But I think you get it at this point.

So a hyperbola, if that's the x, that's the y axis, it has two asymptotes.

That's not the exact word, but it would look something like that where it's below the asymptote here.

And you notice that the sigmoid function, well, it asymptotes at one, and asymptotes at zero as

And here, it's going to keep getting closer and closer to the asymptote on that side and then on that side.

Asymptotes of a Hyperbola

So x is equal to 1 y is equal to minus 1.

100, you end up with about five thousand features.

So that's one asymptote.

So we're always going to be a little bit above the asymptote.

It's going to come down and then go off, and never quite touch the asymptote, but approach it.

When you're taking the positive square root, we're always going to be slightly below the asymptote.

It just gets closer and closer and closer, arbitrarily close to the asymptote.

The two asymptotes of a hyperbola.

Oh woops, not using my line tool.

Let's draw our asymptotes.

Go out there.

I know you can't read that.

Which essentially b over a x, plus or minus b over a x.

Because it will never, a hyperbola will never cross the asymptotes.

Hyperbola by Asymptotes Point

So that line would look like that.

Good enough.

Got to have a steady hand.

And then go like that.

So it'll go 3.

Let's do that.

So if these are the axes, then if I were to draw let me draw some asymptotes.

Construct a conic with this asymptote

Construct the asymptotes of this conic

Square root of 4 over square root of 9, times x.

So that tells us that were always going to be slightly below the asymptote on the positive square root, and we're always going to be slightly above the asymptote on the negative square root.

And this would've been the two lines of the asymptotes if we were centered at 0.

The positive square root is the top line.

It's going to look something like that.

Let me see if I can draw it.

Our equation from before, 2n 2 n just becomes Θ(n).

This is equal to plus or minus b over a x.

So the asymptotes are going to intersect at the center of our hyperbola, so to speak.

The asymptote is this thing, but we're always going to be slightly below it.

So it's going to get really close to the asymptote, and then go off, and go off in that direction.

Select the conic of which you want to construct the asymptotes...

Now, the square root of just this term is 5 2 x.

So it's approximately linear around F of X equals zero, but it levels off to zero and one as we go to the extremes.

If f(n) is in little o(g(n)), that's kind of like saying f(n) is strictly less than g(n).