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## AP®︎/College Calculus AB

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

Lesson 3: Estimating limit values from graphs- Estimating limit values from graphs
- Unbounded limits
- Estimating limit values from graphs
- Estimating limit values from graphs
- One-sided limits from graphs
- One-sided limits from graphs: asymptote
- One-sided limits from graphs
- Connecting limits and graphical behavior
- Connecting limits and graphical behavior

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# Unbounded limits

This video discusses estimating limit values from graphs, focusing on two functions: y = 1/x² and y = 1/x. For y = 1/x², the limit is unbounded as x approaches 0, since the function increases without bound. For y = 1/x, the limit doesn't exist as x approaches 0, since it's unbounded in opposite directions.

## 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.(20 votes)
- '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.(35 votes)

- how do i simple this down?(3 votes)
- 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.(23 votes)

- what is the value of an unbounded limit? what is it equal to?(9 votes)
- Nothing. If a function goes to infinity or negative infinity at a point, then the limit at that point doesn't exist.(9 votes)

- just curious but is there a shortcut or notation to write 'limit does not exist' when writing the limit?(7 votes)
- 3 years late, but yes. Sometimes people write "DNE" for "Does Not Exist."(7 votes)

- Does unbounded situation counts as not exist?(5 votes)
- 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.(8 votes)

- I'm confused why the limit of the first problem would not be infinity.(7 votes)
- because infinity is not something you can define(2 votes)

- what if we use " +∞ , -∞ " to represent the limit is this correct for the first example ?(4 votes)
- 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.

It is not 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.

It is by convention in mathematics to use limit definition that describe the function in the most detail. Hence it is best to use the infinite limit definition in this scenario.(5 votes)

- Could +∞ be an answer to the first question?(4 votes)
- For this case, you could. But, know that unbounded limit does not always equal an infinite limit.

For example, for $f(x) = xsin(x)$, as $x$ tends to infinity, $f(x)$ is unbounded, but doesn't tend to infinity. It oscillates between multiple values.(4 votes)

- The word unbounded is exactly the opposite of limited. Saying it has no limit makes more sense, don't you agree ?(3 votes)
- 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.(5 votes)

- 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.(3 votes)

## 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.