WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about the idea of a limit.
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This lesson marks our entry into an entirely new section of mathematics; we are entering calculus territory.
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From here on, this course will preview some of the topics that you will be exposed to in a calculus class.
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You might wonder what calculus is and why it matters.
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In short, calculus is a new way to look at functions.
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It gives us new tools to analyze function behavior and see how they relate to one another and the world around us.
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As to why it matters: calculus is crucially important to science and engineering.
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If you want to get anything of any real depth done in those fields, you need calculus in your tool belt.
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Plus, I think it is just really cool; it is one of these really cool things that you can do in mathematics.
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It is a really new, interesting idea, and we get to play around with it and do all sorts of cool stuff.
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And it creates...from here one, we get to see a whole bunch of new stuff that lets us explain a whole bunch of phenomena in the real world--really cool stuff.
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For the next few lessons, we will focus on limits.
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Limits allow us to describe functions in ways that were previously impossible for us to describe.
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We will be able to approach the infinite and the infinitesimal (that being the infinitely small) things with them.
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And they make up the heart of calculus; they are pretty much the foundation that the rest of calculus rests upon.
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So, it is really useful to have a good understanding of what a limit is and how it works,
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before we can start moving into other ideas in calculus that have more direct applicability to the daily world.
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All right, let's start with a motivating example to get things started.
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Consider the function f(x) = x/x: what happens if we try to look at f(0)?
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Well, if we plug 0 in, then we are going to get f(0) = 0/0; but since dividing by 0 is not defined, f(0) does not exist.
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Therefore, we cannot evaluate the value of the function at 0; f(0) is not a thing.
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We can't do anything with it, and that is the end of the story--that is it.
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And up until now, that would have been the end of the story.
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But now, when we look at f(0), well, OK, it is kind of the end of the story on what f(0) is.
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But before we entirely refuse the idea of considering f at 0, considering how f and 0 interact, let's look at the function's graph.
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If we look at f(x) = x/x, look at that: the function is always 1.
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Now, right here at x = 0 on our vertical y-axis, we have this hole here.
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That circular hole tells us that it does not actually exist there.
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But anywhere that isn't x = 0, we end up getting a 1 out of it.
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On the one hand, f(0) does not exist; you plug in a 0; you get 0/0; that is bad.
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You are not allowed to divide by 0, so we say that f(0) does not exist.
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But on the other hand, it is obvious where it is headed: look, the thing is going right in towards that 1.
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On the one hand, sure, it doesn't exist; but it totally should be a 1.
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So, we cannot say f(0) = 1, because f(0) does not exist, remember.
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But we still want some way to talk about where it was headed.
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It was clearly going to be a 1 before those pesky rules about dividing by 0 got in the way and stopped us
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from being able to figure out what the real answer should have been--
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what we feel like it was going towards, at least--perhaps not the real answer, because...
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we will talk about it; you will see other things about why we can't really assign a direct value to it.
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But we can talk about this idea of where it was headed; and we talk about that with a limit.
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With this idea in mind, we want to think of a limit as where it was headed.
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A limit is the vertical location that a function is headed towards--what it is going to be at as it gets closer and closer to some horizontal location.
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In that previous example, our motivating one of x/x, we had that, as it got closer and closer to 0, it got closer and closer to being at a height of 1.
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In fact, it was always at a height of 1, no matter where it was.
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We end up seeing this sense of...as it gets closer to 0, it is really, really close to this value, where it is headed towards.
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Equivalently, a limit is what the function output is going to be as we approach some input.
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As we get close to some input, what output seems like it is going to come out of it--what should it be; what is it going to be?
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We are going to be talking about this idea a lot, so we want some notation to describe it.
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We use this notation right here: this part here, the "lim," says that it is the limit we are working with.
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The x arrow c says what we are going towards; and the f(x) is the thing that actually is having the limit be applied to it; that is the function.
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This says "the limit of f(x) as x approaches c"; so as x gets close to c, what happens to f(x).
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You might also hear this spoken aloud as "the limit as x goes to c of f(x)"; I tend to say that a lot.
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Or you might hear some similar variant; in any case, the idea is the same.
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It is this question of what f(x) does as x gets very close to c.
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x is going to c, and our question is, "What will f(x) do in response to x going to c?"
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Let's test out our new idea on an old friend, f(x) = x², the good old parabola.
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Consider if we wanted to find what value x² approaches as x approaches 2.
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As x gets closer and closer to 2, what does x² become?
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Well, if we graph f(x) = x², we see that the answer is exactly what we would expect.
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2: as we get really close to a 2, we end up getting really close to a 4.
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As we come in from the left side, we can see that the value we are getting to, as we get closer and closer
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to a 2 from the left side--our vertical height gets closer and closer to a 4.
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Similarly, if we come in from the right side, as we get closer and closer to 2, and x = 2, from the right side,
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on our horizontal location, we get closer and closer to a vertical location of 4.
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We end up getting 4 as the limit: as x approaches 2, x² gets really close to 4.
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It becomes 4 as x approaches infinitely close to 2.
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So, as x gets closer and closer to a value of 2, f(x) gets closer and closer to a value of 4.
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And that is why we end up getting this limit.
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Now, to expand our ideas, let's modify the function and see what happens.
