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

Lesson 14: Connecting infinite limits and vertical asymptotes

# Introduction to infinite limits

Introducing the notation of infinite limits.

## Want to join the conversation?

• Just curious, but is there a such thing as the opposite of infinity? What I mean is, since infinity is the notion of an incomprehensibly large number that doesn't follow the rules of arithmetic, is there a such thing an an infinitely tiny, minute, incomprehensibly small number?
(9 votes)
• Yes, there are systems with a reciprocal of infinity, such the hyperreal numbers and the surreal numbers.

The hyperreals are basically the reals with 'infinte' and 'infinitesimal' elements appended. The surreals have a more complicated construction, which is walked through in Donald Knuth's novel 'Surreal Numbers'.
(12 votes)
• how do you know where the line is coming from and where it is going?
(7 votes)
• In these scenarios, you can't just plug in the value because the values approach an asymptote. If the line is coming from the negative side/left and plunging down rather than showing a value, it is going infinitely down, or to negative infinity. Same thing goes for the other side.
(0 votes)
• if i say that the limit is infinity and not unbounded , does it help me in math problems ?
(4 votes)
• There are many forms of indeterminate forms and they include forms of infinity. Further we will encounter problems having lim of x tends to infinity where we will have to work ourselves out.
Thus, for the starters, most of the teachers prefer to avoid the concept of infinity, but the concept itself is very versatile in my opinion.
(5 votes)
• One could still say a limit is unbounded or does not exist for a function that grows without bound in a positive direction on one side of an asymptote and a negative direction on the other side, correct?

So am I right in thinking this notation just gives us a more nuanced way to talk about unbounded one-sided limits and limits that grow without bound in the same direction on both sides of an asymptote?
(5 votes)
• Can't we say $\lim_{x \to 0} x^{-1} = \pm \infty$? Is there some shorthand mathy notation for “limit does not exist/is unbounded”?
(2 votes)
• First of all, there's unfortunately no LaTeX here on KA.
"Does not exist" is either ¬∃ (\neg\exists), ~∃, or exists with a slash through it (sorry I couldn't find the symbol, but it's \nexists). Unbounded would just be written out as infinity or the text "is unbounded".
However, in this case, you cannot say that the limit is unbounded. It simply does not exist. If the left hand limit does not equal the right hand limit, or the limit oscillates between two values, you can only say that it is nonexistent.
Let me know if this helps.
(3 votes)
• Didn't we have a previous lesson where evaluating these exact situations as infinity wasn't allowed? Maybe I'm thinking of an IRL class I took, but I remember going on at length about why we wouldn't use infinity as the answer to such questions.
(2 votes)
• Technically, yeah, saying the limit is infinity is not really correct. Infinity is really just a shorthand notation for a much more rigorous definition of the limit, which you'll learn later as the "epsilon-delta (ε-δ) definition" (or in this case, the M-δ definition. Why M? It's because ε and δ are incredibly small numbers, while M here is supposed to be considerably big. So, we changed notations). I'll provide the rigorous definition below.

If I have lim (x-->a) f(x) = infinity, this, more rigorously, means that for some M > 0, there exists a δ > 0 exists such that |f(x)| ≥ M implies |x-a| ≤ δ. If I need to tone down the math lingo here, it basically says that no matter how big you make your value of M, there will always be a value of x, which will be δ units away from a (either to the left or right of a. hence, the absolute value is used), which will give an f(x) value greater than M.

