Extreme auditioning?

I can't take any more!

People were suffering. People were fighting.

Down to the point where they were undetectable.

So as x approaches one from the positive direction it looks like the limit of f... ...it looks like the limit of f(x) as x approaches one from the positive direction isn't zero It looks like it is one

So if x with a infinity would go to zero because it is a constant

So, it's every thing we have been doing leading uptill now brings us to this point.

Same thing, if instead of having a function here we had a constant

So you could have a situation like this, where you have a gap right over there and looks something

So if we have the limit as x approaches c of f(x) divided by g(x), this is the exact same thing as the limit of f(x) as x approaches c divided by the limit of g(x) as x approaches c which is going to be equal to

Because I could actually use some of that real estate.

Actually, let's deal with that right now.

A approaches infinity.

So plus 1 s.

In another presentation, I'll give you the more formal mathematical, you know, the delta epsilon definition of a limit.

So over here it equals zero

Welcome back.

I wrote a little bit bigger than I wanted to, but that's OK.

And so this is equal to 1 a minus s times now we essentially have to evaluate t at infinity.

It's a fairly well established number.

(Music) You can see how the instrument can be pushed to extremes of character.

Now we have to evaluate it at A minus the anti derivative evaluate it at 0, and then take the limit of whatever that ends up being as

but most of these should be fairly intuitive and they're very helpful for simplifying limit problems in the future

Sure enough. OK. I used to work in this field of elementary particles.

So if we just had the limit (I'll do that in the same color)

we can do the same thing with the differences

So if the limit from the right hand side is a different value than the limit from the left hand side then the limit does not exist

This right here is the point (1.5, 1) and if we approach it from the left hand side, from values less than it it's one, the limit seems to be one and if we approach from the right hand side the limit seems to be one

Now, normally, this would be on this line.

I'm just really giving you the properties here

It's just the limit of f(x) as x approaches c minus the limit of g(x) as x approaches c

limit as x approaches zero from the right of f of x.

The question is suppose I run infinitely many motion steps.

Well we've already thought about that

This is just asking about the left side of the diagram.

So what happens if I take the limit as h approaches zero of this?

Now, sand dunes cover only about one fifth of our deserts.

So let's say I had let me make sure I have the right color and my pen works.

BJ

We went in underneath and tapped it, and we found that it had the pH of Drano.

So what is the limit as tau approaches zero of my little d sub tau function?

Taken to its ultrasonic extreme, can be fatal,

To see if you understood this, let me look at the rain sun Markov chain again, and let me ask you for the stationary distribution or the limit distribution for rain to be the case after infinitely many steps.

Let's find a limit as our delta x approaches 0.

Now what happens if you take the product of the functions?