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Consider the piecewise function g(x), which equals x² when x is not equal to 2
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(that is most of our parabola right here) and 1 when x equals 2.
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At the specific value of x = 2, as opposed to following our normal parabolic arc, we end up putting out just a value of 1.
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This is a piecewise function; if this is not sounding familiar, and you have no idea how to interpret this sort of thing,
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go back to the lesson...we saw this a long time ago, near the beginning of this course...the lesson on Piecewise Functions.
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They will show up a lot, especially in the beginning of calculus.
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They are an important thing to have in your understanding.
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So, if this doesn't make any sense, go and check out piecewise functions from the early part of the course, when we were studying functions.
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All right, with this idea in mind, let's ask what the limit is, as x approaches 2, of g(x).
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Well, what we want to know--we might be tempted, first, to say, "Look, it is right here; that is what x is at 2."
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Yes, that is what g(x) is at 2; g(2) is 1.
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Well, yes, but we can also think about it as what we are getting close to.
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Well, as we get close to x = 2 from the left side, we see that we are getting really, really close to what height?--the height of 4.
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As we approach from the right side, we see that we are getting really, really close to what height?--the height of 4.
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So, our first automatic response might be to think that it has to be here, because that is what it is at x = 2.
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But the limit isn't about where you actually are; as a limit, x/x, as x goes to 0, produces 1,
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even though actually the function just fails to produce anything.
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A limit is about where you are headed towards, not what actually happens at that location.
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We don't really care about what happens here; this part isn't important.
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The thing that actually happens at 2 isn't important; it is a question of what happens on our way to 2.
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Well, on our way to 2, the thing that we are getting close to is this location at 4.
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That is the location we are getting to; so the answer for this limit will be 4.
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So, g(2) is not equal to 4; g(2) is equal to 1, because at x = 2, we just put out 1 for that function.
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But as x approaches 2, the value of g(x) approaches 4.
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g(x) jumps; it does this sudden leap off of the parabolic arc, only at x = 2; it only swaps at x = 2.
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What it seems to do, up until that moment, is behave like x², because for everything that isn't x = 2,
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for everything other than x = 2, it behaves just like x².
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Up until that moment, it is behaving just like x²; and because of this, it has the above limit of becoming a 4.
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It is going towards a 4 until that single moment where all of a sudden, when it actually touches the 2, it jumps away.
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But up until that moment, it seems like it is going there; so that ends up being our location.
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The vertical height that it seems to be going to as x approaches 2 is 4.
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In a way, we can visualize what is going on by doing the following:
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Begin by graphing the functions normally; we graph f(x) normally, x/x...oops, that should not say x/x;
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that should be x²; I'm sorry about that; that should be f(x) = x².
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And over here, g(x) = x² when x is not equal to 2, 1 when x equals 2.
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We have this single part that jumps away--this single point that is not on the normal curve.
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Then, what we can do is end up covering up the part that we are going to.
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Notice: in this case, what we are about to consider is the limit, as x approaches 2,
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as we get close to this value, as we get really close to this on both of them.
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With that idea in mind (I'm going to swap this back out; it should be an x²; we are looking at the limit as x goes to 2),
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as we take the limit, what we do is cover up the horizontal location that x is approaching,
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because, since it is a limit, what we are concerned with is what happens on our way to that value.
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But we don't actually care about that value in specific.
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We don't care about that horizontal location; what we care about is our way to the horizontal location.
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It is the journey that matters, not the destination, when it comes to limits.
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So, the limit is the height we expect--what we feel like would happen without peeking under the cover.
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We have that black bar there that keeps us from being able to see what it actually turns out to be.
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But in this case, it seems like what it is going to come out to be is 4.
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If all of the information we have is just the picture in front of us (except that part that has been covered up--
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we are not allowed to look under it), the information that we have makes it seem like it looks like it is going to 4 in both of them.
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The idea is the question of what we expect will end up happening.
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Where does it seem like this function is going to? That is what a limit is about.
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With all of that in mind, we now have the ability to create a more formalized definition.
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The definition of a limit: if f(x) becomes arbitrarily close to some number l, as x approaches some number c,
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but is not equal to c (we don't actually care about that horizontal location;
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we just care about the way to that horizontal location), then the limit of f(x), as x approaches c, is l.
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Symbolically, we write that as "limit as x goes to c of f(x) equals l"; the limit of f(x) as x approaches c is l.
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So, as x gets close to c, what value does our f(x) seem to go to--what value is f(x) getting towards as we get close to that horizontal location?
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There are two important things to note: we are looking at f(x) as x goes to c, but are not concerned with x = c.
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So, for the purposes of a limit, x is never equal to c; we are only concerned with what happens on the way to c, but not actually x at c.
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Remember: it is the journey, not the destination, that matters when it is a limit.
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The other thing to notice is that, when we consider x approaching c, we are considering x approaching from all directions--not just one side.
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It is not just this side; it is not just this side; it is both sides coming together.
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To have a limit, f(x) must go to the same value from both sides.
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The right side and the left side have to agree with each other.
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If they go to totally different things for a horizontal location, if they go to totally different heights, then they don't agree; there is not a limit.
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There is not a sense of what to expect if they are going to totally different places.
