Limits from tables for oscillating functions

The function h is defined over the real numbers. This table gives a few values of h. So they give us for different x values what is the value of h of x. What is a reasonable estimate for the limit of h of x as x approaches one?

So with the table, we can estimate the limit. We won't know 100% for sure, but it's a good way to estimate a limit. And so what they have here is these first three entries show what's happening as we approach x equals one from the left, from values less than x equals one.

We can see what is h of x for those x values as we get closer and closer to x equals one. When we're at 0.9, it's 0.003, it's 3 thousandths. Then when we get even closer to 1, the value of our function is negative 0.4. Then when we get even closer to 1, the value of h is 0.5.

So we have this oscillation in sign, and it seems like the absolute value is getting larger and larger. Now, if we approach from the right, if we approach x equals one from values larger than one, we see we're approaching from the right. At 1.1, we’re at 0.003. At 1.001, we are negative 0.4. At 1.0001, or one and one thousandth, we are at 0.5.

This is interesting. You know, when you have this oscillation of sign, you say, well maybe we might be approaching zero. Maybe something is getting closer and closer and closer and closer to zero. But what's troubling here, the reason why I would say that it's reasonable to think that the limit doesn't exist is that as we're getting closer and closer, the oscillations are getting more and more significant.

So actually think of it this way: as we're getting from the left, the oscillations are getting more significant, and as we're going from the right, as we get closer and closer, the oscillations are getting more significant. Now it looks like there's a little bit of symmetry here. Let me see if I can draw symmetry reasonably well. It looks like something like this is happening.

But this is implying this. As we go to 0.9999, then we're going to oscillate again, and we're going to get even further away from zero. So we're oscillating around zero, but where the oscillations are swinging wilder and wilder. So we're definitely not approaching zero. We're not getting closer and closer. The absolute value of these aren't getting closer and closer to zero.

So I would say that the limit doesn't exist. A takeaway here is when things start to oscillate, you're probably already saying, okay, something fishy is going on. But that doesn't mean that the limit necessarily doesn't exist just because of the oscillation. But if it's oscillating around a value, and as it's getting closer and closer, the oscillations are swinging wilder and further away from that value, then you have to worry, and then the limit probably does not exist.

Let's do another example. The function h is defined over the real numbers. This table gives a few values of h. So a similar situation: what is a reasonable estimate for the limit as x approaches 3 of h of x?

As x approaches 3 from the left, it looks like at 2.9, we're at 2.1. Then at 2.99, we go to negative 0.02. Then at 2.999, we get to positive 0.002. So even though it's oscillating, magnitude-wise, we're getting closer and closer to zero.

This oscillation might look like this: we're oscillating in sign, but it looks like we're getting closer and closer to zero. So this looks like our function might be approaching zero as x approaches three from the left. Let's think about from the right. Well, from the right, we see something similar here. At 3.1, we're at negative 0.2. At 3.01, we’re positive 0.02. At 3.001, or even closer to 3, it's negative 0.002.

It looks like we are definitely oscillating, but the oscillations are getting less and less wild. Their absolute value is getting smaller and smaller. So it actually seems reasonable. We're oscillating around zero, and with every swing as we get closer and closer to x equals three, the oscillations are getting smaller and smaller.

So once again, we don't know for sure; we can't prove it mathematically, but from this table, it is reasonable to estimate that the limit here is zero. Now, these other choices I would call them distractors. This one down here is interesting because the function is actually defined at x equals three, and it equals four.

But this might be a situation where the limit is approaching; we're getting closer and closer, but the function is defined at four. This reinforces the idea that look, the limit can approach a value that is different from the value that the function is defined. That’s actually one of the points of limits: this is to study what is a function approaches even when the function might be defined at a different value, or even when the function isn't defined at that value.

What is it approaching as x approaches that value? Now, as I said, you know, for this situation, it’s reasonable to think that the limit exists. I don't have any good information to think otherwise. We don't know 100% for sure. They will put the 3 here. We're thinking about what is h approaching as x approaches 3.

Now, there’s no information here that would imply that h is approaching 3, and this is 0.22. They're just playing with the digits over here, but this is getting further from zero than what seems like what we're trying to approach. So I’d rule that one out as well.

The big takeaway here is if you start to see us, especially on a table or anything, that as you get closer and closer to a value that the function is oscillating, that should kind of put your spider senses on notice. But that doesn't necessarily mean that the limit doesn't exist. If the swings are getting wilder and wilder as we approach the value that we're oscillating around, then the limit probably doesn't exist.

But if the swings, the oscillations, are getting smaller in magnitude as we approach the thing that we're oscillating around, in this case zero, then it actually is reasonable to think that the thing that we're oscillating around actually is the limit.