Approximating asymptotic limit from table

Function f is defined over the real numbers. This table gives a few values of f.

So when x is equal to -4.1, f of x is 5. f of -4.01 is 55. They give us a bunch of values for different x's of what f of x would be. What is a reasonable estimate for the limit as x approaches -4 of f of x?

So, let's think about what's happening. To think of a reasonable estimate for the limit as x approaches -4, we'd want to consider what it looks like the function is approaching as x approaches -4 from values less than -4. So that's what we have here on the table. Then we'd also want to think about, well, what does it look like our function is approaching as x approaches -4 from values greater than -4, from the right?

This is what f of x approaches as x approaches -4 from the left. This is what f of x is approaching as x approaches -4 from the right. So, let's think about it. When x is -4.1, we're at 5. Then we're at -4.1, we go to -55. And then when we get to -4.001, so we're only a thousandth away from -4, we get to 555. So this just seems to get larger and larger.

Then at -4, it's 5,555. So this is interesting; there could be a scenario where this is getting more and this is getting infinitely negative as we get close to -4, and then jumps back up to 5,555. Or this is a situation where we are at -4 or we're just kind of approaching this value right over here. It's unclear based on the information in the table.

Now, let's think about what's happening as we approach from the right. So at -3.9, we're at 53.99, then we're at 55. As we go to -3.9, we go to 555. So this seems to be getting larger. As we get closer and closer to -4, as x gets closer and closer to -4 from the right, it looks like f of x is getting larger and larger.

It's not going more and more negative. So, as I mentioned, if you wanted to think about scenarios of what might be happening here, and we don't know for sure because remember when we're using a table, we're just taking samples.

So, let me draw what could be happening here. Once again, we don't know for sure just by sampling, but once again, they just want us to get a reasonable estimate. So that's the x-axis, and that is the y-axis. We care about x = -4. So let's say that this is x = -4 right over here.

There are a couple of scenarios. There's one scenario where it's doing something like this, where it's just going to negative 5,555 (this isn't at the same scale). So there's this scenario where this right over here is -5,555. But then from the right, you're doing something completely different; you're going in the opposite direction.

So from the right, it looks like this; you're just getting larger and larger. As you get closer and closer to -4 from the right, maybe you're going to positive infinity, or maybe you're going to… well, we don't know. Based on this data, it looks like it could just be going unbounded to positive infinity.

Then right at -4, you go to 5,555. In this situation, you would have no limit at x = -4, even though the function is defined there. You would have no limit because when you approach from the right, you're going to positive infinity, while when you approach from the left, you're either going to negative infinity or you're going to the value 5,555.

Another scenario that might be happening as we approach from the left might be something like this; it might be approaching negative infinity, and then you just jump back up to this value right over here. So let me erase this one just so you can see what I'm talking about.

So this is another scenario that right at -4; from either side, it looks like you have an asymptote going on. From either side, you have this vertical asymptote right over here. So, as you approach from the left, you're going to negative infinity. As you approach from the right, you're going to positive infinity. But right at -4, you're defined at this value.

So this is another possibility, but in this case as well, there's no limit here. You're not approaching the same finite value from both sides. Now, one thing that you'll sometimes see is if people think that, okay, if you’re approaching the same, you know, either from both sides, you're approaching positive infinity, or from both sides you're approaching negative infinity. Sometimes people say, "Oh, my limit is infinity," or "my limit is negative infinity."

So let's say you have a situation like this where at some value, you're thinking about the limit as you approach as x approaches that value. As you approach from either side, you're going to positive infinity. Well, formally, you still wouldn't say the limit is infinity because a limit, formally, is a finite value that you are actually approaching.

So, in any of these scenarios, especially the scenario that we see here where you're going to positive infinity here, negative infinity here, or you're going to 5,555 here, the limit does not exist. So let me just circle that.