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Where is it going to be? Is it going to be somewhere in the middle?
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Is it going to be the top one? Is it going to be the bottom one?
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We don't have a good sense of which side to trust, so we can't get a limit out of it.
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The two sides have to agree for us to end up having a limit.
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Technically, I want to point out that this isn't actually the formal definition of a limit.
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That said, what we are seeing here is going to certainly be enough for now.
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This idea, the definition of a limit that we just talked about (this whole thing here and all of our commentary that came after it--
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all of the stuff that we have been working through so far in this lesson)--this is plenty for the class you are currently in.
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A precalculus-level course like we are in right now--this is more than enough of an understanding of what a limit is.
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You are doing great at this point.
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Even for a calculus-level class, this is really pretty much enough understanding.
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Some courses will maybe vaguely talk about the formal definition,
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but very, very few will really expect you to fully understand the formal definition of a limit.
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This is really all that they are looking for--this sense of a limit being what you are going towards, but not where you actually end up.
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And for pretty much most science courses, most engineering courses, this is really all you need.
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If you want to talk about the really formal definition of a limit, that is going to show up later on,
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in really advanced math courses, like second--maybe even third-year college math courses, really proof-heavy math stuff.
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And I think that that is really great stuff.
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But for the most part, you will be fine with just this, probably forever.
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However, if you are really interested in mathematics, if you are curious, you might want to check out the next lesson, Formal Definition of a Limit.
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I think that this stuff is really, really cool; and I have a great lesson that will help us explain and understand what the formal definition of a limit is.
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But frankly, probably 99 times out of 100, you are never going to need to know that stuff.
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Pretty much any class that you will be taking in the next two years is never going to actually require you to know the formal definition of a limit.
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So, don't worry too much if you don't feel like watching it; it is totally a fine lesson to end up skipping.
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But if you are interested, it is really cool; and if you are interested in this, you might end up being interested in taking advanced math classes later.
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And it will totally come into play later on, and you will be a step ahead of everybody else in understanding this fairly complicated idea.
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All right, let's keep talking about limits: so far, all of the examples we have seen have had limits.
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But a limit does not always exist: for example, consider the limit as x goes to 0 of 1/x.
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Well, here what we are doing is looking at as x goes to 0 (the y-axis is at x = 0).
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As we come in from the right side, we aren't actually going to a single value.
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We are just going to go up and up and up and up and up and up and up and up, and we don't ever stop going up.
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It is a vertical asymptotes; when we worked on rational functions and vertical asymptotes, we just went on up forever.
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There is no single l value, no single limit number that we are going towards.
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So, there is nothing that can be agreed upon.
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And even worse than that, we end up going in the totally opposite direction when we come from the left side.
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One side is shooting off to positive infinity; the other side is shooting down to negative infinity.
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There is nothing to say for the limit here.
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They are not approaching something where they are agreeing on some value.
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They are not even going to separate values; they are just blasting off to infinity on both sides.
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We don't really have a good way to talk about this; there is no way to assign a specific number value that we expect will happen at x = 0,
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because it is just going to clearly go crazy, and that is that; there is nothing that it is going to go to.
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So, because of that, we say that the limit does not exist.
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This limit does not exist; there can be no limit, as x goes to 0, because there is no single value that 1/x is headed towards.
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There is nothing that we are going to end up seeing them agree on.
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Even one side is not going to agree on anything, because it just keeps going up.
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There is no single value; therefore, there is no limit that we get out of it.
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Still, I want to point out: for x approaching any other value than 0, the limit would exist, because it would approach a single value.
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If we approached 1 as our thing, then we would end up approaching 1.
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If we approached 4 as our thing, then we would end up approaching 1/4.
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If we went to -3, if we were approaching -3 from both sides, then we would be going to -1/3.
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All of those make sense; the only issue is here at x = 0, where, because it has an asymptotic thing, it just goes crazy and shoots off in both directions.
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There is nothing that we can end up pinning it down with, so we have to say that the limit does not exist.
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But anywhere else on 1/x would exist; but x going to 0 does not exist for 1/x.
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In the previous example, the limit as x goes to 0 of 1/x, that didn't exist.
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But at the same time, 1 divided by 0 does not exist, either; f(0) didn't exist, and so limit as x goes to 0 of f(x) didn't exist.
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And there is a connection there; however, it is possible for the function to exist while the limit does not exist.
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So, to see this, take, for example, g(x) equals the piecewise function x² when x is less than 1.
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(we see this parabola on the left side) and 4 - x when x is greater than or equal to 1 (and we see this straight line on the right side).
00:18:40.700 --> 00:18:44.500
We have every single point in the real numbers defined.
00:18:44.500 --> 00:18:47.700
They will end up having some value that the function will put out.
00:18:47.700 --> 00:18:53.900
If you plug in any number, it is going to either be on the straight line portion, or it is going to be on the parabola portion.
00:18:53.900 --> 00:19:08.100
So, g(1) = 3; if we plug in 1, we use 4 - x; 4 - 1 gets us 3; that is this point right here.
00:19:08.100 --> 00:19:13.000
However, the limit as x goes to 1 of g(x) does not exist; why?
