 Now I want to move over to some more advanced topics. We are really going to look at what happens as time goes to infinity. Now that's a very important concept in the life sciences. Depending on what we are dealing with, say for instance we are dealing with some disease and epidemic, that might last for months or years. And in that setting, infinity for us might mean what happens over one or two years. Not nearly infinity mean an infinite number of time. We really can't fathom that. But many times we just want to muddle things at infinity. So that's a very important topic. We'll also look at continuity. When is a function continuous? And that has a lot to do with limits. Have a look at some of these advanced topics. We'll make them quite easy. We know so much about limits now. And we know that we can write an equation. It can help us to see what will happen in the future as the limit as t approaches infinity for instance. But it has proper mathematical uses as well. Imagine we model something and we want to see what the rate of change is. And we all know the rate of change is the first derivative for instance. The first derivative with respect to time in our instance. But you can only take the derivative on a certain domain if a function is continuous in that domain. If it's not continuous in that domain, you can have problems relying on taking a derivative in a certain section. Now that is one of the other beauties of a limit. It really helps us to tell if a function or a graph is continuous at a certain point. So when can we have discontinuities? Well, there's a few examples there. What if I have an f and a that is not defined? So I have this function f. It's modeling something for me, but at a certain point t, or a, I should say, where t equals a, it is not defined. What about the natural log of t? Well, that's an extreme example here. But the natural log is not defined when t is less than or equal to zero. So there's definitely a discontinuity at time equals zero for instance. It's never going to be defined there. So what about when a fiber function in t and t equals a is defined, but that limit does not exist? For instance, if my model requires a piecewise defined function and the limit as t approaches the f of t simply doesn't exist. Or for instance, the left and right end limits are not equal. That would mean that there's a discontinuity in my graph there. Other problems I can think of as vertical asymptotes. For instance, when the f of t equals the tangent of t, there are no various vertical asymptotes. And certainly there is discontinuity of the function and you cannot take the derivative in areas where there is a discontinuity. So certainly limits help us as far as that is concerned. Now certain discontinuities are removable. At the left top here, I have the f of t equals t squared plus 2t minus 15 over t equals 3. What happens? Imagine that models some growth of some bacteria for me over time. Just imagine that. What happens at time equals 3? Well, if I plug 3t equals 3 into my equation, then I'm going to get zero over zero. That doesn't work. There's certainly a hole in my graph at that spot. It turns out to be at 3.8. We'll see now why that is. But there's certainly a discontinuity there. There's ways to get rid of it. I can do the limit as t approaches 3 from the left side and the right side. And if I plug in values closer and closer and closer to 3 from both the left and the right side, I'm going to get to 8. There's certainly a limit exists there. And if I wanted to do this limit properly, I can make use of something that we'll definitely look at a bit later. At the indeterminate forms. Forget that for now. Let's just see one way we can remove discontinuities mathematically. I can certainly factorize the numerator into t minus 3 and t plus 5. The t minus 3's cancel and I'm going to have to t plus 5. So the f of t of t squared plus 2t minus 15 over t minus 3 is just the same as writing the f of t as t plus 5. And if I plug in 3 now, I get 8. I don't get 0 over 0. Be very careful now. Even though I can simplify t squared plus 2t minus 15 divided by t minus 3, even though I can simplify that as t plus 5, those two are not the same thing. Clearly, the t squared plus 2t minus 15 over t minus 3 has a whole, a big hole at t equals 3. f of t equals t plus 5 has no such discontinuity. I've removed that, but you can clearly see although they are algebraically equal, they are not the same function. So if I were to model something, I would have to do this piecewise to find function to remove that discontinuity properly. Not by simplification, but by rewriting my original f of t as such, when t is not equal to 3, I can have the t squared plus 2t minus 15 over t minus 3. But when t equals 3, I have the fact that the f of t equals 8 or t plus 5. That has removed the discontinuity for me. What about non-demovable discontinuities? If you have the f of t equals 1 over t squared, there's no way to remove that. There's nothing I can do to remove that. You can clearly see from the left and the right-hand side, it goes up to infinity. In other words, the limit does not exist at t equals 0. If that limit does not exist, it means there's a discontinuity in my graph. It doesn't just hop over from left side of 