This probably won't make much sense now, but once you learn about epsilon-delta, it'll start to make more sense.
(2 votes)
• What is the limit of x to the xth power as x approaches 0?
lim_(𝑥→0)⁡〖𝑥^𝑥 〗
(2 votes)
• Technically the limit does not exist as the limit from the left side is not well-defined. However, the limit of x^x as x approaches 0 from the right is 1.
(2 votes)
• When Sal mentions negative infinity vs. positive infinity, I understand we are talking about the direction of f(x) as x increases or decreases. However, there is no such thing as negative infinity. Infinity is infinity, so if it is negative or positive in direction why does that matter? Mathematically, it should be the same, right?
(2 votes)
• No, it is not the same. Negative infinity is when a number gets infinitely negative (like -1, -2, -3, -4...) and positive infinity is when a number gets infinitely positive (1, 2, 3, 4...). As you can see, they are not the same. If a function approaches positive infinity, this means that it goes, colloquially speaking, "up". If the function approaches negative infinity, it goes "down". Hope this helps!
(2 votes)
• Does the infinity sign without a plus or minus sign before it automatically mean positive infinity? because I thought that an infinity sign without a plus or minus meant infinity from both sides.
(1 vote)
• In this case, an infinity without a sign is assumed to be positive. Rather than the formal definition of infinity (all numbers), ∞ and -∞ are symbols that represent all numbers in the positive direction and all numbers in the negative direction
(3 votes)
• is it required to put plus sign if it is positive infinity? cause I always see people add it.
(1 vote)
• Like the real numbers, infinity is assumed to be positive unless a negative sign is added before it.
(2 votes)

## Video transcript

- [Instructor] In a previous video, we explored the graphs of Y equals one over X squared and one over X. In a previous video we've looked at these graphs. This is Y is equal to one over X squared. This is Y is equal to one over X. And we explored what's the limit as X approaches zero in either of those scenarios. And in this left scenario we saw as X becomes less and less negative, as it approaches zero from the left hand side, the value of one over X squared is unbounded in the positive direction. And the same thing happens as we approach X from the right, as we become less and less positive but we are still positive, the value of one over X squared becomes unbounded in the positive direction. So in that video, we just said, "Hey, "one could say that this limit is unbounded." But what we're going to do in this video is introduce new notation. Instead of just saying it's unbounded, we could say, "Hey, from both the left and the right it looks like we're going to positive infinity". So we can introduce this notation of saying, "Hey, this is going to infinity", which you will sometimes see used. Some people would call this unbounded, some people say it does not exist because it's not approaching some finite value, while some people will use this notation of the limit going to infinity. But what about this scenario? Can we use our new notation here? Well, when we approach zero from the left, it looks like we're unbounded in the negative direction, and when we approach zero from the right, we are unbounded in the positive direction. So, here you still could not say that the limit is approaching infinity because from the right it's approaching infinity, but from the left it's approaching negative infinity. So you would still say that this does not exist. You could do one sided limits here, which if you're not familiar with, I encourage you to review it on Khan Academy. If you said the limit of one over X as X approaches zero from the left hand side, from values less than zero, well then you would look at this right over here and say, "Well, look, it looks like we're going unbounded in the negative direction". So you would say this is equal to negative infinity. And of course if you said the limit as X approaches zero from the right of one over X, well here you're unbounded in the positive direction so that's going to be equal to positive infinity. Let's do an example problem from Khan Academy based on this idea and this notation. So here it says, consider graphs A, B, and C. The dashed lines represent asymptotes. Which of the graphs agree with this statement, that the limit as X approaches 1 of H of X is equal to infinity? Pause this video and see if you can figure it out. Alright, let's go through each of these. So we want to think about what happens at X equals one. So that's right over here on graph A. So as we approach X equals one, so let me write this, so the limit, let me do this for the different graphs. So, for graph A, the limit as x approaches one from the left, that looks like it's unbounded in the positive direction. That equals infinity and the limit as X approaches one from the right, well that looks like it's going to negative infinity. That equals negative infinity. And since these are going in two different directions, you wouldn't be able to say that the limit as X approaches one from both directions is equal to infinity. So I would rule this one out. Now let's look at choice B. What's the limit as X approaches one from the left? And of course these are of H of X. Gotta write that down. So, of H of X right over here. Well, as we approach from the left, looks like we're going to positive infinity. And it looks like the limit of H of X as we approach one from the right is also going to positive infinity. And so, since we're approaching you could say the same direction of infinity, you could say this for B. So B meets the constraints, but let's just check C to make sure. Well, you can see very clearly X equals one, that as we approach it from the left, we go to negative infinity, and as we approach from the right, we got to positive infinity. So this, once again, would not be approaching the same infinity. So you would rule this one out, as well.