00:19:13.000 --> 00:19:17.600
Well, from the left side, we end up approaching this value here.
00:19:17.600 --> 00:19:20.600
We are approaching this side as we approach from the left side.
00:19:20.600 --> 00:19:26.400
However, as we approach from the right side, we end up approaching this totally different value.
00:19:26.400 --> 00:19:29.600
And so, there is this big gulf between the two limits.
00:19:29.600 --> 00:19:33.100
We are going to two totally different places at this horizontal location.
00:19:33.100 --> 00:19:39.900
So, the function approaches two totally different values from the right and left sides.
00:19:39.900 --> 00:19:45.800
Since they don't agree, we can't say, "Oh, that is what we expect," because we have totally different expectations from the two sides.
00:19:45.800 --> 00:19:50.100
So, that means that the limit does not exist.
00:19:50.100 --> 00:19:52.300
How do we actually go about finding limits in general?
00:19:52.300 --> 00:19:55.000
The first really great way to do this is with graphs.
00:19:55.000 --> 00:19:58.100
One way to find the value of a limit is just to look at a graph of the function.
00:19:58.100 --> 00:20:00.700
If you have a graphing calculator, you can plot it on the graphing calculator.
00:20:00.700 --> 00:20:05.200
If you have some sort of graphing program, like I make these graphs with, you can plot it on a graphing program.
00:20:05.200 --> 00:20:08.000
Or if you are just really good at making graphs, you can draw a graph.
00:20:08.000 --> 00:20:13.900
Figure out if a limit there makes sense: is there something where both sides are going to the same thing?
00:20:13.900 --> 00:20:16.800
And if so, find what value the graph indicates.
00:20:16.800 --> 00:20:22.700
For this case, we have f(x) = (x² + x - 2)/(x² - x);
00:20:22.700 --> 00:20:26.800
(x² + x - 2) on the top, (x² - x) on the bottom of the fraction.
00:20:26.800 --> 00:20:29.700
So, we see that there are some parts where we have issues.
00:20:29.700 --> 00:20:37.100
If we plug in x = 0, we end up having an asymptote; and if we plug in x = 1, we end up having this hole here.
00:20:37.100 --> 00:20:40.900
But we can still ask what the limit is as x goes to 1.
00:20:40.900 --> 00:20:49.600
Well, if we go to the graph, as x goes to 1 from the left side, we end up seeing that we are approaching that height of 3.
00:20:49.600 --> 00:20:55.200
As x goes to 1 from the right side, we see that we are approaching the exact same height.
00:20:55.200 --> 00:21:02.700
We end up getting up a value of 3; either way we approach this, we are going to end up seeing that 3 is what the expected value is.
00:21:02.700 --> 00:21:07.600
The graph shows us that we are working towards 3, or something at least on this graph looks very close to 3.
00:21:07.600 --> 00:21:11.400
So, we could say 3; but of course, we are reading a graph.
00:21:11.400 --> 00:21:15.400
Reading a graph isn't always as precise as we would like.
00:21:15.400 --> 00:21:19.500
Reading a graph, we know that you can sometimes be off by 1/2 or 1 whole thing.
00:21:19.500 --> 00:21:23.500
So, it doesn't give us a perfect answer; but it gives a pretty good idea.
00:21:23.500 --> 00:21:27.300
We have a good sense of what it is going to be, although it is not perfectly precise.
00:21:27.300 --> 00:21:33.100
However, limits do have the massive benefit of being able to allow us to get an intuitive sense of how the thing is working.
00:21:33.100 --> 00:21:37.500
They will let us see what the function is, as a general idea; and sometimes that is the most useful thing of all.
00:21:37.500 --> 00:21:41.200
Graphs are really, really handy, even if they don't let us see precisely what the value is;
00:21:41.200 --> 00:21:45.400
they let us understand what is going on--does it even make sense for it to have a limit here?
00:21:45.400 --> 00:21:49.000
Things like that are what a graph allows us to answer.
00:21:49.000 --> 00:21:54.700
Alternatively, if we want something that is more precise, if we want a more precise sense of where the limit will go,
00:21:54.700 --> 00:21:58.200
or we don't want to graph the function, just because we have some sense of what the graph looks like,
00:21:58.200 --> 00:22:00.600
or we just don't feel like graphing it, because we know it is going to be a pain,
00:22:00.600 --> 00:22:06.400
but we have enough of an idea to know that the limit would exist there; we can use a table of values,
00:22:06.400 --> 00:22:10.700
where x will approach the value that it approaches in the limit.
00:22:10.700 --> 00:22:15.400
Once again, we have the same f(x) = x² + x - 2 over x² - x.
00:22:15.400 --> 00:22:21.200
And now, we are looking at the limit as x goes to 1 of that function, the same limit as before.
00:22:21.200 --> 00:22:29.900
What we can do is: we have 0.9, 0.99, 0.999...we are approaching 1 from the left side there.
00:22:29.900 --> 00:22:36.700
We can approach it from the right side: 1.1, 1.01, 1.001...we are getting closer and closer values.
00:22:36.700 --> 00:22:43.900
Now, of course, we can't actually plug in x = 1, because if we plug in x = 1, it is just going to fail on us.
00:22:43.900 --> 00:22:47.900
The limit isn't about where it actually would be; it is about what happens on the way there.
