Name: Functions with same limit at infinity | Limits and continuity | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T14:04:48.000Z
Duration: 3 min 46 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

is to get an appreciation that you could have many,

in fact, you could have an infinite number

So, if we were to make the general statement

as x approaches infinity, is equal to three.

more and more examples, really an infinite number

of examples where that is going to be true.

So, for example, we could look at this graph over here.

And in other videos, we'll think about why this is the case,

and so it gets closer and closer to three x squared

And you can see, even when x is equal to 10,

we're getting awfully close to three right over there.

Let me zoom out a little bit so you see our axes.

Let me draw a dotted line at the asymptote.

That is y is equal to three, and so you see

But that's not the only function that could do that,

as I keep saying, there's an infinite number

You could have this somewhat wild function

it is getting closer and closer to three.

at a slightly slower rate than the one in green,

As x approaches infinity, this thing is approaching three.

And as we've talked about in other videos,

as long as they're getting closer and closer and closer

to it as x gets larger and larger and larger.

So, for example, that function right over there.

we can see that they're all approaching three.

the other two are approaching three from below.

let me actually zoom out a ways, and then I'll zoom in.

So, actually, even 100 isn't even that large

200 is much larger than numbers we've been looking at.

we have to zoom in an awfully lot, an awful lot,

just to even see that the graphs still aren't quite

that they are a little bit different than the asymptote.

I really zoomed in, I mean look at the scale.

This is, each of these are now 100th, each square.

You can see the calculation, this is up to

we're getting awfully close to three now,

So the green functions got there the fastest,

But the whole point of this is to emphasize the fact

that there's an infinite number of functions

for which you could make the statement that we made.

That the limit of the function as x approaches infinity,

in this case, we said that limit is going

to be equal to three, and I just picked three arbitrarily.

This could be true for any, for any function.

I didn't realize how much I had zoomed in.

So, there we have it, and maybe I can zoom in this way.

Limit of any of these, as x approaches infinity,

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Functions with same limit at infinity | Limits and continuity | AP Calculus AB | Khan Academy

approach

slightly

expression

matter