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  • - [Instructor] What we're going to do in this video

  • is use the online graphing calculator Desmos,

  • and explore the relationship between vertical

  • and horizontal asymptotes, and think about

  • how they relate to what we know

  • about limits.

  • So let's first graph two over x minus one,

  • so let me get that one graphed,

  • and so you can immediately see that something interesting

  • happens at x is equal to one.

  • If you were to just substitute x equals one

  • into this expression, you're going to get

  • two over zero, and whenever you get a non-zero thing,

  • over zero, that's a good sign that you might

  • be dealing with a vertical asymptote.

  • In fact we can draw that vertical asymptote

  • right over here at x equals one.

  • But let's think about how that relates to limits.

  • What if we were to explore the limit as x approaches one

  • of f of x is equal to two over x minus one,

  • and we could think about it from the left

  • and from the right, so if we approach one

  • from the left, let me zoom in a little bit

  • over here, so we can see as we approach from the left

  • when x is equal to zero, the f of x

  • would be equal to negative two,

  • when x is equal to point five, f of x

  • is equal to negative of four, and then it just gets

  • more and more negative the closer we get

  • to one from the left.

  • I could really, so I'm not even that close yet

  • if I get to let's say 0.91, I'm still nine hundredths

  • less than one, I'm at negative 22.222, already.

  • And so the limit as we approach one from the left

  • is unbounded, some people would say

  • it goes to negative infinity, but it's really

  • an undefined limit, it is unbounded

  • in the negative direction.

  • And likewise, as we approach from the right,

  • we get unbounded in the positive infinity direction

  • and technically we would say that that limit

  • does not exist.

  • And this would be the case when we're dealing

  • with a vertical asymptote like we see over here.

  • Now let's compare that to a horizontal asymptote

  • where it turns out that the limit

  • actually can exist.

  • So let me delete these or just erase them for now,

  • and so let's look at this function

  • which is a pretty neat function, I made it up

  • right before this video started

  • but it's kind of cool looking, but let's think

  • about the behavior as x approaches infinity.

  • So as x approaches infinity, it looks like our y value

  • or the value of the expression, if we said y

  • is equal to that expression, it looks like

  • it's getting closer and closer and closer to three.

  • And so we could say that we have a horizontal asymptote

  • at y is equal to three, and we could also

  • and there's a more rigorous way of defining it,

  • say that our limit as x approaches infinity

  • is equal of the expression or of the function,

  • is equal to three.

  • Notice my mouse is covering it a little bit

  • as we get larger and larger, we're getting

  • closer and closer to three,

  • in fact we're getting so close now, well here

  • you can see we're getting closer and closer

  • and closer to three.

  • And you could also think about what happens

  • as x approaches negative infinity and here

  • you're getting closer and closer and closer

  • to three from below.

  • Now one thing that's interesting about horizontal

  • asymptotes is you might see that the function

  • actually can cross a horizontal asymptote.

  • It's crossing this horizontal asymptote

  • in this area in between and even as we approach infinity

  • or negative infinity, you can oscillate

  • around that horizontal asymptote.

  • Let me set this up, let me multiply this times sine of x.

  • And so there you have it, we are now oscillating

  • around the horizontal asymptote,

  • and once again this limit can exist

  • even though we keep crossing the horizontal asymptote,

  • we're getting closer and closer and closer to it

  • the larger x gets.

  • And that's actually the key difference between

  • a horizontal and a vertical asymptote.

  • Vertical asymptotes if you're dealing with a function,

  • you're not going to cross it, while with a horizontal

  • asymptote, you could, and you are just getting

  • closer and closer and closer to it

  • as x goes to positive infinity or as x

  • goes to negative infinity.

- [Instructor] What we're going to do in this video

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B1 中級 美國腔

13-2 無限極限和漸近線(Infinite limits and asymptotes | Limits and continuity | AP Calculus AB | Khan Academy)

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    yukang920108 發佈於 2022 年 07 月 05 日
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