Example reasoning about limits from tables

Told the function G is defined over the real numbers. This table gives a few values of G. So for a few values of X, they give us what G of X is equal to. And they say what is a reasonable estimate for the limit of G of X as X approaches 9 from the left or as X approaches 9.

Remember, when we say x approaches 9 with this little superscript, that little negative sign, that doesn't necessarily mean that we're approaching nine from negative values of X; it means that we're approaching nine from values less than nine.

To think about this, what is a reasonable value for the left-sided or the left-handed limit? We would say, "All right, let's pick some X values that get closer, that are less than nine, but keep getting closer and closer to nine." So I would look at these X values, which they seem to have chosen to do exactly that. They're starting at 8.9, then they get even closer, 8.99, then they get even closer to 8.999.

Once again, they only want us to figure out whether we can estimate the left-handed limit. If we wanted to know the actual limit, we would want to figure out the left-handed limit and the right-handed limit, and to feel good that they seem to be approaching the same thing. But here we're only talking about the left-handed limit.

So when I look at this, at 8.9, G of X is equal to 5.01. As we get even closer to 9, we get to 4.83. As we get even closer to 9, we get to -4.81. So just looking at these three values here for these three X's, it looks like we're getting close to about -4.8. So it's approximately -4.8. We don't know that for sure.

Oftentimes the limit might end up being a nice number like -4.8, and that's not that nice of a number, but the limit might be -4.82, 4.38, where we are approaching. It doesn't always have to go to a nice to look at number.

But something interesting happens when we get exactly to nine. All of a sudden it seems like we're getting further away from -4.8; we jump to -5.3. So what does that tell us? Well, this is actually trying to tackle a key misconception that folks have when trying to think about limits.

A limit is the value that you are approaching, a value that the function is approaching as X approaches a certain value. It isn't necessarily going to be what the function is defined at that value. So you can be approaching a different value than what the function is defined at.

And it looks, at least looking at inspecting this table, it looks like that's what's happening here. So what is a reasonable limit? What is a reasonable estimate for the left-handed limit as X approaches 9 of G of X? Well, I would rule out 5.3. Even though that is what G of 9 is, this is equal to G of 9 as we approach from the left; it does not look like we are approaching -5.3.

So in fact, it looks like we're getting further and further away, or it looks like we're approaching -4.8. And we're actually, as we get closer and closer, we're getting a little bit further from something like 5.3. So I would rule this one out.

4.8 I liked; even before I looked at the choices, it looked like that's what we were approaching. So that seems like a reasonable estimate. I like that. So 4.7—let's see, is that a reasonable result?

Okay, so this is tricky. At first, I was going to say oh yeah, maybe, but I realized—didn't this isn't negative 4.7? If it was negative 4.7, well, maybe that would be reasonable. We don't know for sure, but it's not; it's 4.7. So there's no reason why we would be approaching -4.8 and then all of a sudden jump to 4.7, and that would be the limit.

That's not what we're approaching; we're not approaching a positive 4.7. So definitely rule that out, and we're definitely not approaching nine here. They're just trying to confuse you.

The X is approaching nine from the left, from values less than nine, but that's not what the function itself is approaching. The function itself seems to be getting closer and closer to -4.8. So rule that one out, and the limit seems to exist.

Once again, we are estimating. When we have these tables, it's completely possible that this is some type of weird function that begins to do some type of weird oscillation as we get really close, so it oscillates between -4.8 and 4.7.

So we don't know 100% sure, but it's reasonable—it's not reasonable to think based on this information that the limit doesn't exist, so I would rule that out. I like this choice right over here.

Now let's do one more example. A student created a table to help them reason about the limit as X approaches 3 of F of X. So here we want the actual limit. We want the limit as we approach from the left and from the right and see if we are approaching the same thing.

Based on the table, what can you reason about the limit? So here we see X approaching—we could say from below, even though visually it looks like from above—but we're approaching X from the left. We're approaching X from values less than three. If we're looking at the x-axis, we're approaching three from values less than three.

And it looks like at 2.9, we get 1.28; at 2.99, we get 1.42; at 2.9999, we get 1.49. So it looks like we're getting closer and closer to 1.5. And even though three is undefined, even though the function is undefined at x equals 3, the limit could exist.

In fact, that's one of the main values of a limit—they say what is the function approaching, even in cases where the function is not defined there. So it looks like it's approaching 1.5.

Now, if we approach three from the positive direction, from values greater than three, see when we get 3.1, then our function is 1.8; at 3.01, our function is 1.63; at 3.1, we get 1.51.

So even from the right, or even from values larger than three, it still looks like we're approaching 1.5. So this could be a reasonable estimate, a reasonable guess, based on this table of what the limit actually is.

Now, we have to be very, very careful; you don't know definitively just by trying out these numbers that the limit is definitely 1.5. It might be 1.51, 87 repeating; it might be that number. We don't know for sure.

Now, in a typical calculus class, in a typical exam, you don't typically see numbers like that. The questions are usually engineered to give you nicer numbers, but I just want to make it clear: you might be approaching an irrational number; you could be approaching, you know, pi divided by two or something like that—not necessarily in this case, but in general.

So, this is a reasonable estimate. So let's see what we can reason. The limit as X approaches 3 of F of X is approximately 1.5. Yeah, I think this is a reasonable thing, so I would check that off.

Now, we don't know this for sure. In fact, there's even a scenario where this limit still might not exist, even if it looks like it. Maybe as we get closer and closer, this is some type of weird function that starts oscillating in some strange way or stops getting closer to a value, even though it looks like it's getting closer and closer right over here.

But this is a reasonable—what can you reasonably reason? I'll put that here: what can you reasonably reason about the limit? There appears to be an asymptote at x equals 3. Well, no—an asymptote would be the situation if we were a vertical asymptote, especially if we seem to be approaching infinity or negative infinity. That's definitely not the case here, so I would rule that out.

As we approach x at 3, the values of F of x seem to approach negative 1.5, but we can't say for sure if that's the limit exactly. Yeah, that's the point I was making right over here. It seems like it's getting to negative 1.5, and most tests—most questions on most tests—are kind of engineered to give you numbers that aren't crazy.

But even with this data, even with this information, it might be approaching something a little bit more interesting, or this could be some kind of crazy function that as we get within a thousandth of three, maybe it's not defined even around three. Maybe it's not even defined at 2.99999 or 3.1, or maybe it just doesn't get closer anymore.

So once again, you don't know for sure. But this does seem to be a reasonable thing to say: the values of F of x appear to oscillate, jump back and forth near x equals 3. So it's hard to say what the limit is or if it even exists. Well, we're not seeing that behavior right over here, so I wouldn't pick that. That's not a reasonable thing because you're not observing it in the data.

Now, there are some crazy functions out there where all of a sudden, if you start trying out values that are much closer than what we're seeing here, maybe some of this behavior starts to happen. But we're not seeing it in the data, so I definitely wouldn't pick that.