WEBVTT

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- [Instructor] In a previous
video, we explored the graphs

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of Y equals one over X
squared and one over X.

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In a previous video we've
looked at these graphs.

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This is Y is equal to one over X squared.

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This is Y is equal to one over X.

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And we explored what's the limit

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as X approaches zero in
either of those scenarios.

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And in this left scenario we saw

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as X becomes less and less negative,

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as it approaches zero
from the left hand side,

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the value of one over
X squared is unbounded

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in the positive direction.

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And the same thing happens as
we approach X from the right,

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as we become less and less positive

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but we are still positive,

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the value of one over X squared becomes

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unbounded in the positive direction.

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So in that video, we just said, "Hey,

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"one could say that this
limit is unbounded."

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But what we're going
to do in this video is

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introduce new notation.

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Instead of just saying it's unbounded,

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we could say, "Hey, from
both the left and the right

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it looks like we're going
to positive infinity".

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So we can introduce
this notation of saying,

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"Hey, this is going to infinity",

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which you will sometimes see used.

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Some people would call this unbounded,

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some people say it does not exist

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because it's not approaching
some finite value,

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while some people will use this notation

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of the limit going to infinity.

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But what about this scenario?

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Can we use our new notation here?

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Well, when we approach zero from the left,

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it looks like we're unbounded
in the negative direction,

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and when we approach zero from the right,

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we are unbounded in
the positive direction.

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So, here you still could not say

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that the limit is approaching infinity

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because from the right
it's approaching infinity,

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but from the left it's
approaching negative infinity.

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So you would still say
that this does not exist.

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You could do one sided limits here,

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which if you're not familiar with,

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I encourage you to review
it on Khan Academy.

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If you said the limit of one over X

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as X approaches zero
from the left hand side,

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from values less than zero,

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well then you would look at
this right over here and say,

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"Well, look, it looks like we're going

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unbounded in the negative direction".

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So you would say this is
equal to negative infinity.

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And of course if you said the
limit as X approaches zero

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from the right of one over X, well here

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you're unbounded in the positive direction

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so that's going to be
equal to positive infinity.

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Let's do an example
problem from Khan Academy

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based on this idea and this notation.

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So here it says, consider
graphs A, B, and C.

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The dashed lines represent asymptotes.

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Which of the graphs agree
with this statement,

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that the limit as X approaches 1 of H of X

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is equal to infinity?

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Pause this video and see
if you can figure it out.

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Alright, let's go through each of these.

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So we want to think about
what happens at X equals one.

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So that's right over here on graph A.

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So as we approach X equals one,

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so let me write this, so the limit,

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let me do this for the different graphs.

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So, for graph A, the
limit as x approaches one

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from the left, that looks like

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it's unbounded in the positive direction.

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That equals infinity and the limit

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as X approaches one from the right,

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well that looks like it's
going to negative infinity.

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That equals negative infinity.

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And since these are going
in two different directions,

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you wouldn't be able to say that

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the limit as X approaches one

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from both directions is equal to infinity.

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So I would rule this one out.

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Now let's look at choice B.

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What's the limit as X
approaches one from the left?

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And of course these are of H of X.

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Gotta write that down.

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So, of H of X right over here.

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Well, as we approach from the left,

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looks like we're going
to positive infinity.

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And it looks like the limit of H of X

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as we approach one from the right is

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also going to positive infinity.

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And so, since we're
approaching you could say

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the same direction of infinity,
you could say this for B.

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So B meets the constraints, but

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let's just check C to make sure.

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Well, you can see very
clearly X equals one,

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that as we approach it from the left,

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we go to negative infinity,

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and as we approach from the right,

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we got to positive infinity.

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So this, once again,
would not be approaching

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the same infinity.

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So you would rule this one out, as well.