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## Differential Calculus

### Course: Differential Calculus>Unit 1

Lesson 2: Estimating limits from graphs

# Unbounded limits

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.C (LO)
,
LIM‑1.C.2 (EK)
,
LIM‑1.C.4 (EK)
Introducing the notion of a limit that is unbounded. These limits don't exist in the strict sense, but we can still say something about them that makes clear how they behave.

## Want to join the conversation?

• If I'm not mistaken, in an earlier video, when a function was approaching positive infinity from both left and right, the narrator/instructor still said that, since the limit was "unbounded," he would say that the "limit does not exist." (I believe he actually wrote it out that way.) I would be less anxious if he stuck with that label in these situations, or at least began the earlier video by saying that these situations can be and are described in different ways. Changing the definitions just a few videos later is a bit confusing.
• 'The limit does not exist' and 'the limit is unbounded' are not quite the same thing. If a limit is unbounded, then it does not exist. But a limit may not exist, and still be bounded, e.g. if we have a jump discontinuity.
• how do i simple this down?
• If the graph is approaching the same value from opposite directions, there is a limit. If the limit the graph is approaching is infinity, the limit is unbounded. A limit does not exist if the graph is approaching a different value from opposite directions.
• what is the value of an unbounded limit? what is it equal to?
• Nothing. If a function goes to infinity or negative infinity at a point, then the limit at that point doesn't exist.
• just curious but is there a shortcut or notation to write 'limit does not exist' when writing the limit?
• 3 years late, but yes. Sometimes people write "DNE" for "Does Not Exist."
• Does unbounded situation counts as not exist?
• Yes. A limit is a real number that satisfies the ε-δ definition. Because infinity is not a real number, the limit doesn't exist when the function is unbounded.
• I'm confused why the limit of the first problem would not be infinity.
• because infinity is not something you can define
(1 vote)
• what if we use " +∞ , -∞ " to represent the limit is this correct for the first example ?
• So there is an infinite limit definition. Using that yes. (positive infinity)

I think the main goal would be to describe the behaviour of a function as much as possible. In the video, Sal is using the standard epsilon delta definition of limits which if used would result in the answer undefined.

So the definition he is using will not describe the function behaviour in detail but he makes note of that in the video.

I don't think it meaningful to use the standard epsilon delta definition in this video. Limits are essentially are combinations of definition, standard epsilon delta, infinite limits, limits at infinity, one-sided limits.

From my experience it has been most common in mathematics to use limit definition that describe the function in most detail. Hence it is best to use the infinite limit definition in this scenario.
• The word unbounded is exactly the opposite of limited. Saying it has no limit makes more sense, don't you agree ?
• It is true that there is not limit when the function is unbounded. However, there are cases where a function can be bounded, but still have no limit, like the limit as x goes to 0 of sin(1/x).

So by saying 'unbounded', we are conveying not only that the limit doesn't exist, but the the function exhibits a certain behavior. It carries more information.
• If the function of the graph moved the picture, would it then have a limit? Like if the graph goes to infinity and approaches 4.
• In , can I use the answer Undefined instead of DNE(does not exist)? Is it similar?
(1 vote)
• Unless told otherwise those two answers can be used interchangeably.

## Video transcript

- [Instructor] So right over here, we have the graph of y is equal to one over x squared. And my question to you is what is the limit of one over x squared as x approaches zero? Pause this video, and see if you can figure that out. Well, when you try to figure it out, you immediately see something interesting happening at x equals zero. The closer we get to zero from the left, you take one over x squared, it just gets larger and larger and larger. It doesn't approach some finite value. It's unbounded, has no bound. And the same thing is happening as we approach from the right. As we get values closer and closer to zero from the right, we get larger and larger values for one over x squared without bound. So terminology that folks will sometimes use, where they're both going in the same direction, but it's unbounded, is they'll say this limit is unbounded. In some context, you might hear teachers say that this limit does not exist or, and it definitely does not exist if you're thinking about approaching a finite value. In future videos, we'll start to introduce ideas of infinity and notations around limits and infinity, where we can get a little bit more specific about what type of limit this is. But with that out of the way, let's look at another scenario. This right over here, you might recognize as the graph of y is equal to one over x. So I'm going to ask you the same question. Pause this video, and think about what's the limit of one over x as x approaches zero? Pause this video, and figure it out. All right, so here, when we approach from the left, we get more and more and more negative values. While we, when we approach from the right, we're getting more and more positive values. So in this situation, where we're not getting unbounded in the same direction, the previous example, we were both, we were being unbounded in the positive direction. But here, on the, from the left, we're getting unbounded in the negative direction. While from the right, we're getting unbounded in the positive direction. And so when you're thinking about the limit as you approach a point, if it's not even approaching the same value or even the same direction, you would just clearly say that this limit does not exist, does not exist. So this is a situation, where you would not even say that this is an unbounded limit or that the limit is unbounded. Because you're going in two different directions when you approach from the right and when you approach from the left, you would just clearly say does not exist.