WEBVTT mathematics/ap-calculus-ab/hovasapian
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Hello, welcome back to www.educator.com, and welcome back to AP Calculus.
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Today, we are going to start talking about the limit of a function.
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Very important topic, it is absolutely the foundation of all of calculus.
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Let us jump right on in.
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In the first few lessons, I have mentioned that given some f(x), the derivative involves some limit process.
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I think I will work in blue today.
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In the first few lessons, I mentioned that given f(x),
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the derivative which we symbolize with a little symbol, the derivative f’(x) was found as follows.
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The limit as h approaches 0, let us go ahead and put f’(x) is equal to quotient f(x) + h – f(x)/ h.
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Basically, once we form this quotient, we simplify it.
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Once we form this quotient and simplify algebraically, or whatever else that we need to do,
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that is going to be algebraically, we get some function, let us call it g(x).
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When we form this quotient, we simplify it and it gives us some function of x.
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Then, we take the limit, then we apply this thing.
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We end up taking the limit as h goes to 0 of some g(x).
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More generally, we want to be able to handle things like this.
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More generally, when we take a derivative, we are going to be taking limits.
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But that is not the only time that we are going to be taking limits.
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We want to be able to handle limits whenever they come up, not just in the context of differentiation.
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Or generally, we want to be able to handle any function, the limit as x approaches some number of any function f(x),
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whenever it might come up, whether it is in the context of derivative or not.
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This is what we want to know about and this is what we are going to do for the next couple of lessons.
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What this says is as follows.
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Let me rewrite it so we have it on this page.
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The limits as x approaches a of f(x).
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This symbol says given some function f(x), what happens to this f(x),
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as x itself gets closer and closer to some number a, a can be infinity, x could go off to infinity.
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We ask ourselves, if x is going to infinity, what is the function doing?
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As x approaches 5, what is the function doing, how was it behaving?
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Is it oscillating, is it going off to infinity itself, is it getting close to a number?
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That is what we are asking.
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In other words, y = f(x).
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We know the when we have f(x) = x², y = x².
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f(x) and y are essentially synonymous.
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For y = f(x), again, what this says, what the symbol says is, as x gets closer and closer to a,
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does y get closer and closer to some number itself?
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Does it go off to infinity?
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Does y go off to positive or negative infinity?
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Does y oscillate back and forth between two numbers?
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Does y oscillate?
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Does y bounce back and forth between two numbers?
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What does y do, that is what we are asking, that is what this symbol says.
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As x gets closer to some number, what does f(x) do?
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Notice what happens to y?
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As it turns out, all three of these things happen.
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Sometimes y gets close to a number, sometimes it goes off to infinity, positive or negative infinity.
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Sometimes it just oscillates between two numbers.
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It turns out, all of these happen depending on the function and depending on the number you are approaching, depending on a.
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Depending on what f(x) is, depending on x and f(x), and depending on a.
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I'm going to describe this limit concept.
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I'm going to describe this limit concept by looking at the graphs of several functions.
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Because I want you to have an intuitive understanding of what the limit means.
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I want you to have an intuitive understanding,
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I think understanding is probably not the word I want to use.
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I want you to have an intuitive feel for what the symbol limit as x approaches a of f(x) means.
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If you have an intuitive understanding then any math that we do on a formal level will make sense.
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If we just throw out some definition, some formal definition involving mathematical symbols,
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we can explain but you need to have to get a feel for it.
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It is very important.
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The intuitive is actually more important than the formal mathematical of this level.
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As a matter of fact, let me go ahead and take just a second to discuss this notion between formal and intuition.
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Nowadays in calculus courses, in AP calculus, I believe in most classes, when they talk about this idea of a limit,
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they introduce the formal definition of a limit.
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You may or may not do it in your class, I’m not exactly sure.
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It is my personal belief that at this level, when you are just doing calculus, multivariable calculus, linear algebra, and differential equations,
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your first exposure to these things, you should not be exposed to formal definitions and what we call epsilons and deltas.
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It is more important that you understand what is happening intuitively,
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so that you can actually manipulate your mathematics, based on what you understand.
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Rather than trying to fiddle with really intricate formal definitions.
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For those of you that go on to mathematics, you are going to end up taking a course called analysis
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where you go back and you revisit calculus.
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But instead of actually doing computational problems,
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you actually prove why it is that you can do the things in calculus that we are going to do, for this entire course.
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You are going to see the formal definition.
