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with the idea of a limit, which is a super important idea.

It's really the idea that all of calculus is based upon.

it's actually a really, really, really, really, really, really

So let me draw a function here, actually,

let me define a function here, a kind of a simple function.

So let's define f of x, let's say that f of x

I have the same thing in the numerator and denominator.

Can't I just simplify this to f of x equals 1?

And I would say, well, you're almost true,

and this thing right over here, is that this thing can never

equal-- this thing is undefined when x is equal to 1.

Let me write it over here, if you have f of,

sorry not f of 0, if you have f of 1, what happens.

which is, let me just write it down, in the numerator,

And in the denominator, you get 1 minus 1, which is also 0.

And so anything divided by 0, including 0 divided by 0,

You can say that this is you the same thing as f of x is equal

to 1, but you would have to add the constraint that x cannot be

for all other X's other than one, but at x equals 1,

This is undefined and this one's undefined.

So that, is my y is equal to f of x axis,

y is equal to f of x axis, and then this over here

And then let's say this is the point x is equal to 1.

This over here would be x is equal to negative 1.

This is y is equal to 1, right up there I could do negative 1.

but that matter much relative to this function right over here.

So it's essentially for any x other than 1 f

It's going to look like this, except at 1.

So I'm going to put a little bit of a gap right

over here, the circle to signify that this function is not

We don't know what this function equals at 1.

This definition of the function doesn't tell us

It's literally undefined, literally undefined

And so once again, if someone were to ask you what is f of 1,

you go, and let's say that even though this was a function

definition, you'd go, OK x is equal to 1,

oh wait there's a gap in my function over here.

It's kind of redundant, but I'll rewrite it f of 1 is undefined.

is the function approaching as x equals 1.

And now this is starting to touch on the idea of a limit.

On the left hand side, no matter how close

you get to 1, as long as you're not at 1,

Over here from the right hand side, you get the same thing.

So you could say, and we'll get more and more

familiar with this idea as we do more examples,

that the limit as x and L-I-M, short for limit,

as x approaches 1 of f of x is equal to, as we get closer,

we can get unbelievably, we can get infinitely close to 1,

And our function is going to be equal to 1,

it's getting closer and closer and closer to 1.

So once again, it has very fancy notation, but it's just saying,

Let me do another example where we're dealing with a curve,

f of x, let me just for the sake of variety,

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