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## AP®︎/College Calculus AB

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

Lesson 15: Connecting limits at infinity and horizontal asymptotes

# Introduction to limits at infinity

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.D (LO)
,
LIM‑2.D.3 (EK)
,
LIM‑2.D.4 (EK)
Introduction to the idea and notion of limits at infinity (and negative infinity).

## Want to join the conversation?

• At , the function that Sal has created crosses through horizontal asymptotes. This could not be a real function right?
• Horizontal asymptotes are more for how a function behaves as it heads toward infinity, or a vertical asymptote in the middle. Functions can cross horizontal asymptotes between these sections.
• At , Sal mentions the word, "asymtote." I was just scrolling through the videos to see any new and interesting topics, and I came across this one. I am curious as to what this word means, will someone please enlighten me?
• An asymptote is a line that a function approaches, but never quite touches. For example, the line y=0 is an asymptote to the function e^x, because as x becomes very large and negative, e^x gets arbitrarily close to 0, but is never actually 0.
• Is the limit of "(1/x)+sinx" as x approaches infinity equal to 0? Because technically, it gets closer and closer...
• Are you sure about that?

What is the value that sin(x) approaches as x→∞?

Is that consistent with your conclusion?

• I have a strange question.
Find the limit of (2x/x) as x approaches infinity.
As I interpret the question, as x approaches infinity, the expression becomes (2∞)/∞. Since two times infinity is equal to infinity, my answer will be (∞/∞), which evaluates to 1.
(1 vote)
• Infinity is not a number, so we cannot apply some of the typical math operations to it, such as simplifying ∞/∞ to 1. ∞/∞ is actually one of the indeterminate forms, so it could equal any non-negative number or infinity. The exact value depends on the specific problem. In this case, the indeterminate form is equal to 2.

To actually solve the limit of (2x)/x as x approaches infinity, just simplify the fraction. So, you would have the limit of 2 as x approaches infinity which is clearly equal to 2.
• I am getting confused about whether the limit will be negative or positive when x approves -∞. All my answers, if they are questions related to x approaching -∞, are going wrong. Sometimes the value of the limit is negative when it approaches -∞ and sometimes positive. Please help me with the confusion.
• can a function have a limit at infinity without having an horizontal asymptote ?
(1 vote)
• I don't think so. The whole point of a limit at infinity is that the function approaches a value, which is represented by a horizontal asymptote.
• how could you prove that the limit as n->infinity of 2^(n) = infinity?
• Is it possible for both vertical and horizontal asymptotes in the same graph, because in the graph shown in the video, we already know that there is a horizontal asymptote at x=2. But I also observed a vertical asymptote at y=1
(1 vote)
• Certainly. y=1/x has asymptotes at y=0 and x=0.
(1 vote)
• Is this correct:
1. Horizontal asymptotes just tell us what f(x) approaches as x approaches +/- infinity.
2. Vertical asymptotes just tell us where f(x) approaches +/-infinity.
(1 vote)
• How do we know that we're supposed to look at the limit for the horizontal asymptote and not the vertical asymptote?
(1 vote)
• If x approaches a finite value, you're usually looking at a case of vertical asymptotes. However, if the function is approaching a finite value, it's most likely a horizontal asymptote.
(1 vote)

## Video transcript

- [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.