 Before we start looking at limits, and I teach you how to work with a very exciting world of limits, I think it's always good to start with a tangible example. Can I use limits in real life? Now look at this very busy slide we've got here. We've got seem to have two equations, a P of T and a C of P. So that P would refer to the first P and the P of T. So these equations are somehow locked in and we see one is marked red. So we presume that's the red graph at the bottom and the blue one seems to just go up and up and up. And what's the story behind this? Well, I said we should make it real life. So let's come up with a good story. Aliens are about to invade the Earth. Oh, and everyone is concerned and the president asks what can be done? And the scientist says, well, actually aliens have landed before. We did some experiments on them and we came up with these two equations. What's these two equations? The president says one is we had a population model there and we called a P of T and as time increases that is in this area where we had the aliens and we experimented on them, that's the population of the people there. And this C of P that seems to be the rate at which the alien bugs can multiply and infect people and people turn into aliens. The president asks now what kind of losses are we looking at? The scientist says, not a problem. Let's see what happens as time reaches infinity. Let's see how many people we're going to lose. And so if we take the limit as time goes to infinity of P of T, there's the sum. You don't need to know how to do it. It ends up at 20. The answer is going to be 20 and if we plug that P value of 20 into the C of P, we see all the answers 4.2. So out of a maximum population of 20, we're going to lose 4.2 because the humans are going to fight back. So very useful example, real life example there for you. So let's look at these equations again. We have P of T and there's the red graph. You can see it sort of starts to climb but it reaches this maximum and it seems not to go up any further. It's hit some form of a limit there. It doesn't go up and if we let these bugs just go and we gave them ample humans to feed on these alien bugs, you'll see the infection rate just goes up and up and up and up. The blue line is going to go up to infinity. It's just not going to stop. As time reaches infinity, it seems as if in the y-axis the number of infections we're talking about is going to go up to infinity as well. But if we combine these two, it says to you that while these bugs can't just really infect humans and turn them into aliens ad infinitum, there's going to be some rate limitation here. So the population is only going to get to 20. That means these bugs don't have a resource of food that's limitless and the humans fight back. And so if we combine those two, then we get this little green line here that is the c of the p of t of drawn in green and you see that also reaches some limit. So with this very useful in real life example, I want to show you that we can model things. We can model what might happen in the future and we can see we can end up with these graphs that seem to increase. We might as well decrease, but they reach some sort of limit as opposed to this blue line. This is going to go up and up and up and never stop. But you can get these other equations that seem to go up or down as I said, but they reach some form of limit. And then we know we usually model time to go to infinity and realistically that might mean months or the next couple of years or the next couple of decades, whatever the situation might be. We can now estimate what the result might be under a set of circumstances and you have constrained the circumstances by allowing the population only to reach 20 over time. You can put some constraints on my equations there and there seems to be this natural limit things can get to and that is how things work in medicine in the disease, in the spread of disease. Some limit is reached. So very exciting stuff limits and it's well worth the effort to learn how to do the mathematics of limits. Let's get started.