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.

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."

I'm confused why the limit of the first problem would not be infinity.

because infinity is not something you can define

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.

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. 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.

Could +∞ be an answer to the first question?

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.

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.

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.

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AP®︎/College Calculus AB

Course: AP®︎/College Calculus AB > Unit 1

Lesson 3: Estimating limit values from graphs

Unbounded limits

Name: Unbounded limits
Uploaded: 2018-08-22T14:00:52Z
Description: 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.

Google Classroom

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.

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John McManus
Posted 5 years ago. Direct link to John McManus's post “If I'm not mistaken, in a...”
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.
Button navigates to signup pageComment on John McManus's post “If I'm not mistaken, in a...”
(20 votes)
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- kubleeka
  Posted 5 years ago. Direct link to kubleeka's post “'The limit does not exist...”
  '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.
  Comment on kubleeka's post “'The limit does not exist...”
  (35 votes)
jaden blaske
Posted 5 years ago. Direct link to jaden blaske's post “how do i simple this down...”
how do i simple this down?
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(3 votes)
Answer
- mathjunkie11
  Posted 5 years ago. Direct link to mathjunkie11's post “If the graph is approachi...”
  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.
  Button navigates to signup page
  (23 votes)
thierno bousso
Posted 5 years ago. Direct link to thierno bousso's post “what is the value of an u...”
what is the value of an unbounded limit? what is it equal to?
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(9 votes)
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- kubleeka
  Posted 5 years ago. Direct link to kubleeka's post “Nothing. If a function go...”
  Nothing. If a function goes to infinity or negative infinity at a point, then the limit at that point doesn't exist.
  Button navigates to signup page
  (9 votes)
Bilal Memon
Posted 5 years ago. Direct link to Bilal Memon's post “just curious but is there...”
just curious but is there a shortcut or notation to write 'limit does not exist' when writing the limit?
Button navigates to signup pageComment on Bilal Memon's post “just curious but is there...”
(7 votes)
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- A. Fox
  Posted 2 years ago. Direct link to A. Fox's post “3 years late, but yes. So...”
  3 years late, but yes. Sometimes people write "DNE" for "Does Not Exist."
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  (7 votes)
ZIJIAN CHEN
Posted 5 years ago. Direct link to ZIJIAN CHEN's post “Does unbounded situation ...”
Does unbounded situation counts as not exist?
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(5 votes)
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- kubleeka
  Posted 5 years ago. Direct link to kubleeka's post “Yes. A limit is a real nu...”
  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.
  Comment on kubleeka's post “Yes. A limit is a real nu...”
  (8 votes)
Ehdahnn
Posted 4 years ago. Direct link to Ehdahnn's post “I'm confused why the limi...”
I'm confused why the limit of the first problem would not be infinity.
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(7 votes)
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- 石乐志大师
  Posted 8 months ago. Direct link to 石乐志大师's post “because infinity is not s...”
  because infinity is not something you can define
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  (2 votes)
Anken Aoudia
Posted a year ago. Direct link to Anken Aoudia's post “what if we use " +∞ , -∞ ...”
what if we use " +∞ , -∞ " to represent the limit is this correct for the first example ?
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(4 votes)
Answer
- cossine
  Posted a year ago. Direct link to cossine's post “So there is an infinite l...”
  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.
  Comment on cossine's post “So there is an infinite l...”
  (5 votes)
oliverreiter507
Posted 4 months ago. Direct link to oliverreiter507's post “Could +∞ be an answer to ...”
Could +∞ be an answer to the first question?
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(4 votes)
Answer
- Venkata
  Posted 4 months ago. Direct link to Venkata's post “For this case, you could....”
  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.
  Button navigates to signup page
  (4 votes)
Mohamed Ibrahim
Posted 4 years ago. Direct link to Mohamed Ibrahim's post “The word unbounded is exa...”
The word unbounded is exactly the opposite of limited. Saying it has no limit makes more sense, don't you agree ?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- kubleeka
  Posted 4 years ago. Direct link to kubleeka's post “It is true that there is ...”
  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.
  Button navigates to signup page
  (5 votes)
MaspnW
Posted 5 years ago. Direct link to MaspnW's post “If the function of the gr...”
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.
Button navigates to signup pageComment on MaspnW's post “If the function of the gr...”
(3 votes)
Answer

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.