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## Connecting limits at infinity and horizontal asymptotes

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# Functions with same limit at infinity

AP Calc: LIM‑2 (EU), LIM‑2.D (LO), LIM‑2.D.3 (EK), LIM‑2.D.4 (EK)

## Video transcript

- [Instructor] The goal of this video is to get an appreciation
that you could have many, in fact, you could have an infinite number of functions that have the same limit as x approaches infinity. So, if we were to make
the general statement that the limit of some function f of x, as x approaches infinity,
is equal to three. What I wanna do in this video is show some examples of that. And to show that we can keep creating 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, but just think about what happens when you have very, very large Xs. When you have very, very large Xs, the plus five doesn't matter as much, and so it gets closer and
closer to three x squared over x squared, which is equal to three. And you could see that right over here, it's graphed in this green color. 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. So that is three. Let me draw a dotted
line at the asymptote. That is y is equal to
three, and so you see the function's getting closer and closer as x approaches infinity. But that's not the only
function that could do that, as I keep saying, there's
an infinite number of functions that could do that. You could have this somewhat wild function that involves natural logs. That too, as x approaches infinity, it is getting closer and closer to three. It might be getting closer to three at a slightly slower rate
than the one in green, but we're talking about infinity. As x approaches infinity, this
thing is approaching three. And as we've talked about in other videos, you could even have things that keep oscillating around the asymptote, 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. Let me zoom in. So, let's zoom in. Let's say when x is equal to 14, we can see that they're
all approaching three. The purple one is oscillating around it, the other two are
approaching three from below. But as we get much larger, let me actually zoom out a
ways, and then I'll zoom in. So let's get to really large values. So, actually, even 100
isn't even that large if we're thinking about infinity. Even a trillion wouldn't be that large if we're thinking about infinity. But let's go to 200. 200 is much larger than
numbers we've been looking at. And let me zoom in when x is equal to 200, and you can see, we have to zoom in an
awfully lot, an awful lot, just to even see that the
graphs still aren't quite stabilized around the asymptote, 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. And so we've gotten much, much, much closer to the asymptote. In fact, the green function, we still can't tell the difference. You can see the calculation, this is up to three or four decimal places, we're getting awfully close to three now, but we aren't there. So the green functions
got there the fastest, is an argument. 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 let me now go back to the origin where we had our original expression. So, there we have it, and
maybe I can zoom in this way. So there you have it. Limit of any of these,
as x approaches infinity, is equal to three.