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  • - [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.

- [Teacher] The function g is defined over the real numbers.

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A2 初級 美國腔

用表格估計極限(Estimating limits from tables | Limits and continuity | AP Calculus AB | Khan Academy)

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    yukang920108 發佈於 2022 年 06 月 30 日
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