Read and Understand Graphs: Limits and Infinity

Welcome to the third video in this series, where we learn how to read and understand graphs – a skill that makes it a lot easier to link graphical representations to their real-world meanings.
If you feel you’ve reached your limit with mathematical limits, this is the video to watch. But if your learning material has not tended to that subject, you can get a nice head-start today. Some of the concepts in Mathematics may seem like its problem children. They don’t fit nicely into the basic calculations, standard methods or more general characteristics. Concepts like infinity, undefined, division by zero, discontinuity. In many of these seemingly problematic concepts limits play a very important role in handling the strange cases and outliers. And when they’re managed correctly, you realize that they actually form an unmissable subsection of Mathematics – one without which Mathematics would have been a lot poorer when it comes to applicability, depth and variety.

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Let’s consider a relatively simple example of a limit, and what each element of the formula represents, followed by a practical scenario that can be represented by this limit. We use the shortened “lim” to indicate a limit. The limit is accompanied by some kind of variable, which is associated with the point at which the problematic concept rears its head. Often, this point corresponds to extremes, for instance when the variable tends to infinity. Something feels strange when one tries to think about infinity as a specific number, since whatever that specific number is, you could always add one to it get a larger number, which means that specific number was not infinity. You could almost describe infinity as the smallest number to which we cannot count, but it’s better to not think about infinity as a number at all. We use the arrow to indicate “tends to”, which means the variable gets closer and closer to the target, but don’t actually ever precisely hits it.

We are then interested in what happens to some outcome or answer as this variable approaches the target or point at which the problem occurs. The outcome is usually represented by a function, in this case ½ to the power 1/5 of the variable. For each 5 units with which the variable x increases, the power of ½ increases by 5/5 = 1. But let’s visualize it. Sam is back, and he’s once more heading to his car to fetch his backpack. But this time around the following restriction is placed on his movement: every 5 seconds he has to halve the distance between him and his car. At any point in our example, he can therefore only ever walk half the distance between himself and the car at that time. Let’s say that the current distance, which we’ll display on the y-axis of the graph, is one unit. We say that the point in time, which we’ll display on the x-axis in seconds, is the point 0, the start. Sam can therefore walk half this distance during the next 5 seconds. We can see how the function in our limit starts forming on the graph as it moves 1 on the y-axis to ½, as the 5 seconds played out on the x-axis. Now Sam can walk half of the remaining distance, and he can continue like this. Each time the distance between him and the car gets smaller and smaller, but each time he’s only allowed to take a smaller and smaller step towards the car. Theoretically, he can continue like this for ever, but never actually really get there. The distance simply remains slightly larger than zero. If it wasn’t for limits, this picnic would be doomed. It goes to show what Mathematics can do for you! Limits allow us to solve that problematic point. Rather than trying to say the value of the function in the point infinity is 0, which does not make technical sense, we can say that the value of the function y tends to 0 as the input value x tends to infinity. It doesn’t sound like much of a difference, but technically it is. And other limits could of course be a lot more elaborate than our example.