Introduction to Limits Introduction to the intuition behind limits
Introduction to Limits
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- Welcome to the presentation on limits.
- Let's get started with some-- well, first an explanation
- before I do any problems.
- So let's say I had-- let me make sure I have the right
- color and my pen works.
- OK, let's say I had the limit, and I'll explain what a
- limit is in a second.
- But the way you write it is you say the limit-- oh, my color is
- on the wrong-- OK, let me use the pen and yellow.
- OK, the limit as x approaches 2 of x squared.
- 13 00:00:42,55 --> 00:00:46,34 Now, all this is saying is what value does the expression x
- squared approach as x approaches 2?
- Well, this is pretty easy.
- If we look at-- let me at least draw a graph.
- I'll stay in this yellow color.
- So let me draw.
- x squared looks something like-- let me use
- a different color.
- x square looks something like this, right?
- 23 00:01:10,73 --> 00:01:20,04 And when x is equal to 2, y, or the expression-- because
- we don't say what this is equal to.
- It's just the expression-- x squared is equal to 4, right?
- 27 00:01:27,65 --> 00:01:33,1 So a limit is saying, as x approaches 2, as x approaches 2
- from both sides, from numbers left than 2 and from numbers
- right than 2, what does the expression approach?
- And you might, I think, already see where this is going and be
- wondering why we're even going to the trouble of learning this
- new concept because it seems pretty obvious, but as x-- as
- we get to x closer and closer to 2 from this direction, and
- as we get to x closer and closer to 2 to this
- direction, what does this expression equal?
- Well, it essentially equals 4, right?
- The expression is equal to 4.
- The way I think about it is as you move on the curve closer
- and closer to the expression's value, what does the
- expression equal?
- In this case, it equals 4.
- You're probably saying, Sal, this seems like a useless
- concept because I could have just stuck 2 in there, and I
- know that if this is-- say this is f of x, that if f of x is
- equal to x squared, that f of 2 is equal to 4, and that would
- have been a no-brainer.
- Well, let me maybe give you one wrinkle on that, and hopefully
- now you'll start to see what the use of a limit is.
- Let me to define-- let me say f of x is equal to x squared
- when, if x does not equal 2, and let's say it equals
- 3 when x equals 2.
- So it's a slight variation on this expression right here.
- So this is our new f of x.
- So let me ask you a question.
- What is-- my pen still works-- what is the limit-- I used
- cursive this time-- what is the limit as x-- that's an x--
- as x approaches 2 of f of x?
- 60 00:03:29,59 --> 00:03:30,21 That's an x.
- It says x approaches 2.
- It's just like that.
- OK, so let me graph this now.
- So that's an equally neat-looking graph as
- the one I just drew.
- Let me draw.
- So now it's almost the same as this curve, except something
- interesting happens at x equals 2.
- So it's just like this.
- It's like an x squared curve like that.
- But at x equals 2 and f of x equals 4, we
- draw a little hole.
- We draw a hole because it's not defined at x equals 2.
- This is x equals 2.
- This is 2.
- This is 4.
- This is the f of x axis, of course.
- And when x is equal to 2-- let's say this is 3.
- When x is equal to 2, f of x is equal to 3.
- This is actually right below this.
- I should-- it doesn't look completely right below it,
- but I think you got to get the picture.
- See, this graph is x squared.
- It's exactly x squared until we get to x equals 2.
- At x equals 2, We have a grap-- No, not a grap.
- We have a gap in the graph, which maybe
- could be called a grap.
- We have a gap in the graph, and then we keep-- and then after x
- equals 2, we keep moving on.
- And that gap, and that gap is defined right here, what
- happens when x equals 2?
- Well, then f of x is equal to 3.
- So this graph kind of goes-- it's just like x squared, but
- instead of f of 2 being 4, f of 2 drops down to 3, but
- then we keep on going.
- So going back to the limit problem, what is the
- limit as x approaches 2?
- Now, well, let's think about the same thing.
- We're going to go-- this is how I visualize it.
- So as x approaches 2 from this side, from the left-hand side
- or from numbers less than 2, f of x is approaching values
- approaching 4, right? f of x is approaching 4 as x
- approaches 2, right?
- I think you see that.
- If you just follow along the curve, as you approach f of 2,
- you get closer and closer to 4.
- Similarly, as you go from the right-hand side-- make sure
- my thing's still working.
- As you go from the right-hand side, you go along the
- curve, and f of x is also slowly approaching 4.
- So, as you can see, as we go closer and closer and
- closer to x equals 2, f of whatever number that is
- approaches 4, right?
- So, in this case, the limit as x approaches
- 2 is also equal to 4.
- Well, this is interesting because, in this case, the
- limit as x approaches 2 of f of x does not equal f of 2.
- Now, normally, this would be on this line.
- In this case, the limit as you approach the expression is
- equal to evaluating the expression of that value.
- In this case, the limit isn't.
- I think now you're starting to see why the limit is a slightly
- different concept than just evaluating the function at
- that point because you have functions where, for whatever
- reason at a certain point, either the function might not
- be defined or the function kind of jumps up or down, but as you
- approach that point, you still approach a value different than
- the function at that point.
- Now, that's my introduction.
- I think this will give you intuition for what a limit is.
- In another presentation, I'll give you the more formal
- mathematical, you know, the delta-epsilon
- definition of a limit.
- And actually, in the very next module, I'm now going to
- do a bunch of problems involving the limit.
- I think as you do more and more problems, you'll get more and
- more of an intuition as to what a limit is.
- And then as we go into drill derivatives and integrals,
- you'll actually understand why people probably even invented
- limits to begin with.
- We'll see you in the next presentation.
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