Analyzing limits numerically
Estimating limits from tables
- [Teacher] The function g is defined over the real numbers. This table gives select values of g. What is a reasonable estimate for the limit is x approaches five 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? Alright 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 3.37, 4.9? It's a little higher. Is it 3.5, 4.99? Is it 3.66? 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 five from values greater than five. At six we're at 3.97; at 5.1 we're at 3.84. 5.01; 3.7; 5.001; we're at 3.68. So a thousandth below five and a thousandth above five we're at 3.68, but then at five all of sudden we're at 6.37. So my most reasonable estimate would be, well it look like we're approaching 3.68. When we're approaching from values less than five. And we're approaching 3.68 from values as we approach five from values greater than five. It doesn't matter that the value of five is 6.37. The limit would be 3.68 or 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 five; what is g of five? 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 five right over here, At the point five 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 this, that's 6.37; then at four, 3.37 is about here and it looks like it's approaching 3.68. So 3.68, actually let me draw that. So 3.68 is gonna be roughly that. 3.68 is gonna be roughly that. So the graph, the graph might look something like this. We can infer it looks like it's doing something like this. Where it's approaching 3.68 from values less than five and values greater than five, but right at five, our value is 6.37. I don't know for sure if this is what the graph look 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.