0 to the right side of there. That limit does not exist, therefore there is a definite discontinuity in my function f of t equals 1 over t squared. I can generalize that 1 over t squared is the following. f of t equals some constant a. In my example at the top, that was 1, and it didn't even be t squared. It just can be t minus b to the power n. A and b are just members of the set of real functions. As long as n is more than 0, we're going to get the same kind of vertical asymptote at b. At the top it was t minus 0, squared b was 0, hence my vertical asymptote being at b. Now let's look at another non-removable discontinuity, and that's infinite oscillation. You can zoom in on the cosine of 1 over t, closer and closer to 0. It just starts moving up and down even more. It goes absolutely crazy. It never settles on a solution, a certain y value. That is certainly a non-removable discontinuity. It's one that you'll come across often. Put it in your memory banks. The cosine of 1 over t as t approaches 0 does not exist. You can clearly see that that's a beautiful graph actually, and you're never going to get a solution there. There's a non-removable discontinuity, and certainly in the domain on either side of 0, you really can't take a derivative there. You have a discontinuity. Now one that we're going to come across very often. I'm going to include it in this lecture because you're going to use it so many times. Not of 1 over t, but 1 over t squared, but 1 over t. What happens to the limit as t approaches infinity of 1 over t? You can clearly see the graph there. You can clearly see we're going to approach 0. Plugging in values, large and larger values for t there, you see we are just going to get closer and closer to 0. Again, I can generalize this to say that the limit as t approaches plus or minus infinity of some constant a, which is a member of the element of the set of real values, divided by t to the power k. For k larger than 0, you can't have t to the power negative 2. It's got to be larger than 0. The power there and t of k must be defined. t of k can't be 0, or the natural log of something that's less than 0. It's got to be defined, obviously. All of those will equal 0. Anyway, I can write the limit as t approaches positive infinity of 3 million, divided by t to the power 6. It's going to be 0. Are we going to make use of that? Yeah. Again, you see there on the left-hand side, if I just plugged in infinity there, it's going to be infinity over infinity. Again, it's indeterminate form. We're going to deal with indeterminate form. Before we get there, there's another way to look at this. I've got t squared plus 2 over t to the power 3 plus 2t to the power 2 plus t minus 2. The limit as t approaches infinity of that. Now look at what I can do. Remember I said that the limit of t as t approaches plus or minus infinity of some constant over t to the power k, as long as k is equal 0 and t of k is t to the power k is defined equal 0. I can change that numerator and denominator, so I have quite a few examples of that. If I were to divide the numerator by the largest powered variable in the denominator, now look at the denominator, it's t cubed is the largest power. If I divide the numerator and the denominator by t cubed, I'm going to end up with all of those. Look at it. Plus I have to skip the step by making use of the limit laws. Remember if it's something like this, I can just take the limit individually in there. But look what has happened. If I were to divide t squared plus 1 by t to the power 3, I'm left with 1 over t, and I know the limit of that as t approaches infinity is just 0. I'm also going to be left in the numerator with 1 over t cubed, that still fits the bill of a over t to the power k. And now in the numerator, in the denominator, I should say I have the limit of 1 because t cubed divided by t cubed is 1, and the limit of a constant is that constant. So suddenly I have a constant in the denominator. All the others are just examples of a over t to the power k in some form, and I know those limits will always be 0. In other words, I have 0 over 1, which is the 0. So again, that limit is going to approach 0 as t approaches infinity. It's going to be 0 as t approaches infinity. So there's some important limits to remember. The one we've just seen is t approaches positive and negative infinity of some constant over t to the power k. With the constraints that I said t to the power k must be defined, and k must be larger than 0, it's always going to be 0. The limit is t approaches 0 of a over t to the power k. So this is something else. Now that's the empty set. That is just going to blow up to infinity. So that will never work under those constraints, and that's why I put the open-closed little curly braces, meaning the empty set. Just another way to write that that limit does not exist. We've seen the trigonometric functions before. Remember that the limit is t approaches 0 of sine of t over t, or the sine of t, those are both 1. We've also seen that the limit is t approaches 0 of the cosine of 1 over t. Remember that's the one that just jiggles up and down, even more. Same happens for the sine of 1 over t. Again, that limit does not exist. Therefore I put the empty set there. Excellent.