Estimating limits from tables | Limits and continuity | AP Calculus AB | Khan Academy

The function g is defined over the real numbers. This table gives select values of g. What is a reasonable estimate for the limit as x approaches 5 of g of x? So pause this video, look at this table. It gives us the x values as we approach five from values less than five and as we approach five from values greater than five. It even tells us what g of x is at x equals five. And so given that, what is a reasonable estimate for this limit?

All right, now let's work through this together. So let's think about what g of x seems to be approaching as x approaches five from values less than five. Let's see, at four is it 3.374, at 4.9 it's a little higher, it's at 3.5. At 4.99 is it 3.66? At 4.999, so very close to five, we're only a thousandth away, we're at 3.68. But then at five, all of a sudden, it looks like we're kind of jumping to 6.37.

And once again, I'm making an inference here; I don't, these are just sample points of this function. We don't know exactly what the function is. But then if we approach 5 from values greater than 5, at 6 we're at 3.97, at 5.1 we're at 3.84, at 5.01 we're at 3.7, and at 5.001, we are at 3.68. So a thousandth below five and a thousandth above five, we're at 3.68. But then at five, also at 6.37.

So my most reasonable estimate would be, well, it looks like we are approaching 3.68 when we are approaching from values less than 5 and we're approaching 3.68 from values as we approach 5 from values greater than 5. It doesn't matter that the value of 5 is 6.37; the limit would be 3.68. A reasonable estimate for the limit would be 3.68.

And this is probably the most tempting distractor here, because if you were to just substitute 5, if you're, what is g of 5, it tells us 6.37. But the limit does not have to be what the actual function equals at that point. Let me draw what this might look like.

So an example of this. So if this is 5 right over here, at the point 5, the value of my function is 6.37. So let's say that this right over here is 6.37. So that's the value of my function right over there, so 6.37. But as we approach five, so that's four, actually let me spread out a little bit. This obviously is not drawing to scale, but as we approach five, so if that's 6.37, then at 4, 3.37 is about here and it looks like it's approaching 3.68.

So 3.68—actually, let me draw that—3.68 is going to be roughly that. So the graph might look something like this. We could infer it looks like it's doing something like this, where it's approaching 3.68 from values less than 5 and values greater than 5. But right at 5, our value is 6.37.

I don't know for sure if this is what the graph looks like; once again, we're just getting some sample points. But this would be a reasonable inference. And so you can see our limit; we are approaching 3.68 even though the value of the function is something different.