00:22:47.900 --> 00:22:55.400
So, we don't plug x = 1 into our table; all we are concerned about is what happens to the numbers as they get really close to x = 1.
00:22:55.400 --> 00:23:05.500
We calculate this with a calculator: .9 comes out to be 3.222; .99 comes out to be 3.020; .999 comes out to be .002.
00:23:05.500 --> 00:23:13.100
Going from the other side, 1.1 is 2.818; 1.01 is 2.980; 1.001 is 2.998.
00:23:13.100 --> 00:23:23.600
So, we can see that the value that we are getting close to, 2.998, 3.002...we are clearly tending pretty close to the value of 3.
00:23:23.600 --> 00:23:29.100
So, we can assign that this limit is going to end up having a 3 as we get closer and closer and closer.
00:23:29.100 --> 00:23:36.200
Now, maybe we are off by .0001 or some small number; but we can be pretty sure that,
00:23:36.200 --> 00:23:41.500
unless it does some really sudden jump there, 3 is probably going to be pretty close to it.
00:23:41.500 --> 00:23:49.400
That lets us get a good approximation--probably a good approximation within many decimal places, but we are not absolutely, precisely sure.
00:23:49.400 --> 00:23:58.100
Still, that is really, really close; and the closer our table has x approach the limit, the more sure we can be.
00:23:58.100 --> 00:24:08.300
If we, instead of using 1.001, would use 1.0000000001, we would be that many more decimal places sure of where we are headed.
00:24:08.300 --> 00:24:15.900
The same happens with 0.999999999; by plugging in more and more decimal places, we get more and more accuracy.
00:24:15.900 --> 00:24:20.100
So, we can be more and more sure of what the value that we will end up getting out of that limit is.
00:24:20.100 --> 00:24:25.900
If you have a graphing calculator, this is a great use for the table of values feature,
00:24:25.900 --> 00:24:28.900
where you can just set up a function, then go the table of values,
00:24:28.900 --> 00:24:31.500
and have it be an independent thing where you plug in each number.
00:24:31.500 --> 00:24:36.100
You plug in .9, .99, .99999; you hit them all in, and it will just put out values for you.
00:24:36.100 --> 00:24:40.100
And you won't have to type in the entire expression over and over.
00:24:40.100 --> 00:24:45.000
So, check out the course appendix; there is that appendix on graphing calculators as part of this.
00:24:45.000 --> 00:24:47.500
It is a great thing to check out if you have access to a graphing calculator.
00:24:47.500 --> 00:24:53.900
That table of values will make your life so much easier when you are working through this sort of thing--this is a really good use here.
00:24:53.900 --> 00:24:59.300
Precise methods: the lesson after the next is finding limits (not the very next lesson,
00:24:59.300 --> 00:25:04.300
which will be the formal definition of limits, which you can totally skip, if you are not particularly interested;
00:25:04.300 --> 00:25:07.700
but the one after that--you will want to watch it, and that is Finding Limits).
00:25:07.700 --> 00:25:12.500
In it, we will see ways to precisely find the limit of a function through algebraic methods.
00:25:12.500 --> 00:25:16.900
Graphing and a table of values--those two things give us really good approximations.
00:25:16.900 --> 00:25:21.600
They give us a good sense of what is going on, but they don't tell us what the value has to be, precisely.
00:25:21.600 --> 00:25:26.300
We will figure out algebraic methods in the next, next lesson, Finding Limits.
00:25:26.300 --> 00:25:32.200
For right now, though, we will just stick to the methods of graphing and making tables: they are pretty good methods, actually.
00:25:32.200 --> 00:25:38.600
Even after you learn those more precise methods about how to figure out algebraically, precisely, don't forget about these.
00:25:38.600 --> 00:25:43.500
They can be really handy when you can't figure out how to figure it out precisely, algebraically,
00:25:43.500 --> 00:25:46.500
but you still need to evaluate it and get a good sense of where it is going.
00:25:46.500 --> 00:25:49.100
You can always use these methods.
00:25:49.100 --> 00:25:55.100
Now, often you will end up having problems where the problem says that you have to get it precisely, and you won't be able to use these methods.
00:25:55.100 --> 00:25:59.000
But sometimes, you will have some really, really complicated thing, and you won't be able to work it out.
00:25:59.000 --> 00:26:06.200
You can just put it in a table of values, something like this, and you will be able to get a good sense of where it is headed; and that can be useful.
00:26:06.200 --> 00:26:07.700
All right, we are ready for some examples.
00:26:07.700 --> 00:26:11.600
For each limit below, if it exists, determine the value using the associated graph.
00:26:11.600 --> 00:26:15.800
Our first one is the limit as x goes to 0 of 1/x².
00:26:15.800 --> 00:26:25.500
Well, what we can see is that, as we get closer and closer to x = 0, this thing just shoots off; both sides fly off to infinity.
00:26:25.500 --> 00:26:27.400
They shoot off forever and ever and ever.
00:26:27.400 --> 00:26:33.500
So, even though they are both going towards positive infinity, do they ever end up establishing at a single value?
00:26:33.500 --> 00:26:37.700
Does it ever settle on a single value? No, they are always going to keep going up.
