One-sided limits from graphs

So if we were to ask ourselves, what is the value of our function approaching-- as we approach x equals 2 from values less than x equals 2. So as you imagine, as we approach x equals 2-- So x equals 1, x equals 1.5, x equals 1.9, x equals 1.999, x equals 1.99999999. What is f of x approaching? And we see that f of x seems to be approaching this value right over here. It seems to be approaching 5. And so the way we would denote that is the limit of f of x, as x approaches 2-- and we're going to specify the direction-- as x approaches 2 from the negative direction-- we put the negative as a superscript after the 2 to denote the direction that we're approaching. This is not a negative 2. We're approaching 2 from the negative direction. We're approaching 2 from values less than 2. We're getting closer and closer to 2, but from below-- 1.9, 1.99, 1.99999 . As x gets closer and closer from those values, what is f of x approaching? And we see here that it is approaching 5. But what if we were asked the natural other question-- What is the limit of f of x as x approaches 2 from values greater than 2? So this is a little superscript positive right over here. So now we're going to approach x equals 2, but we're going to approach it from this direction-- x equals 3, x equals 2.5, x equals 2.1, x equals 2.01, x equals 2.0001. And we're going to get closer and closer to 2, but we're coming from values that are larger than 2. So here, when x equals 3, f of x is here. When x equals 2.5, f of x is here. When x equals 2.01, f of x looks like it's right over here. So in this situation, we're getting closer and closer to f of x equaling 1. It never does quite equal that. It actually then just has a jump discontinuity. This seems to be the limiting value when we approach when we approach 2 from values greater than 2. So this right over here is equal to 1. And so when we think about limits in general, the only way that a limit at 2 will actually exist is if both of these one-sided limits are actually the same thing. In this situation, they aren't. As we approach 2 from values below 2, the function seems to be approaching 5. And as we approach 2 from values above 2, the function seems to be approaching 1. So in this case, the limit-- let me write this down-- the limit of f of x, as x approaches 2 from the negative direction, does not equal the limit of f of x, as x approaches 2 from the positive direction. And since this is the case-- that they're not equal-- the limit does not exist. The limit as x approaches 2 in general of f of x-- so the limit of f of x, as x approaches 2, does not exist. In order for it to have existed, these two things would have had to have been equal to each other. For example, if someone were to say, what is the limit of f of x as x approaches 4? Well, then we could think about the two one-sided limits-- the one-sided limit from below and the one-sided limit from above. So we could say, well, let's see. The limit of f of x, as x approaches 4 from below-- so let me draw that. So what we care about-- x equals 4. As x equals 4 from below-- So when x equals 3, we're here where f of 3 is negative 2. f of 3.5 seems to be right over here. f of 3.9 seems to be right over here. f of 3.999-- we're getting closer and closer to our function equaling negative 5. So the limit as we approach 4 from below-- this one-sided limit from the left, we could say-- this is going to be equal to negative 5. And if we were to ask ourselves the limit of f of x, as x approaches 4 from the right, from values larger than 4, well, same exercise. f of 5 gets us here. f of 4.5 seems right around here. f of 4.1 seems right about here. f of 4.01 seems right around here. And even f of 4 is actually defined, but we're getting closer and closer to it. And we see, once again, we are approaching 5. Even if f of 4 was not defined on either side, we would be approaching negative 5. So this is also approaching negative 5. And since the limit from the left-hand side is equal to the limit from the right-hand side, we can say-- so these two things are equal. And because these two things are equal, we know that the limit of f of x, as x approaches 4, is equal to 5. Let's look at a few more examples. So let's ask ourselves the limit of f of x-- now, this is our new f of x depicted here-- as x approaches 8. And let's approach 8 from the left. As x approaches 8 from values less than 8. So what's this going to be equal to? And I encourage you to pause the video to try to figure it out yourself. So x is getting closer and closer to 8. So if x is 7, f of 7 is here. If x is 7.5, f of 7.5 is here. So it looks like our value of f of x is getting closer and closer and closer to 3. So it looks like the limit of f of x, as x approaches 8 from the negative side, is equal to 3. What about from the positive side? What about the limit of f of x as x approaches 8 from the positive direction or from the right side? Well, here we see as x is 9, this is our f of x. As x is 8.5, this is our f of 8.5. It seems like we're approaching f of x equaling 1. So notice, these two limits are different. So the non-one-sided limit, or the two-sided limit, does not exist at f of x or as we approach 8. So let me write that down. The limit of f of x, as x approaches 8-- because these two things are not the same value-- this does not exist. Let's do one more example. And here they're actually asking us a question. The function f is graphed below. What appears to be the value of the one-sided limit, the limit of f of x-- this is f of x-- as x approaches negative 2 from the negative direction? So this is the negative 2 from the negative direction. So we care what happens as x approaches negative 2. We see f of x is actually undefined right over there. But let's see what happens as we approach from the negative direction, or as we approach from values less than negative 2, or as we approach from the left. As we approach from the left, f of negative 4 is right over here. So this is f of negative 4. f of negative 3 is right over here. f of negative 2.5 seems to be right over here. We seem to be getting closer and closer to f of x being equal to 4, at least visually. So I would say that it looks-- at least, graphically-- the limit of f of x, as x approaches 2 from the negative direction, is equal to 4. Now, if we also asked ourselves the limit of f of x, as x approaches negative 2 from the positive direction, we would get a similar result. Now, we're going to approach from when x is 0, f of x seems to be right over here. When x is 1, f of x is right over here. When x is negative 1, f of x is there. When x is negative 1.9, f of x seems to be right over here. So once again, we seem to be getting closer and closer to 4. Because the left-handed limit and the right-handed limit are the same value. Because both one-sided limits are approaching the same thing, we can say that the limit of f of x, as x approaches negative 2-- and this is from both directions. Since from both directions, we get the same limiting value, we can say that the limit exists there. And it is equal to 4.