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  • - This video we're going to try to get

  • a sense of what the limit as x approaches three

  • of x to the third minus three x squared

  • over five x minus 15

  • is.

  • Now when I say get a sense,

  • we're gonna do that by seeing what

  • values for this expression we get as x gets

  • closer and closer to three.

  • Now one thing that you might wanna try out is,

  • well what happens to this expression

  • when x is equal to three?

  • Well then it's going to be three to the third power

  • minus three times three squared,

  • over five times three

  • minus 15.

  • So at x equals three, this expression's gonna be,

  • let's see in the numerator you have 27 minus 27, zero.

  • Over 15 minus 15, over zero.

  • So this expression is actually not defined

  • at x equals three.

  • We get this indeterminate form,

  • we get zero over zero.

  • But let's see, even though the function,

  • or even though the expression is not defined,

  • let's see if we can get a sense of what the limit might be.

  • And to do that, I'm gonna set up a table.

  • So let me set up a table here.

  • And actually I wanna set up two tables.

  • So this is x and this is x to the third

  • minus three x squared

  • over

  • five x

  • minus

  • 15.

  • And actually, I'm gonna do that again.

  • And I'll tell you why in a second.

  • So this is gonna be x and this is x to the third

  • minus three x squared

  • over

  • five x minus 15.

  • The reason why I set up two tables,

  • I didn't have to do two tables,

  • I could have done it all in one table,

  • but hopefully this will make it

  • a little bit more intuitive what I'm trying to do.

  • Is on this left table, I'm gonna, let's try out x values

  • that get closer and closer to three from the left.

  • From values that are less than three.

  • So for example, you can go to two point nine

  • and figure out what the expression equals

  • when x is two point nine.

  • But then we can try to get even a little bit closer

  • than that, we could go to two point nine nine.

  • And then we could go even closer than that.

  • We could go to two point nine nine nine.

  • And so one way to think about it here

  • is as we try to figure out what this expression equals

  • as we get closer and closer to three,

  • we're trying to approximate the limit from the left.

  • So limit

  • from

  • the left.

  • Now why do I say the left?

  • Well if you think about this on a coordinate plane,

  • these are the x values that are to the left of three,

  • but we're getting closer and closer and closer.

  • We're moving to the right,

  • but these are the x values that

  • are on the left side of three, they're less than three.

  • But we also, in order for the limit to exist,

  • we have to be approaching the same thing from both sides.

  • From both the left and the right.

  • So we could also try to approximate

  • the limit from

  • the right.

  • And so what values would those be?

  • Well those would be, those would be x values

  • larger than three.

  • So we could say three point one,

  • but then we might wanna get a little bit closer,

  • we could go three point zero one.

  • But then we might wanna get even closer to three.

  • Three point zero zero

  • one.

  • And every time we get closer and closer to three,

  • we're gonna get a better approximation for,

  • or we're gonna get a better sense

  • of what we are actually approaching.

  • So let's get a calculator out and do this.

  • And you could keep going, two nine nine nine nine nine.

  • Three point zero zero zero zero one.

  • Now one key idea here to point out,

  • before I even calculate what these are going to be,

  • sometimes when people say the limit from both sides,

  • or the limit from the left or the limit from the right,

  • they imagine that the limit from the left

  • is negative values and the limit from the right

  • are positive values.

  • But as you can see here, the limit from the left

  • are to the left of the x value that you're trying

  • to find the limit at.

  • So these aren't negative values,

  • these are just approaching the three

  • right over here from values less than three.

  • This is approaching the three

  • from values larger than three.

  • So now let's fill out this table,

  • and I'm speeding up my work so that you don't

  • have to sit through me typing everything

  • into a calculator.

  • So based on what we're seeing here,

  • I would make the estimate that this looks

  • like it's approaching one point eight.

  • So is this equal to one point eight?

  • As I said, in the future, we're gonna

  • be able to find this out exactly.

  • But if you're not sure about this

  • you could try an even closer and closer and closer value.

- This video we're going to try to get

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A2 初級 美國腔

用表格近似極限(Approximating limits using 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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