00:26:37.700 --> 00:26:42.700
It is going up asymptotically, so there is never a number that they get steady out on.
00:26:42.700 --> 00:26:46.300
They never decide to land on 47 or some other number.
00:26:46.300 --> 00:26:49.400
Since they keep going on forever, it doesn't have a limit.
00:26:49.400 --> 00:26:59.100
The limit here does not exist, because it never settles on a single value.
00:26:59.100 --> 00:27:08.300
It shoots up to infinity forever, and since it never ends up staying around at a single value, it never settles on a single value.
00:27:08.300 --> 00:27:12.200
So, we end up not being able to get a limit out of it.
00:27:12.200 --> 00:27:19.300
The next one is the limit as x goes to -3 of (x + 3)/(x² + 5x + 6).
00:27:19.300 --> 00:27:25.000
In this one, if we plug in -3, look: there is a hole, so we can't just directly plug in -3.
00:27:25.000 --> 00:27:28.000
But from the graph, we can see that, yes, there is nothing wrong.
00:27:28.000 --> 00:27:30.900
We are approaching the same thing from the left and the right side.
00:27:30.900 --> 00:27:39.200
What value are we approaching? We are approaching a value of -1, so the limit is -1; great.
00:27:39.200 --> 00:27:44.600
The next one: Evaluate the limit as x goes to -2 of 3x² + 5x + 1.
00:27:44.600 --> 00:27:49.400
We could graph this; this is just a parabola--it wouldn't be too tough for us to graph.
00:27:49.400 --> 00:27:55.700
And if we graphed it, we would end up seeing that it looks something like this.
00:27:55.700 --> 00:28:01.800
And we could figure out what it is--draw a really careful graph; but drawing a really, really careful graph takes a fair bit of effort.
00:28:01.800 --> 00:28:05.000
And we have a good sense that there are going to definitely be limits everywhere,
00:28:05.000 --> 00:28:10.300
because it never does anything weird; it never jumps around; everything that we expect to happen is what happens.
00:28:10.300 --> 00:28:13.000
So, we don't have to worry about that; but that is what the graph gets us.
00:28:13.000 --> 00:28:15.600
At this point, now we want to actually figure out what the value is.
00:28:15.600 --> 00:28:22.300
The easiest way to figure out what the value is: we can just plug in values; we make a table of values.
00:28:22.300 --> 00:28:28.000
We are going to have x on the left side, and the f(x) that comes out on the right side; what value are we approaching?
00:28:28.000 --> 00:28:42.200
We are approaching -2; so x going to -2...if we are a little below -2, we will be at -2.1, then -2.01, then -2.001.
00:28:42.200 --> 00:28:52.000
On the other side, we will be coming away from -2: -1.999, -1.99, -1.9...
00:28:52.000 --> 00:29:08.100
If we plug these things into our calculator, -2.1 gets us 3.73; -2.01 gets us 3.0703; -2.001 gets us 3.007003.
00:29:08.100 --> 00:29:22.800
Flipping to the other side, -1.9 would get us 2.3; -1.99 would get us 2.9303; -1.999 gets us 2.993003.
00:29:22.800 --> 00:29:25.200
So, it is pretty clear what we are approaching.
00:29:25.200 --> 00:29:34.100
As we go in from each side, -2.001, -1.999...they are getting really, really close to this middle value of 3.
00:29:34.100 --> 00:29:41.500
So, that is what the limit ends up being; the limit comes out to be equal to 3; that is the value that it is approaching.
00:29:41.500 --> 00:29:48.700
Also, remember how we talked about...well, look at that graph: the way that that graph is there--it doesn't do anything weird.
00:29:48.700 --> 00:29:54.400
Everything that we expect to come out of it is what is going to be that function's location.
00:29:54.400 --> 00:29:59.500
So, another thing that we could do is: because, in this specific case, the graph doesn't do anything weird,
00:29:59.500 --> 00:30:05.800
the function doesn't jump around in any weird ways; there are no breaks; and there is nothing strange about it.
00:30:05.800 --> 00:30:08.300
So, if there is nothing strange about it, what we could do is: we could also say,
00:30:08.300 --> 00:30:13.300
"Well, that means that the limit has to be what the function actually ends up going to at that point."
00:30:13.300 --> 00:30:21.000
So, we could also just evaluate it by plugging it in: 3(-2)², plus 5 times -2, plus 1.
00:30:21.000 --> 00:30:30.400
3 times...-2 squared gets us 4, plus 5 times -2 gets us -10, plus 1...3 times 4 is 12, plus -10...so 2 + 1...
00:30:30.400 --> 00:30:34.300
That came out to be 3 as well, so that checks out, as well.
00:30:34.300 --> 00:30:38.000
So, what we just saw there was actually one of the precise methods of doing this stuff.
00:30:38.000 --> 00:30:42.600
We will talk about this more in Finding Limits; but it seemed like a really easy one for us to see...
00:30:42.600 --> 00:30:46.400
we are going to start to get a sense of how this stuff works, and we want to find the precise stuff.
00:30:46.400 --> 00:30:51.200
So, we will see more, and we will also understand exactly why we can do this, in the coming lessons.