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It is going to depend on your teacher, the extent to which they actually want to emphasize it.
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But it is my personal belief that it does not belong at this level.
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For that reason, I'm not going to present a formal definition.
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In some of the problems that I do later, I may mention it in passing.
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But I want you to have the intuitive feel, before we do a formal definition.
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Let us look at the following function.
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Let us look at y = x² and ask the following question.
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What is the limit as x approaches 3 of f(x) or same thing, because f(x) is x², what is the limit as x approaches 3 of x²?
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What we are asking is, as x gets close to 3, what does x² do?
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Write that down.
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What we are asking is, as x gets closer and closer to 3, what is happening with y?
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What is happening to y?
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You know the answer already, but let us take a look at it.
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Here is the function y = x² and here is our value of 3, right over there.
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The limit, this symbol, the limit as x approaches 3 of x² is actually 2 symbols.
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There are two things going on here and we have to deal with both.
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That is fine, I will just do it here.
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The first one is, what is the limit as x approaches 3, when you see a little negative sign to the top right of that number,
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it means what is the limit as x approaches 3 from below 3?
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In other words, 1, 2, 2.5, 2.6, 2.7, from the negative, from the bottom, of x².
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We write that as the limit as x approaches , with the + from above 3, 5, 4, 3.5, 3.4, 3.1, as you get close from above.
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This we call approaching from below.
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This we call approaching from above.
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When you see the symbol, limit as x approaches some number,
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if it is not specified, whether you are approaching that number from below or from above,
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you have to assume that you are approaching it from both.
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You have to actually solve two limits, every time that is the case.
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If they specify ready what the ±, then you just have to solve that one limit.
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This is also called the left hand limit, this is called the right hand limit.
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Left hand limit because you are approaching the number from the left.
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In other words, you are approaching 3 from the left.
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Right hand limit, you are approaching 3 from the right.
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That is all that means.
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Let us see what is happening.
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Let us do this one over here.
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The limit as x approaches 3 from the left, 2, 2.5.
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It looks like as we get close to 3, y itself, what is y doing?
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We just follow the path.
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It looks like it is getting close to 9.
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Now let us see what happens as x approaches 3 from above.
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As x approaches 3 from above, this way, the function itself y, it looks like the y value is also approaching 9.
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We did from below, it looks like it is approaching 9.
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It looks like both from above and from below, this is 9 and this is 9.
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As x approaches 3, it appears that f(x) or y is approaching 9.
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That is what our little arrow means.
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Arrow means it is approaching 9.
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Let us confirm this with a table of values.
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We have the graph, the graph is one way to actually deal with a limit.
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Let us see what happens, let us see what the graph does.
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It will tell us something about what is happening.
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Let us confirm this with an actual table of values.
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Here is the graph and here is the table of values.
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Here, from here to here, here is x approaching a from below.
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Notice 2.5, 2.7, 2.9, 2.99, 2.999, 2.9999.
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When we say gets closer and closer, that is really what we mean.
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We mean this part right here and it gets really close.
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We see that as it approaches 3 not equals 3, as it approaches 3, the function y which is x² is going 6.25, 7.29, 8.41, 8.9, 8.99, 8.999.
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Yes, it looks like it is approaching 9 from below.
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From above, this is x approaching 3 from above, 3.5, 3.3, 3.1, 3.01, 3.001, 3.001.
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You notice the y values, they descend and they come down to about 9.
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Sure enough, the table of values confirms that as we approach 3 from below and from above,
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the function itself approaches 9, approaches 9.
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The table of values confirms what we thought.
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The table values confirms our intuition, confirms our graphical intuition.
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The limit from below, what we call the left hand limit.
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I will often just call it 'lh'.
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The limit from above, what we call the right hand limit, are converging to the same number.
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That number is 9.
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When the left hand limit and the right hand limit converge to the same number, we say that limit exists.
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We say that the limit as x approaches 3 of x² exists.
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We call this limit the number they converge to.
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We call 9 the limit.
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It is very important.
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When you see a limit and it asks to specify whether it is a left hand limit or a right hand limit, you have to calculate both.
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If the left hand limit and the right hand limit converge to the same number which they do, 9 and 9,
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we call that number that they are converting to the limit.
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We say the limit of x approaches 3 of x² = 9.
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We actually write the limit as x approaches 3 of x² = 9.
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That is our final statement, that the left and right hand limits are the same and they converge.
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Let us see what we have got.
