So, in this lesson we're going to be trying to figure out what is a limit -- what does it mean to take a limit of a function. So, we're going to start by looking at this picture right here. And, we have a function but it's behind this window. So, you have your window and your curtains and everything and you look out the window and you see a function. And, so, we're asked this question, which is what is happening when x is two? So, if we look on the graph right there, we can't actually see when x equals two. There happens to be a bar over the window right there. But, we could make a really good guess. What do you guys think is happening there? What do you think the y value is when x is two? Yep. It looks like its four. Here's your graph. If it went like that, it would be four, right? There'd be a point right there. But, the thing is, we don't actually know that for sure. If we look, if we move the window, it could be that we have a point here at four, which is what we guessed. But there could actually be a hole at four, or there could be a point somewhere else that's not at four. So, what we've done by saying it looks like it ought to be at four, even though I don't really know exactly what's happening at two, because at two there's this bar in the way, right? What we've done is we've found the limit as x goes to two of our function is four. So, that means if I looked on either side, but I can't see what's going on exactly what's going on right when I'm at two, I'm going to guess that the answer should be four. And if we look at what our definition is here for the limit, we would say the limit of f of x as x approaches, in our case two, equals in our case four means that we can make our function value -- our y value -- as close as we want to four by taking x as close as we want to two. So, if we go back and look at our picture again, with that bar in the way, as long as I can look and I can see if we're really close to two on the x axis, I'm looking at the function on either side of the bar and I'm saying the closer I am to two, my function's getting closer and closer to having a y value of four. And if I move that bar out of the way and all three of these situations, whether there's a hole or there's a dot somewhere else. In all of these three situations on either side of that actual point, the closer and closer the x value gets to two, the closer and closer the y value gets to four. So, we would say that in this case, in all three of these cases no matter what's actually happening at two, the limit as x goes to two of f of x is four. Now, there's one other way to think of limits other than the windowpane that also can be helpful and that's to think of your function as actually like a rollercoaster or a road that you're driving on. And so in this example we're looking at the function x minus one over x squared minus one as x gets close to one. So, I imagine that I'm this car right here, and I'm driving along the function. And if I drive so my x value gets closer and closer to one, where is my car going? Where is the road going -- that's what we're asking ourselves. And road is going to be at a height of 0.5. And it doesn't matter which direction I'm coming from. If I'm coming from this side over here, as I'm driving along with my little car, I'm still getting close to a height of 0.5. Now, in our analogy here, we seem to have a pothole or something exactly at that spot, but it doesn't matter because that's where the road is headed. And so we would say that the limit here according to this graph is 0.5. So, that's the idea of a limit. Where is the graph headed as you get close to the point? What's going on near the point? But the big idea here is we don't care what's actually happening at the point. So, for example, if we have f of x being x minus one over x squared minus one, then if we let, if actually try to plug in one here, we're going to get one minus one over one minus one. We're going to get zero over zero, which doesn't make any sense we can't divide by zero, right? So, we can't actually plug the point in but we can still say something about it. We can say that its getting close to 0.5.