Introduction to infinite limits | Limits and continuity | AP Calculus AB | Khan Academy

In a previous video, we've looked at these graphs. This is y equal to one over x squared. This is y is equal to one over x. We explored what's the limit as x approaches zero in either of those scenarios.

In this left scenario, we saw as x becomes less and less negative, as it approaches 0 from the left-hand side, the value of 1 over x squared is unbounded in the positive direction. The same thing happens as we approach x from the right; as we become less and less positive, but we are still positive, the value of one over x squared becomes unbounded in the positive direction.

So in that video, we just said, "Hey, one could say that this limit is unbounded." But what we're going to do in this video is introduce new notation. Instead of just saying it's unbounded, we could say, "Hey, from both the left and the right, it looks like we're going to positive infinity."

So we can introduce this notation of saying, "Hey, this is going to infinity," which you will sometimes see used. Some people would call this unbounded; some people say it does not exist because it's not approaching some finite value, while some people will use this notation of the limit going to infinity.

But what about this scenario? Can we use our new notation here? Well, when we approach zero from the left, it looks like we're unbounded in the negative direction. When we approach zero from the right, we're unbounded in the positive direction.

So here, you still could not say that the limit is approaching infinity because from the right it's approaching infinity, but from the left, it's approaching negative infinity. So you would still say that this does not exist. You could do one-sided limits here, which, if you're not familiar with, I encourage you to review it on Khan Academy.

If you said the limit of 1 over x as x approaches 0 from the left-hand side, from values less than zero, well then you would look at this right over here and say, "Well, look, it looks like we're going unbounded in the negative direction." So you would say this is equal to negative infinity.

And of course, if you said the limit as x approaches 0 from the right of 1 over x, well here you're unbounded in the positive direction, so that's going to be equal to positive infinity. Let's do an example problem from Khan Academy based on this idea and this notation.

So here it says, "Consider graphs A, B, and C. The dashed lines represent asymptotes. Which of the graphs agree with this statement that the limit as x approaches 1 of h of x is equal to infinity?" Pause this video and see if you can figure it out.

All right, let's go through each of these. So we want to think about what happens at x equals 1. So that's right over here on graph A. As we approach x equals 1, let me write this so the limit—let me do this for the different graphs.

For graph A, the limit as x approaches 1 from the left looks like it's unbounded in the positive direction; that equals infinity. The limit as x approaches 1 from the right? Well, that looks like it's going to negative infinity; that equals negative infinity. Since these are going in two different directions, you wouldn't be able to say that the limit as x approaches 1 from both directions is equal to infinity, so I would rule this one out.

Now, let's look at choice B. What's the limit as x approaches 1 from the left, and of course, these are of h of x—got to write that down. So of h of x right over here: well, as we approach from the left, it looks like we're going to positive infinity. It looks like the limit of h of x as we approach 1 from the right is also going to positive infinity.

Since we're approaching, you could say, the same direction of infinity, you could say this for B, so B meets the constraints. But let's just check C to make sure. Well, you can see very clearly at x equals 1 that as we approach it from the left, we go to negative infinity, and as we approach from the right, we go to positive infinity.

So this once again would not be approaching the same infinity, so you would rule this one out as well.