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Very important idea, that is this symbol, the limit as x approaches a of f(x) does not say what happens when x = a.
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It is asking you what is happening to f, as x gets close to a.
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Not what is happening when x = a.
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Distinguish between the two, that is probably going to be the most difficult thing, when you are starting out.
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It does not say what happens when x = a.
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It says what happens when x is near a.
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y can appear to approach a value but that does not mean that,
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Let me try this again.
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y can appear to approach a value as x approaches a, just like we saw a moment ago.
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As x approaches 3, it appears that y was approaching 9.
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But that does not mean that f(x) is defined, it has to be defined.
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It does not mean that f(x) has to be defined at a.
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Now the previous example, it is defined at 3.
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We know that 3² is 9.
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There is some value 9 at when x = 3.
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That is what the limit is asking.
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The limit is asking what does it look like it is getting close to?
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It looks like it is getting close to 9.
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If I wanted to, I can take that 9 out and say the function is not defined there.
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If I wanted to and I can do whatever I want with the functions.
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The limits would still exist, the limit is still 9 from below and from above.
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But at 3, the function does not exist.
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We will see an example of that in just a moment.
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y can appear to approach a value, as x approaches a.
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But that does not mean that f(x) has to be defined at a.
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Those are two independent things.
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The limit of a number and the value of the function at the number are independent.
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The limit as x approaches a of f(x) does not have to be f(a).
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When that is the case, it is a special property, which we will talk about later called continuity.
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In other words, it is a nice smooth curve, there are no gaps or breaks in it.
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But it does not have to be that way.
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Limit of f(x) does not have to be f(a).
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It can be like the last example.
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We had y which = f(x), which = x².
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We said to that the limit as x approaches 3 of x² = 9, because the left hand limit and the right hand limit appear to approach 9.
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In this particular case, f(3) which is equal to 3² = 9, they happen to correspond.
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They do not have to correspond.
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They happen to correspond, in this case.
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They are actually independent.
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The limit as x approaches a of f(x), sorry if I keep repeating myself, this was very important,
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does not say, it does not say what is f(a).
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If we want to know what f(a) is, we will ask you what is f(a).
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This is asking you what is the limit as x gets close to a?
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It says what happens to f(x) as x gets infinitely close to a, gets very close to a.
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What is the behavior of f near a, not at a?
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Let us do another example.
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This one, I’m going to draw it out myself.
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Another example, we notice that it looks like it is not defined.
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I’m going to go ahead and draw a graph, because we want to develop some intuition.
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Here is a graph, empty, and there.
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We have a graph like this.
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Let us say that this is 5 and let us say where that little point has been removed.
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Let us say the y value is 7.
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We ask the following.
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This right here is our f(x), we ask what is the limit as x approaches 5 of f(x)?
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It is not specified whether this is a left hand or right hand limit.
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We have to approach 5 from below, the left hand limit.
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We have to approach it from above to see what f(x) is getting close to.
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We do the limit as x approaches 5 from below of f(x).
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Let us see what happens as we approach 5, the function looks like it approaches 7.
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The left hand limit is 7.
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We will do the limit as x approaches 5 from above of f(x), what is that equal?
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As we approach 5 from above, the function gets closer and closer.
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We take the y values, it also looks like it is approaching 7.
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The limit of f(x), since this corresponds, this is the left hand and the right hand, the limits are equal.
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We say that the limit as x approaches 5 = 7.
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The limit exists, we say that the limit exists and that this limit = 7.
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Notice f(5) is not defined.
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5, there is a hole here, it is not defined.
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We do not know what it is.
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The limit exists, the limit is 7 but f(5) does not exist.
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They are very independent things and this is an example.
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They are completely independent.
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F(5) is not defined.
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The limit as x approaches 5 of this particular function = 7 but f(5) does not exist.
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'dne' means does not exist.
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Let us see what we have got here.
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Let us do another example.
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I’m going to go ahead and draw this one out as well.
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Another example.
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Let us do this and let us do that.
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This is our coordinate system.
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We have some function like this.
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This is some arbitrary function.
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This is our f(x), and let us go ahead and say that this x value is 2.
00:31:06.800 --> 00:31:11.300
Let us go ahead and say that this y value up here is 4.5.
00:31:11.300 --> 00:31:19.700
Let us say that this y value down here is 1.5.
00:31:19.700 --> 00:31:34.600
In this particular case, we ask what is the limit as x approaches 2 of f(x).
