Unbounded limits | Limits and continuity | AP Calculus AB | Khan Academy

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.

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 contexts, you might hear teachers say that this limit does not exist, 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 is the graph of y is equal to 1 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 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, 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.