 Nu jy weet jy nog alles about limits, het was virally nie zo'n moeilijk. Nu gaan we het probleem doen dat we in die lege siense in die lege siense doen. Gaan we praatig doen en nie die zombees probleem. So, let's have a look. We gaan use some mathematical models. They're not going to be real life realistic mathematical models. So some algebraic expressions that I want to use just to cement some of the techniques that we've learned up till now. But it will put things into perspective to show you that we can use mathematical models in the life sciences to model some something realistic or meaningful. Here we have an equation. The p of t equals 35t over t square plus 9 minus 45 over t plus 1 plus 60. And p stands for a number of people in thousands and t is in years. The first question we can ask, what is the population size at the start? At the start for us means usually in the life sciences t equals zero. So if I were to simply plug in zero where every where I see a t, I very quickly get to about 15, well exactly 15. So we're going to have 15,000 people when we start. We can ask what is or is this population size going to expand and increase for ever and ever or some limit going to reach. The natural thing in the life sciences is then for us to model this as a limit as t approaches infinity. So if we were to look at that, if we were just to plug in infinity, of course we're going to get infinity over infinity. The one technique that we know now how to get rid of that, we're going to divide both enumerator and denominator by the largest powered variable in the denominator. So for 35t over t square plus 9 that's easy, that's t squared. And if I divide both enumerator and denominator by the same thing, I'm not changing anything algebraically there. And that means I end up with 35 over t and 1 plus 9 over t squared. I'm going to do the same with 45 over t plus 1. Yeah, I'm just going to divide both enumerator and denominator by t, because t to the power 1 would be the largest powered variable in the denominator. Then 60 is a constant, doesn't matter. Now I can simplify things, I'm going to end up with 35 over t and 1 over 9 plus t squared. I'm going to end up with 45 over t and 1 plus 1 over t and my 60. Now I'm going to make use of a couple of limit laws. I can take individually the limit of each of the parts of the expression. And if the denominator is defined at that limit, I can take the limit of the enumerator and denominator separately. And lastly I can bring a constant outside of the limit. So I've done all of that in a step. Now I end up with 35 times the limit as t approaches infinity of 1 over t. Now we know that 1 over a constant as t approaches over a variable, as that variable approaches infinity. We know that's zero. The same for the 9 times the limit as t approaches infinity of 1 over t squared. We know that's zero. So very quickly we can just make use of what we know. Plug in the zeros now. We know those limits to be zero. And we end up with a easy answer of 60. So this population will stabilize at about 60,000 people. We can ask, what is the absolute population size during the third year? What is that size increase or change in the third year? We might like to plan something. We might need to plan sanitation or something for the population. We need to know what is this size increase going to be between year two and three? Of during the third year? Very easy. We're going to do this as t approaches three and as t approaches two. Which in this simple example, this means we need to even take the limit. We just plug in three and two. But in essence we're taking the limit as t approaches three minus taking the limit as t approaches two. And we see we're going to get an increase of just over 4000 people during that year and we can now plan for that increase. Let's look at this, the number of months patients spend on a waiting list. We've been booked for theatre. We want to know as the booking list is going to get, the waiting list is going to get longer and longer and longer. And we were given this model. Now this is not a model that we'll actually use, but I wanted to just use the arc tangent just so as to remind us of something we know before. So I have this variable over a constant and I have the four on the other side. My limit law says I can take the limit of both separately. And we know for the arc tangent as t approaches infinity. We've seen this before. That's always going to be pi over two. So I can just replace that with pi over two. The limit as t approaches infinity four is just four. And I can see that waiting list is going to stable at around 5.6 months. We can now put in place structures for these patients so that everyone knows. It's going to be five, about five and a half months. If that's not acceptable over the long run now, we can say look, this is not going to be acceptable for patients in our area. And for the disease that we're treating surgically here, we better do something 5.6 months is too long. So fantastic use here of mathematical models and limits to help us to plan things.