Connecting limits and graphical behavior (more examples)

So we have a function, f of x, graphed right over here. And then we have a bunch of statements about the limit of f of x, as x approaches different values. And what I want to do is figure out which of these statements are true and which of these are false. So let's look at this first statement. Limit of f of x, as x approaches 1 from the positive direction, is equal to 0. So is this true or false? So let's look at it. So we're talking about as x approaches 1 from the positive direction, so for values greater than 1. So as x approaches 1 from the positive direction, what is f of x? Well, when x is, let's say 1 and 1/2, f of x is up here, as x gets closer and closer to 1, f of x stays right at 1. So as x approaches 1 from the positive direction, it looks like the limit of f of x as x approaches 1 from the positive direction isn't 0. It looks like it is 1. So this is not true. This would be true if instead of saying from the positive direction, we said from the negative direction. From the negative direction, the value of the function really does look like it is approaching 0. For approaching 1 from the negative direction, when x is right over here, this is f of x. When x is right over here, this is f of x. When x is right over here, this is f of x. And we see that the value of f of x seems to get closer and closer to 0. So this would only be true if they were approaching from the negative direction. Next question. Limit of f of x, as x approaches 0 from the negative direction, is the same as limit of f of x as x approaches 0 from the positive direction. Is this statement true? Well, let's look. Our function, f of x, as we approach 0 from the negative direction-- I'm using a new color-- as we approach 0 from the negative direction, so right over here, this is our value of f of x. Then as we get closer, this is our value of f of x. As we get even closer, this is our value of f of x. So it seems from the negative direction like it is approaching positive 1. From the positive direction, when x is greater than 0, let's try it out. So if, say, x is 1/2, this is our f of x. If x is, let's say, 1/4, this is our f of x. If x is just barely larger than 0, this is our f of x. So it also seems to be approaching f of x is equal to 1. So this looks true. They both seem to be approaching the limit of 1. The limit here is 1. So this is absolutely true. Now let's look at this statement. The limit of f of x, as x approaches 0 from the negative direction, is equal to 1. Well, we've already thought about that. The limit of f of x, as x approaches 0 from the negative direction, we see that we're getting closer and closer to 1. As x gets closer and closer to 0, f of x gets closer and closer to 1. So this is also true. Limit of f of x, as x approaches 0 exists. Well, it definitely exists. We've already established that it's equal to 1. So that's true. Now the limit of f of x as x approaches 1 exists, is that true? Well, we already saw that when we were approaching 1 from the positive direction, the limit seems to be approaching 1. We get when x is 1 and 1/2, f of x is 1. When x is a little bit more than 1, it's 1. So it seems like we're getting closer and closer to 1. So let me write that down. The limit of f of x, as x approaches 1 from the positive direction, is equal to 1. And now what's the limit of f of x as x approaches 1 from the negative direction? Well, here, this is our f of x. Here, this is our f of x. It seems like our f of x is getting closer and closer to 0, when we approach 1 from values less than 1. So over here it equals 0. So if the limit from the right-hand side is a different value than the limit from the left-hand side, then the limit does not exist. So this is not true. Now finally, the limit of f of x, as x approaches 1.5, is equal to 1. So right over here. So everything we've been dealing with so far, we've always looked at points of discontinuity, or points where maybe the function isn't quite defined. But here, this is kind of a plain vanilla point. When x is equal to 1.5, that's maybe right over here, this is f of 1.5. That right over there is the point, well, this is the value f of 1.5. We could say f of, we could see that f of 1.5 is equal to 1, that this right here is the point 1.5 comma 1. And if we approach it from the left-hand side, from values less than it, it's 1, the limit seems to be 1. When we approach from the right-hand side, the limit seems to be 1. So this is a pretty straightforward thing. The graph is continuous right there, and so really, if we just substitute at that point, or we just look at the graph, the limit is the value of the function there. You don't have to have a function be undefined in order to find a limit there. So it is, indeed, the case that the limit of f of x, as x approaches 1.5, is equal to 1.