 This lesson is on an introduction to limits. Now this is just an introduction to get you used to what a limit can possibly be in mathematics. Let's start off with a question. What is meant by the limit? Question. As you enter a room, a spider is spinning a web from the top of the doorway down towards you. Will you and the spider collide? Well, as you think about this, there are many answers. Think of the spider. Here's the doorway. Here's the spider coming down. If you walk into the room and the spider has not reached you yet, no problem. If you walk into the room and the spider has come down, it can just touch your hair, right? Or, as the spider comes down, you walk in the room. It can hit you somewhere else. Or, as the spider comes down, it might be way up here and you're over there. So there are lots of answers to the question, will it hit you? Well, the idea of it's hitting you would mean you and the spider hit the same limit at the same time. But if it doesn't hit you in any way, then you are not being at the same place at the same time. So it's not limited by what you're doing or you're not limited by what it's doing. So let's go on and look at this in mathematics. What is a limited math? Think of yourself and your best friend to be riding on a function. You're on one side of the function, let's say, coming up this way. And your friend is on the other side of the function coming down this way. And as you get closer and closer to each other, let's say in and around, x is equal to 0 and here, will you collide? Well, of course you'll collide because you're both on the same function and you will hit each other. That's all well and good. So a limit means you have to look at coming in from the left and coming in from the right and see whether you and your friend in this case or the function reaches the same point. What happens though with you and your friend if there's a point of discontinuity or what we call a whole right at that point? You're coming in from the left, your friend is coming in from the right. Will you reach that same point? And the answer is very definitely yes. You will reach that same point. You may not have to hit that point, but as long as you are reaching the same point, you have a limit. What about if the function is discontinuous and doesn't have a whole? For instance, like this rational function, you're coming in from the left, your friend is coming in from the right. Do you meet? Of course not. So then we say no and the limit does not exist. So what are we talking about mathematically? Well, we are saying that if you approach a function limit from the left, which is this minus right here, and you approach the limit from the right of your function and they are equal, then you have a limit. And thus we say it is the limit as x approaches a of your function f of x and that equals f of the value a. What happens when they are not equal? That is, when the limit as x approaches a from the left-hand side does not equal the limit as x approaches a from the right-hand side, it means the limit does not exist. This concludes our short lesson on the introduction to limits.