About this video
Sal looks at two functions near x = 0. For 1/x^2, both sides grow larger and larger without bound in the positive direction, so the limit is called unbounded (and does not exist as a finite value). For 1/x, the left side heads to large negative values and the right to large positive values, so the limit simply does not exist.
You'll be able to
- describe what happens to 1/x^2 and 1/x as x approaches 0 from each side.
- explain what an unbounded limit is.
- explain why the limit of 1/x as x approaches 0 does not exist.
Key words
- unbounded
- Growing without any limit, getting larger and larger (or more and more negative).
- does not exist
- Said of a limit when the function doesn't approach one finite value.
- finite value
- An actual number, not infinity.
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.
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