, which is like sine of 1000,
well that's 0.8 and some more, right? So the question is these numbers aren't
really getting close to anything in particular.
Can you really say that if you evaluate f at values which are close to but not
equal to zero, that the outputs are actually getting close to anything in
particular. I mean this is positive, negative,
negative, positive, negative, positive, negative, it's looking pretty bad.
Instead of a table, let's look at a graph.
Here, I've got a graph of the funtcion f (x) = sin 1 / x.
And you see the middle of this graph is just that horrible green blob.
Right? It's really hard to make out any detail.
You might think that's just a consequence of the fact that I'm drawing this graph
with such thick lines. You know, and if I used thinner lines to
draw my graph, maybe I could, you know get rid of this green blob and really see
some detail. Even if I dial down the size of the lines
that I'm using to draw this graph, the blob thing is still there, you know.
And it's really there in the graph of the function.
Even if these lines were true lines, zero thickness, it wouldn't be possible to fit
even a single atom next to the Y axis without touching the graph of this
function. The graph is oscillating wildly near
zero. Even if your input is very close to zero
your output could be anything between -1 and 1.
So in light of this evidence, the limit of sine 1 / x as x approaches 0 does not.
[SOUND] exist. Which sometimes I'll abbreviate DNE, for
does not exist. what does it even mean to say it doesn't
exist? What do we mean by the definition of
limit? To say the limit equals something means
that I can make the output as close as I want to l by making x close to a.
So when I say this limit doesn't exist, I mean it's not the case that this limit is
equal to anything, okay? If you tell me this limit is some
positive number, well look. When I evaluate the function at a number
very close to 0, the output is negative. So the limit is probably not some
positive number but there's also inputs very close to zero that give positive
outputs so that the limit is pulling out a negative number either.
Limit is pulling out zero either cause none of these numbers are getting close
to zero. So in this sense.
This limit just doesn't exist because it's not the case that this limit is
equal to anything in particular. If you tell me this limit is equal to l,
I'm going to show you numbers close to zero which aren't close to l.
Let's see another example along the same lines.
This is a particularly confusing example because in the function f(x)x) = sine pi
/ x. The function evaluated at 1 is 0.
That's pretty clear because that is sine of pi and sine of pi is zero.
About the function at 0.1, I'm counting that's also equal to zero.
The function at 0.01, that is also zero. This can be kind of confusing.
When you take a look here, I typed in sin pi divided by 0.01 on to my calculator.
This is calculating the function's value at 01..
If I ask my calculator to do this, it is not telling me the answer zero, right?
The calculator's giving me this, admittedly, a very small number, right, E
-11 here. But it's still not actually zero.
So, can I convince you that this is even true?
That the functions value at 01. actually is equal to zero.
What is f (0.01)? Well it's the same as sine, of pi / 0.01.
Now here I'm taking pi and I'm dividing it by 0.01.
That's the same thing as what? That's the same thing as multiplying a
100. I'm dividing by a hundredth, that's the
same as multiplying by a 100. So this function at.01 is sine of 100 pi.
What's sine of 100 pi? Well, think back to what the graph of
sine looks like. Here's a graph of sine, zero to two pi.
If I do it again. Here it is at four pi, and I drew it
again. Here it is at six pi, and I'm going to
keep on going. And eventually, I'm going to get to 100
pi. And at that point, sign really is going
to be equal to zero. Calculators are great, they're also
terrible. This calculator can't really calculate
with pi, all it can do is calculate with some approximation to pi.
We can use our human mind to evaluate this function exactly.
In light of this evidence, you might be tricked into believing that the limit of
f(x) as x approaches 0 is equal to 0. After all these points are approaching
zero, and function evaluating each of these points is zero.
So maybe that means that this is true. So it looks like the limit is equal to
zero. But, what happens if I look at some other
points? We'll take a look at this example.
Here's the same function, f of x equals sign of pi over x.
This function, if I evaluate it at 75. is this, maybe a little bit mysterious
number, negative 866. and so forth. If I evaluate this function at 075. you
get the same thing. If I evaluate the function at 0075,. I
get the same thing. At 00075. I get the same thing, .000075,
I get the same thing. So, what's going on here.
Well what is this number? I mean 0.8666.
This isn't just some sort of random number.
Right? This is in fact negative the square root
of three over two. And it looks like this function at all of
these points has the same value, negative the square root of three over two.
So does that mean that the limit as X approaches zero of F of X, [SOUND] is
equal to negative the square root of three over two?
[SOUND] I mean again all of these values, .75.075.0075, these input values are
approaching zero, and the functions value at all of those inputs is the same.
So, what gives, is the limit zero, is a negative point A, which is it?.
Okay, okay, I've been little bit too tricky in picking my input points.
Since the same function, F of X equals Sin pie over X, and here, I am picking a
collection of points, again approaching zero, .7, .07, .007, .0007.
Its getting closer and closer to zero. But now, my output values are looking
pretty random. I mean, they are not over the same, for
instance. So this is, maybe some evidence, that,
the limit. Of sine pi over x as x approaches zero.
Doesn't exist. Here I've got a bunch of input points
that are getting closer and closer to zero but my output values at least don't
appear to be getting close to. We're not just learning.
We're exploring. [SOUND] I encourage you to cook up you
own examples. We've seen a couple examples now of where
limits don't exist but can you come up with more?
[SOUND]