00:31:34.600 --> 00:31:37.500
X approaches 2, let us not specify whether it is a left hand or right hand.
00:31:37.500 --> 00:31:39.000
We have to do both.
00:31:39.000 --> 00:31:44.600
The left hand limit is, when we approach 2 from the left, from below, let me go ahead and do this in blue.
00:31:44.600 --> 00:31:49.100
When we approach 2 this way, the x values get closer and closer and closer to 2,
00:31:49.100 --> 00:31:50.000
what does the function doing?
00:31:50.000 --> 00:31:59.500
The function is getting close to 4.5.
00:31:59.500 --> 00:32:10.000
The limit as x approaches 2 from below of f(x), it equals 4.5.
00:32:10.000 --> 00:32:11.200
Let us do the limit from above.
00:32:11.200 --> 00:32:16.200
We are going to approach 2 from above, from numbers that are bigger than 2.
00:32:16.200 --> 00:32:22.200
We get closer and closer and closer and closer to 2, that means the function is going this way.
00:32:22.200 --> 00:32:27.900
It looks like it is approaching the number 1.5.
00:32:27.900 --> 00:32:34.700
The limit as x approaches 2 from above is equal to 1.5.
00:32:34.700 --> 00:32:38.300
4.5 and 1.5 do not equal each other.
00:32:38.300 --> 00:32:44.000
This limit does not exist.
00:32:44.000 --> 00:33:00.900
The limit as x approaches 2 from below of f(x) which is 4.5, does not equal to the limit as x approaches 2 from above which = 1.5.
00:33:00.900 --> 00:33:06.000
This means that the limit does not exist.
00:33:06.000 --> 00:33:13.200
Notice f(2) does exist, the value of f(2) is that one right there, the solid dot.
00:33:13.200 --> 00:33:15.600
It is actually 4.5.
00:33:15.600 --> 00:33:19.000
The left hand limit exists, it is 4.5.
00:33:19.000 --> 00:33:21.900
The right hand limit exists, it is 1.5.
00:33:21.900 --> 00:33:27.000
But because the left hand and right hand limits are not equal, the limit does not exist.
00:33:27.000 --> 00:33:28.800
We say that the limit does not exist.
00:33:28.800 --> 00:33:34.500
Again, we can ask for a left hand, we can ask for a right hand, or we can ask for both simultaneously.
00:33:34.500 --> 00:33:41.900
In order for the limit, when it is not specified to exist, the left and the right hand limits have to equal.
00:33:41.900 --> 00:33:44.300
You see, you can have a left hand limit, you have a right hand limit.
00:33:44.300 --> 00:33:45.500
You can have it be defined.
00:33:45.500 --> 00:33:55.600
Three completely independent things.
00:33:55.600 --> 00:34:01.900
Let us see what we have got here.
00:34:01.900 --> 00:34:06.100
Let us do another example.
00:34:06.100 --> 00:34:17.100
Let us try another example and I’m going to draw this one out as well.
00:34:17.100 --> 00:34:22.500
This time we are going to go ahead and use the function 1/x, an actual function.
00:34:22.500 --> 00:34:23.800
We know what this function looks like.
00:34:23.800 --> 00:34:32.100
It is a hyperbola, it looks something like this.
00:34:32.100 --> 00:34:47.900
We ask what is the limit as x approaches 0 of f(x)?
00:34:47.900 --> 00:34:50.100
Let us see what happens.
00:34:50.100 --> 00:34:54.800
As x approaches 0, here is our 0.
00:34:54.800 --> 00:34:58.100
We need to the a left hand limit and we need to do a right hand limit.
00:34:58.100 --> 00:35:03.200
We need to do, go ahead and do this in red.
00:35:03.200 --> 00:35:10.300
The limit as x approaches 0 from below of 1/x, which is our function.
00:35:10.300 --> 00:35:22.200
Let us see what happens as we approach 0 from below the function goes off to negative infinity.
00:35:22.200 --> 00:35:26.400
That is what the symbol means, that is all it means.
00:35:26.400 --> 00:35:29.700
It says as x gets close to some number, what does f(x) do?
00:35:29.700 --> 00:35:33.600
It is very intuitive.
00:35:33.600 --> 00:35:34.800
What is happening to the function?
00:35:34.800 --> 00:35:36.600
We see what is happening to the function.
00:35:36.600 --> 00:35:39.900
The function is just dropping down into negative infinity, that is the answer.
