Introduction to limits Introduction to limits
Introduction to limits
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- In this video I want to familiarize you with the idea of a limit, which is a super important idea.
- It's really the idea that all of calculus is based upon.
- But despite being so super important, it's actually a really really simple idea.
- So let me draw a function here - actually, let me define a function
- here. A kind of a simple function. So let's define f(x) - let's say that f(x) is going to be (x-1)/(x-1).
- And you might say, "Hey Sal, look, I have the same thing in the numerator and the denominator.
- If I have something divided by itself, that would just be equal to one! Can't I just simplify this to f(x)=1?"
- And I would say, "Well, you're almost true, the difference between f(x)=1 and this thing right over
- here is that this thing is undefined when x=1. So if you set - let me write it over here - if you have
- f(1), what happens? In the numerator, you get (1-1), which is... let me just write it down...
- in the numerator you get 0, and in the denominator you get (1-1), which is also 0. And so anything divided
- by 0, including 0/0, this is undefined. So you can make the simplification - you can say that this is
- the same thing as f(x)=1, but you would have to add the constraint that x cannot be equal to 1. Now this
- and this are equivalent. Both of these are going to be equal to 1, for all other x'es other than 1. But
- at x=1, it becomes undefined. This is undefined and this one's undefined. So how would I graph this function?
- So let me graph it... That is my y=f(x) axis, and then this over here is my x-axis, and then let's say
- this is the point x=1, this over here would be x=-1, this is y=1, right up there I can do -1 but that
- doesn't do much relative to this function right over here, and and let me graph it. So it's essentially for
- any x other than 1, f(x)=1. So it's gonna look like this... except at 1. At 1, f(x) is undefined, so
- I'm gonna put a little bit of a gap right over here, this circle, to signify that this function
- is not defined - we don't know what this function equals at 1, we never defined it.
- This definition of the function doesn't tell us what to do at 1 - it's literally undefined when x=1.
- So this is the function right over here, and so once again, if someone were to ask you what is f(1), you'd go...
- and let's say, well this was a function definition, you would go x=1. Oh wait, there is a gap in my function
- over here, it is undefined. So let me write it again... well, it's kind of redundant but I'll rewrite it.
- f(1) is undefined. But what if I were to ask you, what is the function approaching
- as x=1? And now, this is starting to touch on the idea of a limit. So as x gets closer and closer to 1...
- what is the function approaching? Well this entire time, what is it getting closer and closer to?
- On the left hand side, no matter how close you get to 1, as long as you're not at 1, f(x)=1.
- Over here from the right hand side, you get the same thing. So you could say - and you'll get
- more and more familiar with this idea as we do more examples - that the limit as
- x (and lim, short for limit) - as x approaches 1 of f(x) is equal to...
- As we get closer we can get unbelievably, infinitely close to 1 as long as we're not at 1...
- And our function is going to be equal to 1, it's getting closer and closer to 1,
- it's actually at 1 the entire time. So in this case, we can say the limit as x approaches 1 of f(x)
- is 1. So once again, has very fancy notation, we're just saying, "Look, what is the function approaching
- as x gets closer and closer to 1?"
- Let me do another example where we're dealing with a curve, just so that you have the general idea.
- So let's say that I have the function f(x) - let me, just for the sake of variety, let me call it g(x).
- Let's say that we have g(x) is equal to - I can define it this way, we can define it as x²
- when x does not equal 2, and let's say that when x=2, it is equal to 1. So once again, kind of an interesting
- function that - as you'll see - is not fully continuous. It has a discontinuity. Let me graph it.
- So this my y=f(x) axis, this is my x-axis right over here. Let's say this is x=1, this is x=2,
- this is -1, this is -2... So everywhere except x=2, it's equal to x². So let me draw it like this,
- this is gonna be a parabola, it looks something like this... It's gonna look something...
- Let me draw a better version of the parabola. So it looks something like this, not the most beautifully
- drawn parabola in the history of drawing parabolas, but I think it will give you the idea of what a parabola
- looks like, hopefully. It should be symmetric... Let me redraw it, because that's kinda ugly.
- That's looking better, okay, alright, there you go. Alright.
- Now, this should be the graph of just x², but it's not x² when x=2. So once again, when x=2,
- we should have a little bit of a discontinuity here, so I'll draw a gap right over there,
- because when x=2, the function is equal to 1.
- I'm not doing them on the same scale... On the graph of f(x)=x² this would be 4, this would be 2,
- this would be 1, this would be 3. So, x=2, our function is equal to 1.
- So this is a bit of a bizarre function, but you can define it this way, you can define a function however
- you like to define it! And so, notice, it's just like the graph of f(x)=x² except when you get to 2,
- it has this gap, because you don't use the "g(x)=x² when x=2", you use "g(x)=1".
- If I've been saying f(x), I apologize for that.
- You use g(x)=1, so then just exactly at 2, it drops down to 1, and then it keeps going along x².
- So there is a couple of things. If I were to just evaluate the function - g(2),
- well you look at this definition. Okay, when x=2, I use this situation right over here,
- and it tells me it's going to be equal to 1. Let me ask a more interesting question, or perhaps a more
- interesting question. What is the limit as x approaches 2 of g(x)? Once again, fancy notation, but
- it's asking something pretty pretty simple. It's saying "as x gets closer and closer to 2...
- as you get closer and closer - and this isn't a rigorous definition, we'll do that in future videos -
- as x gets closer and closer to 2, what is g(x) approaching? So if you get to 1.9, and then 1.999, and then 1.999999
- and then 1.9999999, what is g(x) approaching? If you were to go from the positive direction,
- if you were to say 2.1, what's g(2.1)? What's g(2.01)? What's g(2.001)?
- What is that approaching as we get closer and closer to it?
- And you can see it visually just by drawing the graph. As g gets closer and closer to 2...
- And if we were to follow it along the graph, we see that we're approaching 4,
- even though that's not where the function is - the function drops down to 1 - the limit of g(x) as
- x approaches 2 is equal to 4. You can even do this numerically using a calculator.
- And let me do that, because I think that will be interesting. So let me get a calculator out...
- Let me get my trusty TI-85 out... So here is my calculator... And you can numerically say,
- okay, what's it gonna approach as you approach x=2? So let's try 1.9. For x=1.9, you would use this
- top clause, right over here. So you'd have 1.9², and so you would get 3.61.
- Well, what if you get even closer to 2? So 1.99, and once again let me square that,
- well I'm at 3.96. What if I do 1.999 and I square that?
- I'm gonna get 3.996. Notice, I'm getting closer and closer and closer to our point.
- If I got really close - 1.999999999999²? What am I gonna get to? It's not actually going to be
- exactly 4 - this calculator just rounded things up - because we're gonna get to a number really really
- really really close to 4. And we can do something from the positive direction, too, and it actually
- has to be the same number when we approach from the below, what we're trying to approach,
- and above what we're trying to approach. So if we try 2.1², we get 4.4...
- Let me go a couple of steps ahead...
- 2.0001². So this is much closer to 2 now. Now we're getting much closer to 4.
- So the closer we get to 2, the closer it seems like we're getting to 4.
- So once again that's a numeric way of seeing that the limit as x approaches 2 from either direction
- of g(x) - even though right at 2, the function is equal to 1, because it's discontinuous -
- the limit as we're approaching 2, we're getting closer and closer and closer to 4.
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