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Welcome to the presentation on limits.

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Let's get started with some-- well, first an explanation

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before I do any problems.

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So let's say I had-- let me make sure I have the right

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color and my pen works.

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OK, let's say I had the limit, and I'll explain what a

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limit is in a second.

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But the way you write it is you say the limit-- oh, my color is

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on the wrong-- OK, let me use the pen and yellow.

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OK, the limit as x approaches 2 of x squared.

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Now, all this is saying is what value does the expression x

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squared approach as x approaches 2?

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Well, this is pretty easy.

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If we look at-- let me at least draw a graph.

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I'll stay in this yellow color.

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So let me draw.

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x squared looks something like-- let me use

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a different color.

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x square looks something like this, right?

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And when x is equal to 2, y, or the expression-- because

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we don't say what this is equal to.

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It's just the expression-- x squared is equal to 4, right?

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So a limit is saying, as x approaches 2, as x approaches 2

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from both sides, from numbers left than 2 and from numbers

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right than 2, what does the expression approach?

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And you might, I think, already see where this is going and be

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wondering why we're even going to the trouble of learning this

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new concept because it seems pretty obvious, but as x-- as

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we get to x closer and closer to 2 from this direction, and

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as we get to x closer and closer to 2 to this

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direction, what does this expression equal?

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Well, it essentially equals 4, right?

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The expression is equal to 4.

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The way I think about it is as you move on the curve closer

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and closer to the expression's value, what does the

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expression equal?

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In this case, it equals 4.

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You're probably saying, Sal, this seems like a useless

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concept because I could have just stuck 2 in there, and I

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know that if this is-- say this is f of x, that if f of x is

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equal to x squared, that f of 2 is equal to 4, and that would

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have been a no-brainer.

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Well, let me maybe give you one wrinkle on that, and hopefully

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now you'll start to see what the use of a limit is.

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Let me to define-- let me say f of x is equal to x squared

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when, if x does not equal 2, and let's say it equals

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3 when x equals 2.

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Interesting.

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So it's a slight variation on this expression right here.

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So this is our new f of x.

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So let me ask you a question.

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What is-- my pen still works-- what is the limit-- I used

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cursive this time-- what is the limit as x-- that's an x--

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as x approaches 2 of f of x?

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That's an x.

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It says x approaches 2.

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It's just like that.

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I just-- I don't know.

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For some reason, my brain is working functionally.

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OK, so let me graph this now.

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So that's an equally neat-looking graph as

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the one I just drew.

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Let me draw.

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So now it's almost the same as this curve, except something

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interesting happens at x equals 2.

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So it's just like this.

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It's like an x squared curve like that.

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But at x equals 2 and f of x equals 4, we

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draw a little hole.

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We draw a hole because it's not defined at x equals 2.

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This is x equals 2.

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This is 2.

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This is 4.

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This is the f of x axis, of course.

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And when x is equal to 2-- let's say this is 3.

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When x is equal to 2, f of x is equal to 3.

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This is actually right below this.

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I should-- it doesn't look completely right below it,

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but I think you got to get the picture.

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See, this graph is x squared.

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It's exactly x squared until we get to x equals 2.

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At x equals 2, We have a grap-- No, not a grap.

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We have a gap in the graph, which maybe

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could be called a grap.

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We have a gap in the graph, and then we keep-- and then after x

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equals 2, we keep moving on.

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And that gap, and that gap is defined right here, what

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happens when x equals 2?

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Well, then f of x is equal to 3.

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So this graph kind of goes-- it's just like x squared, but

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instead of f of 2 being 4, f of 2 drops down to 3, but

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then we keep on going.

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So going back to the limit problem, what is the

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limit as x approaches 2?

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Now, well, let's think about the same thing.

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We're going to go-- this is how I visualize it.

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I go along the curve.

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Let me pick a different color.

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So as x approaches 2 from this side, from the left-hand side

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or from numbers less than 2, f of x is approaching values

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approaching 4, right? f of x is approaching 4 as x

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approaches 2, right?

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I think you see that.

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If you just follow along the curve, as you approach f of 2,

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you get closer and closer to 4.

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Similarly, as you go from the right-hand side-- make sure

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my thing's still working.

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As you go from the right-hand side, you go along the

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curve, and f of x is also slowly approaching 4.

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So, as you can see, as we go closer and closer and

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closer to x equals 2, f of whatever number that is

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approaches 4, right?

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So, in this case, the limit as x approaches

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2 is also equal to 4.

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Well, this is interesting because, in this case, the

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limit as x approaches 2 of f of x does not equal f of 2.

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Now, normally, this would be on this line.

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In this case, the limit as you approach the expression is

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equal to evaluating the expression of that value.

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In this case, the limit isn't.

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I think now you're starting to see why the limit is a slightly

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different concept than just evaluating the function at

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that point because you have functions where, for whatever

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reason at a certain point, either the function might not

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be defined or the function kind of jumps up or down, but as you

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approach that point, you still approach a value different than

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the function at that point.

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Now, that's my introduction.

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I think this will give you intuition for what a limit is.

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In another presentation, I'll give you the more formal

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mathematical, you know, the delta-epsilon

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definition of a limit.

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And actually, in the very next module, I'm now going to

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do a bunch of problems involving the limit.

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I think as you do more and more problems, you'll get more and

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more of an intuition as to what a limit is.

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And then as we go into drill derivatives and integrals,

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you'll actually understand why people probably even invented

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limits to begin with.

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We'll see you in the next presentation.

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