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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 1

Lesson 15: Connecting limits at infinity and horizontal asymptotes

# 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)
A limit at infinity (like any other limit) describes the behavior of a function but it isn't unique to that function. Many different functions can have the same limit at infinity.

## Want to join the conversation?

• Why is this important to understand?
(24 votes)
• From the author:The important thing is to understand limits at infinity. We hope that this video helps understanding limits at infinity a little better.

We see in this video how limits at infinity say something about the behavior of the function as the x-values increase to infinity, but they don't say exactly how the function behaves. If the limit at infinity is 3, all we know is that the function approaches 3. We don't know how the function approaches 3. In fact, it can approach 3 in many different ways, as we can see in Sal's multiple examples.
(18 votes)
• If the limit of x-->infinity is 3, then does that mean that the functions will never actually reach 3?
(3 votes)
• Yes that is correct. What the statement means is that as x gets bigger and bigger, y will get closer and closer to 3. At infinity, y will equal 3. So obviously this means that the function will never actually reach 3. Hope this helps!
(7 votes)
• Why isn't the purple function oscillating around y=3 when Sal is zoomed in at 200?
(2 votes)
• Think about it. The purple function is 1/x*sin(x) + 3. As x approaches infinity, 1/x becomes extremely close to 0. Since sin(x) is the only oscillating part, if 1/x*sin(x) becomes about 0, so does the oscillating. If you don't understand why sin(x) oscillates, I encourage you to watch the videos about it on Khan Academy.
(5 votes)
• Isn't it true that graph of Sin(x)/x is oscillating around asymptote upto infinity?
If wee look at Desmos, graph is again crosses asymptote after 200.
Ref : https://www.desmos.com/calculator/sr7dge3tyu
(2 votes)
• Yes, but since each oscillation has a smaller magnitude than the one before, the function is still approaching the asymptote.
(4 votes)
• If a graph has both a vertical and horizontal asymptote and the question is asking us to find the limit of the graph as x approaches infinity, which one do we use to determine the answer?
(1 vote)
• Be careful though, unlike vertical asymptotes, a function can cross its horizontal asymptotes at certain points in its domain, and function might not approach the horizontal asymptote as x approaches infinity, it really depends on the function.
(1 vote)
• I don't get how the oscillating function gets closer and closer to 3. Doesn't the function crossing 3 mean it is equal to 3 at certain points?
(1 vote)
• Yes, it does. But that doesn't mean the limit at infinity isn't also 3.

The limit being 3 means that, for any error bound ε, there is an x-value (call it M) beyond which f(x) stays within ε of 3.

So if we want the function to remain within 0.1 of 3, we just need to find a point after which the amplitude of the sine function is less than 0.1. This is always the case whenever x≥10. Because when x≥10, 1/x≤0.1, and the function is
[something ≤0.1]·[something between -1 and 1]+3. That is, the function is within 0.1 of 3.
(1 vote)
• how did he find the limit at 3?
(1 vote)
• Find the y value where x approaches 3 from the left and right. If the left limit and right limit equal, the limit exists. However, if they are not equal, the limit does not exist.
Hope this helps! If you have any questions or need help, please ask! :)
(1 vote)
• evaluate:
3x^2-x-2/5x^2+4x+1 when limit approches infinity
please help me out to solve diz sir
(0 votes)

## 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.