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## Infinite limits

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# Introduction to infinite limits

AP Calc: LIM‑2 (EU), LIM‑2.D (LO), LIM‑2.D.1 (EK), LIM‑2.D.2 (EK)

## Video transcript

- [Instructor] In a previous
video, we explored the graphs of Y equals one over X
squared and one over X. In a previous video we've
looked at these graphs. This is Y is equal to one over X squared. This is Y is equal to one over X. And we explored what's the limit as X approaches zero in
either of those scenarios. And in this left scenario we saw as X becomes less and less negative, as it approaches zero
from the left hand side, the value of one over
X squared is unbounded in the positive direction. And 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, and when we approach zero from the right, we are 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 one over X as X approaches zero
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 zero from the right of one 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. Alright, let's go through each of these. So we want to think about
what happens at X equals one. So that's right over here on graph A. So as we approach X equals one, so let me write this, so the limit, let me do this for the different graphs. So, for graph A, the
limit as x approaches one from the left, that looks like it's unbounded in the positive direction. That equals infinity and the limit as X approaches one from the right, well that looks like it's
going to negative infinity. That equals negative infinity. And since these are going
in two different directions, you wouldn't be able to say that the limit as X approaches one 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 one from the left? And of course these are of H of X. Gotta write that down. So, of H of X right over here. Well, as we approach from the left, looks like we're going
to positive infinity. And it looks like the limit of H of X as we approach one from the right is also going to positive infinity. And so, 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 X equals one, that as we approach it from the left, we go to negative infinity, and as we approach from the right, we got to positive infinity. So this, once again,
would not be approaching the same infinity. So you would rule this one out, as well.