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• - [Instructor] We now have a lot of experience taking

• limits of functions, if I'm taking limit of f of x.

• What we're gonna think about, what does f of x approach as x

• approaches some value a?

• And this would be equal to some limit.

• Now everything we've done up till now

• is where a is a finite value.

• But when you look at the graph

• of the function f right over here,

• you see something interesting happens.

• As x gets larger and larger, it looks like our function f

• is getting closer and closer to two.

• It looks like we have a horizontal

• asymptote at y equals two.

• Similarly, as x gets more and more negative,

• it also seems like we have a horizontal asymptote

• at y equals two.

• So is there some type of notation we can use to think about

• what is the graph approaching as x gets much larger

• or as x gets smaller and smaller?

• And the answer there is limits at infinity.

• So if we want to think about what is this graph,

• what is this function approaching

• as x gets larger and larger,

• we can think about the limit of f of x

• as x approaches positive infinity.

• So that's the notation, and I'm not going

• to give you the formal definition of this right now.

• There, in future videos, we might do that.

• But it's this idea, as x gets larger and larger and larger,

• does it look like that our function

• is approaching some finite value,

• that we have a horizontal asymptote there?

• And in this situation, it looks like it is.

• It looks like it's approaching the value two.

• And for this particular function, the limit

• of f of x as x approaches negative infinity

• also looks like it is approaching two.

• This is not always going to be the same.

• You could have a situation, maybe we had,

• you could have another function.

• So let me draw a little horizontal

• asymptote right over here.

• You could imagine a function that looks like this.

• So I'm going to do it like that,

• and maybe it does something wacky like this.

• Then it comes down, and it does something like this.

• Here, our limit as x approaches infinity is still two,

• but our limit as x approaches negative infinity,

• right over here, would be negative two.

• And of course, there's many situations where,

• as you approach infinity or negative infinity,

• you aren't actually approaching some finite value.

• You don't have a horizontal asymptote.

• But the whole point of this video is just

• to make you familiar with this notation.

• And limits at infinity

• or you could say limits at negative infinity,

• they have a different formal definition

• than some of the limits that we've looked at in the past,

• where we are approaching a finite value.

• But intuitively, they make sense,

• that these are indeed limits.

- [Instructor] We now have a lot of experience taking

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# Introduction to limits at infinity | Limits and continuity | AP Calculus AB | Khan Academy

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yukang920108 posted on 2022/07/05
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