00:35:39.900 --> 00:35:45.500
The limit as x approaches 0 from below, the 1/x is negative infinity.
00:35:45.500 --> 00:35:50.900
Let us do the other one, let us approach 0 from above.
00:35:50.900 --> 00:35:59.800
The limit as x approaches 0 from above of 1/x = positive infinity, because we see as we get close to 0,
00:35:59.800 --> 00:36:06.400
the function, the function is going off to positive infinity.
00:36:06.400 --> 00:36:09.400
Negative infinity and positive infinity are definitely not the same thing.
00:36:09.400 --> 00:36:13.200
The limit does not exist.
00:36:13.200 --> 00:36:20.700
If the function were different, if the function were both going like that, the left hand limit is going off to positive infinity.
00:36:20.700 --> 00:36:23.400
The right hand limit is going off to positive infinity.
00:36:23.400 --> 00:36:24.300
They are the same.
00:36:24.300 --> 00:36:41.300
We say that the limit of the function is positive infinity.
00:36:41.300 --> 00:36:44.300
Let me actually formalize what I just said.
00:36:44.300 --> 00:36:49.000
We will do another example.
00:36:49.000 --> 00:36:54.400
This time we will take the function f(x) is equal to 1/ x².
00:36:54.400 --> 00:36:56.700
We know what that one looks like.
00:36:56.700 --> 00:37:00.300
It is exactly what we just described.
00:37:00.300 --> 00:37:07.500
There, and it is there, this are coordinate axis.
00:37:07.500 --> 00:37:15.200
In this case, we want to know what the limit is as x approaches 0 of 1/ x².
00:37:15.200 --> 00:37:23.100
What is it, we see as we approach 0, this is our 0, from the left, it goes to positive infinity.
00:37:23.100 --> 00:37:27.300
As we approach it from the right, the function goes to positive infinity.
00:37:27.300 --> 00:37:32.400
Therefore, this limit is equal to positive infinity.
00:37:32.400 --> 00:37:38.000
It is very simple, graphs are really great.
00:37:38.000 --> 00:37:41.600
They tell you exactly what is happening to a function.
00:37:41.600 --> 00:37:44.300
A table of values gives you more information.
00:37:44.300 --> 00:37:48.800
It actually gives you specific numbers, if the graph is not all that great.
00:37:48.800 --> 00:37:52.800
Of course the last thing we are going to do, we are going to learn how to calculate limits analytically,
00:37:52.800 --> 00:37:56.300
mathematically, to get a precise value for what it is.
00:37:56.300 --> 00:37:59.000
You are going to use all three of these tools.
00:37:59.000 --> 00:38:07.400
The graphical, the tabular, and the function itself, the calculus itself.
00:38:07.400 --> 00:38:17.800
Let us take a look at this one.
00:38:17.800 --> 00:38:19.000
This function right here.
00:38:19.000 --> 00:38:38.400
This function is f(x) = 4x² + 2x - 5 divided by x² + x – 1.
00:38:38.400 --> 00:38:42.000
I chose some random function that looks like, I wanted to be a rational function.
00:38:42.000 --> 00:38:44.700
It would look something like this.
00:38:44.700 --> 00:38:47.200
What we ask is the following.
00:38:47.200 --> 00:38:53.700
This is our f(x), this is a graph of f(x).
00:38:53.700 --> 00:38:59.100
Let me do it in blue actually, the little differentiation.
00:38:59.100 --> 00:39:10.100
What is the limit as x approaches infinity of f(x).
00:39:10.100 --> 00:39:16.400
What is the limit as x approaches negative infinity of f(x)?
00:39:16.400 --> 00:39:22.600
That is it, we are just asking what happens when x gets really big, what does f do?
00:39:22.600 --> 00:39:26.300
When x gets really big in the negative direction, what does f do?
00:39:26.300 --> 00:39:29.000
That is all we are asking.
00:39:29.000 --> 00:39:30.000
Let us take a look.
00:39:30.000 --> 00:39:33.900
Based on the graph alone, let us do the first one.
00:39:33.900 --> 00:39:35.100
We will do it over here.
00:39:35.100 --> 00:39:52.200
The limit as x approaches positive infinity of f(x), it looks like as x gets really huge, it looks like the graph is approaching 4.
00:39:52.200 --> 00:39:54.600
We are going to say it = 4.
00:39:54.600 --> 00:39:57.900
It is approaching 4 from below.