00:30:51.200 --> 00:30:56.500
All right, the third example is the limit as x goes to 0 of sin(x)/x.
00:30:56.500 --> 00:31:05.100
If we were to graph this, if we used a graphing calculator to graph what comes out of this, we would see that it is going to look something like this.
00:31:05.100 --> 00:31:16.700
Oops, I actually made a mistake with that graph; we would see that it was going to look something like...there we go; that is better.
00:31:16.700 --> 00:31:20.200
It is going to look something like that.
00:31:20.200 --> 00:31:30.200
The easiest way to do this is to just do a table of values; we have x, and f(x) is coming out of it, as we plug in various values for x.
00:31:30.200 --> 00:31:42.000
So, our x is approaching 0; if we are approaching 0 from under it, we are going to be at -0.1, then -0.01, then -0.001.
00:31:42.000 --> 00:31:45.600
And you could use different numbers, as long as they continue to get closer and closer to 0.
00:31:45.600 --> 00:31:47.800
But I think those are pretty easy ones to use.
00:31:47.800 --> 00:31:56.900
From the other side, we would be coming away: 0.01 on the positive side now; 0.01, and 0.1, now that we are past the 0.
00:31:56.900 --> 00:32:02.100
We plug these into a calculator; we figure out what it is; we get...-0.1 going in gets us 0 point...
00:32:02.100 --> 00:32:05.100
Also, remember: this has to be in radians for us to...
00:32:05.100 --> 00:32:07.400
Actually, it doesn't have to be in radians because of this specific problem.
00:32:07.400 --> 00:32:11.200
But any time we end up seeing a function, we should assume that it is in radians,
00:32:11.200 --> 00:32:14.600
unless we have been explicitly told otherwise--that it is in degrees.
00:32:14.600 --> 00:32:17.800
But normally, assume that it is in radians when you are doing math.
00:32:17.800 --> 00:32:36.700
0.998334...plug in -0.01: 0.999983; -0.001: 0.9999999.
00:32:36.700 --> 00:32:55.800
On the other side, there is 0.1: we have 0.998334; for 0.001, it is 0.999983; and for 0.001, 0.999999.
00:32:55.800 --> 00:32:57.700
It is pretty clear what we are ending up approaching here.
00:32:57.700 --> 00:33:01.600
As we get closer and closer, the value that we are approaching is 1.
00:33:01.600 --> 00:33:07.700
The value that this is getting really close to is 1, and it seems to show that it is going to get really close to it.
00:33:07.700 --> 00:33:12.000
So, we see that the value of this limit is 3.
00:33:12.000 --> 00:33:22.600
The fourth example: Using the associated graph, explain why the limit below does not exist: the limit as x goes to 0 of sin(1/x).
00:33:22.600 --> 00:33:29.000
Just glancing at this thing, the answer is simply that it goes crazy.
00:33:29.000 --> 00:33:33.700
Look at this thing that is happening here--it is going crazy!
00:33:33.700 --> 00:33:38.800
This graph is doing some really weird stuff as x gets close to 0.
00:33:38.800 --> 00:33:41.000
So, as x gets close to 0, what is going on?
00:33:41.000 --> 00:33:46.000
Well, notice how we can see that it is going up, and then it goes all the way down.
00:33:46.000 --> 00:33:48.800
And then, it goes all the way up, and then it goes all the way down.
00:33:48.800 --> 00:33:51.800
And then it goes all the way up and all the way down; the same thing happens on the other side:
00:33:51.800 --> 00:33:55.300
up and then down, and up and then down, and then up and then down.
00:33:55.300 --> 00:34:00.400
Well, what it seems to be doing is going up/down, up/down, up/down, faster and faster, as it gets closer and closer.
00:34:00.400 --> 00:34:12.800
It is going crazy; it is bouncing up and down forever as it gets closer to this x going to 0.
00:34:12.800 --> 00:34:20.000
It is just bouncing so, so fast; so as we get closer and closer to that x going to 0, we don't have any idea what to say,
00:34:20.000 --> 00:34:25.300
because we are flying around; the closer you get to x = 0...it constantly changes.
00:34:25.300 --> 00:34:28.200
The function is constantly changing, constantly going up and down.
00:34:28.200 --> 00:34:33.800
Because it is constantly bouncing up and down forever and ever and ever and ever, we end up saying that it doesn't have a limit.
00:34:33.800 --> 00:34:38.300
It doesn't exist; the limit does not exist, because it doesn't settle on anything.
00:34:38.300 --> 00:34:43.500
A limit has to be settling towards some value, and it has this craziness there.
00:34:43.500 --> 00:34:48.300
So, we end up not being able to say it has a limit; the limit does not exist.
00:34:48.300 --> 00:34:52.600
If you want a better idea of what is going on here (and I think it is always cool to have a better idea of what is going on),
00:34:52.600 --> 00:34:59.600
we can break this into two pieces: the graph of 1/x ends up graphing (let me put that in black,
00:34:59.600 --> 00:35:09.700
just so we can easily see it)...here is our axis...it ends up looking like this.
00:35:09.700 --> 00:35:12.300
We are used to that nice vertical asymptote there.
00:35:12.300 --> 00:35:24.500
And then, the graph of sin...let's say t, just some dummy variable t that we plug in...it is going to end up having that nice periodic graph.
