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OK, hopefully, my tool is working now.

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But anyway, so we were saying when x is equal to minus 0.001,

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so we're getting closer and closer to 0 from the negative

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side, f of x is equal to minus 1,000, right?

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You can just evaluate it yourself, right?

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And as you see, as x approaches 0 from the negative direction,

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we get larger and larger-- or I guess you could say smaller and

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smaller negative numbers, right?

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You get-- you know, if it's minus 0.0001, you'd get minus

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10,000, and then minus 100,000, and then minus 1 million, you

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could imagine the closer and closer you get to zero.

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Similarly, when you go from the other direction, when you say

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what is-- when x is 0.01, there you get positive 100, right?

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When x is point-- the thing is frozen again-- when it's 0.001,

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you get positive 1,000.

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So as you see, as you approach 0 from the negative direction,

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you get larger and larger negative values, or I guess

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smaller and smaller negative values.

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And as you go from the positive direction, you get larger

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and larger values.

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Let me graph this just to give you a sense of what this graph

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looks like because this is actually a good graph to know

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what it looks like just generally.

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So let's say I have the x-axis.

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This is the y-axis.

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Change my color.

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So when x is a negative number, as x gets really, really,

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really negative, as x is like negative infinity, this

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is approaching zero, but it's still going to be a

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slightly negative number.

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And then as we see from what we drew, as we approach x is equal

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to 0, we asymptote, and we approach negative

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infinity, right?

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And similarly, from positive numbers, if you go out to

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the right really far, it approaches 0, but it's

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still going to be positive.

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And as we gets closer and closer to 0, it spikes up, and

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it goes to positive infinity.

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You never quite get x is equal to 0.

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So in this situation, you actually have as x approaches--

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so let me give you a different notation, which you'll

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probably see eventually.

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I might actually do a separate presentation on this.

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The limit as x approaches 0 from the positive direction,

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that's this notation here, of 1/x, right?

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So this is as x approaches 0 from the positive direction,

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from the right-hand side, well, this is equal to infinity.

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And then the limit as x-- this pen, this pen-- the limit as x

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approaches 0 from the negative side of 1/x.

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This notation just says the limit as I approach

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from the negative side.

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So as I approach x equal 0 from this direction, right, from

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this direction, what happens?

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Well, that is equal to minus infinity.

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So since I'm approaching a different value when I

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approach from one side or the other, this limit

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is actually undefined.

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I mean, we could say that from the positive side, it's

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positive infinity, or from the negative side, it's negative

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infinity, but they have to equal the same thing for

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this limit to be defined.

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So this is equal to undefined.

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So let's do another problem, and I think this should

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be interesting now.

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So let's say, just keeping that last problem we had in mind,

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what's the limit as x approaches 0 of 1/x squared?

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So in this situation, I'll draw the graph.

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That's my x-axis.

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That's my y-axis.

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So here, no matter what value we put into x, we get a

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positive value, right?

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Because you're going to square it.

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If you put minus-- you could actually-- oh, let me do it.

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It'll be instructive, I think.

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Once again, obviously you can't just put x equal to 0.

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You'll get 1/0, which is undefined.

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But let's say 1 over x squared.

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What does 1 over x squared evaluate to?

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So when x is 0.1, 0.1 squared is 0.01, so 1/x is 100.

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Similarly, if I do minus 0.1, minus 0.1 squared is positive

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0.01, so then 1 over that is still 100, right?

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So regardless of whether we put a negative or positive number

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here, we get a positive value.

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And similarly, if I put-- if we say x is 0.01, if you evaluate

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it, you'll get 10,000, and if we put minus 0.01, you'll get

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positive 10,000 as well, right?

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Because we square it.

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So in this graph, if you were to draw it, and if you have a

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graphing calculator, you should experiment, it

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looks something like this.

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I can see this dark blue.

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So from the negative side, it approaches infinity, right?

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You can see that.

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As we get to smaller and smaller-- as we get closer and

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closer to 0 from the negative side, it approaches infinity.

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As we go from the positive side-- these are actually

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symmetric, although I didn't draw it that symmetric-- it

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also approaches infinity.

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So this is a case in which the limit-- oh, that's

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not too bright.

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I don't know if you can see -- the limit as x approaches 0

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from the negative side of 1 over x squared is equal to

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infinity, and the limit as x approaches 0 from the positive

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side of 1 over x squared is also equal to infinity.

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So when you go from the left-hand side, it

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equals infinity, right?

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It goes to infinity as you approach 0.

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And as you go from the right-hand side, it

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also goes to infinity.

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And so the limit in general is equal to infinity.

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And this is why I got excited when I first started

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learning limits.

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Because for the first time, infinity is a legitimate answer

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to your problem, which, I don't know, on some metaphysical

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level got me kind of excited.

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But anyway, I will do more problems in the next

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presentation because you can never do enough limit problems.

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And in a couple of presentations, I actually give

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you the formal, kind of rigorous mathematical

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definition of the limits.

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