One, you say, well this is an approximation.

So what's the limit of this as A approaches infinity?

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

I'm just really giving you the properties here

And you can't really simplify this anymore.

Because you have this extra plus 4 there.

So we get our Laplace transform of t to the n is equal to this evaluated at infinity and evaluated at 0.

And, once again, I always forget the formulas.

And then you have to evaluate that from 0 to infinity.

Well we've already thought about that

Well, lucky for us this is going to be equal to the limit of f(x) as x approaches c times the limit of g(x) as x approaches c

we can do the same thing with the differences

And the limit is just what happens as you approach something.

So when you've evaluate what's the limit of this as t approaches infinity?

And to figure out the asymptotes you just have to think about well what happens as x approaches positive or negative infinity.

And then on the top side it's going to do the same thing, it's going to get really close, and as you approach infinity it's never going to touch it.

Well x is, you know, infinity minus any constant is still going to be infinity, or if the limit is t approaches infinity, x is still going to be infinity here.

You take the square root of that and so this is going to dominate.

this is going to be the limit of f(x) as x approaches c plus the limit of g(x) as x approaches c which is equal to well, this right over here is (we'll do that in that same color) this right here is just equal to L it's going to be equal to L plus M this right over here is equal to M

People, this is the biggest investment of their life.

The approach employed in this analysis was as follows.

And now what's the limit... ...the limit of f(x) as x approaches one from the negative direction?

But we can think about, what does this approach as x approaches positive or negative infinity.

Look at Penn State.

Because these are what this graph or this equation or this function approaches as x approaches positive or negative infinity.

Let me do that right now, so we could say that this is the same thing.

You can kind of view it as a substitution, so this is equal to well, let me write it this way.

And like always, the asymptotes, the hyperbola will never equal the asymptotes or intersect the asymptotes, but it's what the graph approaches as x approaches positive and negative infinity.

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

But this is this is a sign of progress, I mean we are halfway there at least for some of these categories.

Well, it definitely exists we've already established that it's equal to one so that's true

Hopefully you see that this is the same thing as this.

The approaches used in those two designs are exactly alike.

And you hear the words scarce resources a lot when people talk about economics.

And there are really two games in town, according to most people, with regard to postponing aging.

A third thing we are looking at and very interested in is simulation.

So this is not true

Because this thing right here, this evaluates to 1.

Oh, for Al Gore to come and be in the real world as president.

And this is the 16 centimeter area.

So that's all of the values that this function can take on, given that we've restricted it to only these inputs.

That's a good approach.

Thenso,by thepowerinvestedinme by the state of Louisiana

Viper 791 Galactica, you are cleared for approach.

And this should also be intuitive, because if we saw here when we did ths approximation, as x approaches positive or negative infinity, we saw that we always did have this plus 4 sitting here.