00:35:24.500 --> 00:35:30.300
That is how sine works: it goes up and down and up and down and up and down and up and down.
00:35:30.300 --> 00:35:38.200
That is what happens; however, what we have here, the t that is going into this, is 1/x.
00:35:38.200 --> 00:35:44.100
So, what happens is that the value, as we get closer and closer to 0...
00:35:44.100 --> 00:35:47.700
well, what happens to that asymptote as we get closer and closer to 0?
00:35:47.700 --> 00:35:51.100
As we get closer and closer to 0, they shoot off to infinity.
00:35:51.100 --> 00:36:00.400
So, what we have is: we effectively have an infinitely long amount of stuff that we are plugging into sin(t) in a very small, compact space.
00:36:00.400 --> 00:36:02.700
That is why we end up seeing that it is going slowly.
00:36:02.700 --> 00:36:13.100
And then, as it gets closer and closer to 0, 1/0 (that is not really formally accurate, but)...1/0 is effectively shooting off to infinity.
00:36:13.100 --> 00:36:17.700
1/0 shoots off to infinity; so if we are shooting off to infinity, and then we are plugging something
00:36:17.700 --> 00:36:23.800
that is going all the way out to infinity into sine, well, that means our periodic thing is going to go up and down forever.
00:36:23.800 --> 00:36:30.400
But because we are doing all of this forever-ness in all of this very tiny distance of .1 to 0,
00:36:30.400 --> 00:36:34.200
that means we end up having it go faster and faster and faster and faster, because it has to manage
00:36:34.200 --> 00:36:40.100
to do all of infinity in 0 to 0.1--pretty crazy.
00:36:40.100 --> 00:36:42.200
That is why we end up seeing this behavior.
00:36:42.200 --> 00:36:44.600
And if that still doesn't quite make sense, don't worry.
00:36:44.600 --> 00:36:49.100
Honestly, you will be fine in a precalculus class, and even a calculus class, without fully understanding this idea.
00:36:49.100 --> 00:36:54.700
But if you let this rattle around in your head--think about it for a while--you might start to think, "Oh, I am seeing it."
00:36:54.700 --> 00:36:59.700
There are some really cool ideas in math, and lots of the cool ideas in math end up taking a little while to fully understand.
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So, even things that you don't understand the first time--the second time you look at it, it might make a lot more sense.
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I think this stuff is really cool.
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All right, we are ready for our final example.
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For each limit below, if it exists, determine the value using the associated graph.
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Our first one is limit as x goes to 0 of 2x/x; well, that one is pretty easy.
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It is clearly going towards 2, because it is just in a steady state at 2 all the time.
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While actually plugging in 0...if we plug in 0, 2(0)/0 gets us 0/0; we can't plug in 0.
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So, it doesn't exist at 0/0; but on the way to that x = 0, it exists just fine.
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We can see what the limit that it is going towards is.
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The limit that it is going towards is 2.
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Similarly, over here, the limit as x goes to -2 of (-3x - 6)/(x + 2): what is it going towards?
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It is going towards that thing, which is -3; so it is going to -3.
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If we actually plugged -2 in, well, x + 2, so -2 + 2, would get us dividing by 0; things would fall apart; we are not allowed to do that.
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But as long as we are not plugging in -2 directly, everything is fine.
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We always end up having -3, as we can see from this picture right here.
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So, since it is always -3, then the limit of what it is going towards (we don't have to worry about that thing that it is actually at) is -3--as simple as that.
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Also, I want to point out to you something that we are going to see in the finding of the values of limits.
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We can actually get an early start on what is going to happen, to get that idea percolating through your head.
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Look: 2x/x...well, at x = 0, it ends up doing something different.
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But with the exception of x = 0, 2x/x behaves exactly like 2.
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So, because it behaves exactly like 2, we can effectively say, "Well, what would it be at 0, if we were using this other alternative way of talking about it?"
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The other alternative is that you are always 2; so since it is always 2, we end up getting a limit of 2 out of it.
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The same thing is going on over here with (-3x - 6)/(x + 2).
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Well, we can rewrite that top as -3 times (x + 2) over (x + 2), which means we can then write an equivalent thing,
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with the exception of x = -2; everywhere else, we won't have that issue.
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It would just be the same as -3 forever and always.
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So, with the exception of that x = -2, it works just fine.
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But we don't actually care about -2; remember, there is that idea of blacking out, where you are covering up that chunk,
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because we are not allowed to peek underneath the cover.
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So, if we cover up that chunk, and then we try to figure out where it is headed towards,
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we don't have to worry about the fact that, if we had plugged in -2 here, it wouldn't work,
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because we are not worried about what happens at -2.
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We are just worried about what happens everywhere else; and what it is equivalent to everywhere else is -3.
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And so, that is why we end up seeing it.
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This is just a quick preview of the ideas we will end up getting when we start talking about finding limits precisely with more algebraic methods.
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All right, that gets us a really good idea of how limits work.
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Just remember: it is the idea of where you are going and if it matches from both sides;
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the question is of where you are going, but it doesn't matter what it actually is there.
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It is about the journey, not the destination.
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All right, we will see you at Educator.com later--goodbye!