 So in this first lecture, let's have a proper look at what a limit is. We're going to start with a polynomial, a nice normal polynomial that you must be so familiar with. And we're going to see what happens to the y value as the x value reaches or tends to reach a certain spot. Now, I warn you, most of the time I'm going to make use of the t variable and not x. I'm just replacing x with t because in the life sciences, most of the time we're interested in what happens over time, not what happens over space. So I'm not interested in an xy, but in a ty. So as time goes on, what happens to some y value? So just watch out for that. So we're going to start with a normal, normally easy polynomial and then I'm going to change things ever so slightly. Just change that polynomial and then you'll really see what a limit is all about. And so here we have a graph that I'm going to use to explain to you what a limit is. It seems as if this graph starts at 0.0 and then goes all the way up. It's actually one that you know pretty well. Let's write it down. We're going to say it's the f of t equals t squared. Now you might be used to x's there, y equals x squared. But in the life sciences, we are very interested in what happens over time. So instead of the x-axis here, we're really just going to have in this direction time increasing. This might be day one, day two, minute one, minute two, minute three. We're usually interested in time. So let's have a look. Y equals t squared and we've got this very special point here. We're letting t obviously equal 2 here at the bottom as if x was 2 but let's write it there. So we have that the f of 2, well that just means we're just going to replace 2 in there. So that'll be 2 squared and 2 squared is 4. And lo and behold the value there is 4. What we're interested in when we talk about a limit though is something quite different. Now you might have seen this already. This is how we write it. We want to know the limit as t approaches 2. What does that mean? Well it means look down here on the, let's call it the old x-axis, the t-axis down here, the time axis. We can come from this side and we can get closer and closer and closer to 2 but we can also come from this side and get closer and closer and closer to 2. In other words we can have 1.9, 1.99, 1.999 and on this side we can have 2.0001, 2.0001, etc. Closer and closer and closer until we get, try at least and get to 2. But we don't and let's just finish this. The limit is t approaches 2 of the f of t. So we just want to know what happens to the f of t as t approaches 2. Very importantly you've got to get this concept down. t never reaches 2 when we talk about limits. When we talk about limits, t never reaches 2. So whatever this value is, whether it's x or t, we say the limit as t approaches 2 from the left-hand side. Let's do that right now. The left-hand side and the right-hand side as it approaches 2 is a certain value reached. As a certain solution to f of t reached, in other words as I get closer and closer to 2 and lo and behold we see as we get closer and closer to 2 we can get arbitrarily closer and closer here to 4. We're going to get to 4. So as I go up here, as I move up here, up here, up here I'm going to get to 4. As I move down, down, down here I'm going to get to 4. So I have that the limit as t approaches 2 is an actual value. It is equal to 4. As t approaches 2 it is 4. The answer is going to be 4. So there is a definite value that is reached as we get arbitrarily closer and closer to 2. So this is what we would call a limit. So here we have a tabular view of what we just tried to attempt. So we have our t values and you see we started 1.9, then 1.99, 1.999, 1.9999 and we get closer and closer to 2. We never get to 2. That's the limit. The limit as t approaches 2. We get closer and closer to it but we never actually get there and then from the other side you see we are very close to 2 and we move further away and on this side we have the f of t, remember that was just t squared. So you can see what's happening here. From this side we are approaching 2 and from this side we are approaching 2 and look at this very clearly 3.6, 3.99, 9.99, 9.99, 9.99, 9.99 and then 4.00. Very clearly here we are trying to reach the value 4. So there is this definite solution to this problem. The limit as t approaches 2 equals an exact value. As we get closer and closer to 2 we get closer and closer to 4 and we can actually get to 4. So there is a solution to this limit. So this was one way of doing it. You can always just plug in values closer and closer and closer and see, well am I getting towards a certain value? Clearly we are getting closer and closer to 4 on this side. So an absolute value is reached. So this is the first way that we can solve a limit as we can start plugging in values that get closer and closer and closer. Now here we have a new function f of t. So let's just do this. We're still increasing time in this direction. So as t carries on we've got this new function. t squared minus 4 over t minus 2. Now I want to draw your attention to the fact that there is some problem here. What happens to the limit? Let's write that down as t approaches 2 of the f of t. Well in the denominator, if I were to plug in 2, I have 2 minus 2 and that gives me 0 and I cannot have 0 in a denominator. It is not a defined function if we divide by 0. So what happens now on this axis? Remember we approaching 2 from this side. We approaching 2 from this side. What is happening right here? What is happening at that point? Well it's still a straight line. I've drawn it for you here and you can do some algebra on this and you'll see why that is so. And perhaps we'll do that later. But there's definitely a hole here. The function is not defined at t equals 2. But very clearly, and this is how limits work, it's one of the beauties of limits, as we get closer and closer to this side, from that side and as we get closer and closer, clearly we are reaching the value 4. So even though 2 is not defined, and here it's not a problem for us, remember we said in a limit t approaches 2, we never actually have to get to 2 and in fact we don't. But as we go from the left hand side, as we go from the right hand side, in these values, so we get closer and closer to this undefined point, we do get a value of 4. So even though 2 is not defined in this function, we are still going to say that the limit as t approaches 2 of the f of t in this instance is still 4, even though the function is not defined at t equals 2. But you can clearly see this graph is trying to reach the value of 4. And really it's trying to get there. So even though there's a gaping hole, there definitely is not a 2,4 here. The function is not defined there. The graph is still absolutely trying to get to 4 and therefore this limit, this function, does have a limit as t approaches 2. And indeed, if you did what we did before by the table that we had, as you plug in values closer and closer and closer to 2, closer and closer and closer to 2 from both sides, you'll see that indeed the solution is trying to get to 4. It does arbitrarily get closer and closer to 4. So the limit here definitely still exists. Now it stood to bring the point home. Let's have a look at what's happening here. We have a piecewise defined function. You'll be familiar with this. And the f of t can take one of two forms. It's still t squared minus 4 over t minus 2 if t does not equal 2, but 3 if t equals 2. Now the function is defined at t equals 2. And here we have it. It takes on the value of 3. Now I ask you, what is the limit as t approaches 2? As t approaches 2, remember it never ever is 2 of the f of t. Now what is it going to be? Is it going to be 4? Because clearly it's still trying to reach this value of 4 as we get closer and closer to 2 here on the t-axis. But at 2 it really is 3. There's my value there. If I go along here, it's 3 on the y-axis. So what is the answer here? As I said with the lecture before, just think about it. If I were to plug in a little table, values closer and closer and closer to 2, the answer is going to get closer and closer to 4. It is of no consequence that the function is defined in equal to 3 or whatever else that constant might be. Even if it is defined there and there clearly is a solution there, the solution at t equals 2 is not the limit. It is what the graph is trying to reach, what value it is trying to reach as we get closer and closer to this limit. And clearly here the answer is still 4. Plug it again. Plug it into a table and get values closer and closer to 2. The answer is going to get closer and closer to 4. So clearly this limit, even though the solution to f of t is 3 at t equals 2, as t approaches to the solution to the limit, is still going to be 4.