00:39:57.900 --> 00:40:06.800
Here, this limit, the limit as x goes to negative infinity, as it gets bigger in that direction,
00:40:06.800 --> 00:40:11.300
the same thing, it looks like the function itself is dropping down.
00:40:11.300 --> 00:40:15.200
It is getting close to 4.
00:40:15.200 --> 00:40:17.300
It also equals 4.
00:40:17.300 --> 00:40:24.100
In this particular case, when you are dealing with infinities, it is the same thing.
00:40:24.100 --> 00:40:26.200
There are two basic conventions regarding infinity.
00:40:26.200 --> 00:40:35.800
When you see the limit as x approaches infinity of f(x), some people take this to mean positive infinity.
00:40:35.800 --> 00:40:38.800
They separate that from negative infinity.
00:40:38.800 --> 00:40:47.400
Or when you see infinity, it means do both, do the positive and negative just like you would for x approaches 3.
00:40:47.400 --> 00:40:52.200
You are going to have to do the x approaches 3 from below, x approaches 3 from above.
00:40:52.200 --> 00:40:55.300
With infinities, we generally tend to keep them separate.
00:40:55.300 --> 00:41:04.500
When you see the limit as x approaches infinity, it generally means positive infinity.
00:41:04.500 --> 00:41:08.100
The limit when x approaches negative infinity, it is the negative infinity.
00:41:08.100 --> 00:41:11.100
We definitely keep these separate.
00:41:11.100 --> 00:41:17.300
Sometimes when I see the limit as x approaches infinity, I tend to just assume that it is both.
00:41:17.300 --> 00:41:19.800
We are going to do both and we will specify which one we are doing,
00:41:19.800 --> 00:41:22.500
when we are actually dealing with the specific problems.
00:41:22.500 --> 00:41:25.500
In general, we handle the infinity separately.
00:41:25.500 --> 00:41:30.900
Do a negative, do a positive.
00:41:30.900 --> 00:41:35.600
They both happen to equal 4 but they are separate limits.
00:41:35.600 --> 00:41:38.900
That is why we handle them separately.
00:41:38.900 --> 00:41:46.700
You might have this limit be 5 and you might have this limit be 9, or it might be infinity itself.
00:41:46.700 --> 00:41:50.000
The function might do something very different.
00:41:50.000 --> 00:41:55.200
The right side of the graph, as opposed to the left side of the Cartesian coordinate system.
00:41:55.200 --> 00:42:01.700
Just because the limit is 4 and the limit is 4, this is not the same as a left hand or a right hand limit.
00:42:01.700 --> 00:42:09.500
Our left or a right hand limit is you are approaching a number from the left and from the right.
00:42:09.500 --> 00:42:17.200
With infinities, they are separate because you are actually going to the right infinitely and to the left infinitely.
00:42:17.200 --> 00:42:19.000
From your perspective, that is the right and that is the left.
00:42:19.000 --> 00:42:22.600
They are separate limits.
00:42:22.600 --> 00:42:28.900
We do not actually say because this is the case, we do not say the limit as x approaches infinity = 4.
00:42:28.900 --> 00:42:36.000
The limit as x approaches positive infinity = 4 and the limit as x approaches negative infinity = 4.
00:42:36.000 --> 00:42:38.100
They are separate limits, we do not combine these.
00:42:38.100 --> 00:42:45.900
For infinite limits, again, treat them separately.
00:42:45.900 --> 00:42:47.400
Let us look at another function.
00:42:47.400 --> 00:43:03.500
This function right here is f(x) is equal to the sin of 5/x, that is what I have written here.
00:43:03.500 --> 00:43:14.200
The question is, what is the limit as x approaches 0 of f(x).
00:43:14.200 --> 00:43:21.300
Here is our 0 mark, we want to ask, we are going to approach 0 from the left,
00:43:21.300 --> 00:43:24.500
we are going to approach 0 from the right because it was not specified.
00:43:24.500 --> 00:43:26.600
You have to do both.
00:43:26.600 --> 00:43:30.700
0 is a specific number, it is not an infinity.
00:43:30.700 --> 00:43:32.800
This is a really wild function.
00:43:32.800 --> 00:43:44.900
We see that it gets closer and closer and closer, the limit as x approaches 0 from below of f(x),
00:43:44.900 --> 00:43:49.500
it just wildly jumping back and forth.
00:43:49.500 --> 00:43:56.400
As we see that even if we move a little bit, like an infinitesimal amount, that function just jumps up and then jumps down.
