Connecting limits at infinity and horizontal asymptotes
Introduction to limits at infinity
- [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.
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