00:43:56.400 --> 00:44:00.600
It does not seem to be converging to anything.
00:44:00.600 --> 00:44:04.100
This is -1 and this is +1.
00:44:04.100 --> 00:44:05.900
It seems to be oscillating.
00:44:05.900 --> 00:44:09.500
This is an example of a function that as you get closer and closer to a number,
00:44:09.500 --> 00:44:16.100
the function itself starts bouncing back and fourth, oscillating between two numbers, +1 and -1.
00:44:16.100 --> 00:44:20.500
It cannot decide, the limit does not exist.
00:44:20.500 --> 00:44:26.400
In order for a limit to exist, it has to be a number and it has to get close to that number and stay close to that number.
00:44:26.400 --> 00:44:29.700
The closer you get to a, that is the whole idea.
00:44:29.700 --> 00:44:33.800
It converges, that word convergence in mathematics is huge.
00:44:33.800 --> 00:44:40.400
It is everything in calculus, it is about convergence.
00:44:40.400 --> 00:44:44.700
The same thing from the other side, when we approach 0 from the right, the same thing happens.
00:44:44.700 --> 00:44:51.900
Here it is reasonable but as we get closer and closer to 0, it starts oscillating really crazy back and forth.
00:44:51.900 --> 00:45:01.400
The limit as x approaches 0 from above also does not exist.
00:45:01.400 --> 00:45:06.200
Here, the limit does not exist because it is oscillating between +1 and -1.
00:45:06.200 --> 00:45:17.500
It is not converging to some single number or it is not going off to positive or negative infinity.
00:45:17.500 --> 00:45:36.000
The idea of a limit, very important, that converges,
00:45:36.000 --> 00:45:54.500
is that as x gets closer and closer to some a, that f(x),
00:45:54.500 --> 00:46:06.800
the function itself, gets closer and closer to some number, to a finite number.
00:46:06.800 --> 00:46:14.700
To an actual number that we can actually say 2, √6, 9, 4000, some finite number that we can point to.
00:46:14.700 --> 00:46:25.200
Some finite number and stays close to that number.
00:46:25.200 --> 00:46:31.500
Here it approaches 1, but then it jumps off to -1, then it jumps of to +1, it jumps back to -1.
00:46:31.500 --> 00:46:33.000
It is no saying staying close to one of these.
00:46:33.000 --> 00:46:40.500
There is no convergence, very wild function.
00:46:40.500 --> 00:46:42.000
Let us see what else we have.
00:46:42.000 --> 00:46:47.000
Let us go ahead and round this out.
00:46:47.000 --> 00:46:56.000
Let me go back to blue here. We have seen the following.
00:46:56.000 --> 00:47:16.000
We have seen the limit as x approaches a of some f(x) = l, some finite number, some finite actual number.
00:47:16.000 --> 00:47:18.300
That was one thing that we have seen.
00:47:18.300 --> 00:47:25.500
We saw the limit as x approaches a of f(x).
00:47:25.500 --> 00:47:30.000
We see it go off to positive or negative infinity, like the function for 1/x.
00:47:30.000 --> 00:47:38.300
The a was 0, it is approaching some specific number but the function itself is flying off to positive or negative infinity.
00:47:38.300 --> 00:47:56.600
We also saw an example of the limit as x itself approaches infinity of f(x) = l, some actual number, that was the rational function.
00:47:56.600 --> 00:48:05.800
We saw that as x gets really big positive, really big negative, the function itself got close to 4.
00:48:05.800 --> 00:48:07.100
4 is an actual number.
00:48:07.100 --> 00:48:13.000
This time x was approaching infinity.
00:48:13.000 --> 00:48:21.700
Maybe from pre-calculus you remember, any time we take x to be going positive or negative infinity, we called it end behavior.
00:48:21.700 --> 00:48:28.500
The limit as x approaches positive infinity of f(x) is asking what is the end behavior of the function.
00:48:28.500 --> 00:48:41.300
We also saw that as an example of the limit as x approaching a from below, not equaling the limit as x approaches a from above.
00:48:41.300 --> 00:48:48.800
Here the limit did not exist.
00:48:48.800 --> 00:48:55.300
In order for a limit to exist, an actual finite number or positive or negative infinity,
00:48:55.300 --> 00:49:03.500
the left hand limit and the right hand limit, as x gets close to a single number, f has to go to the same number.
00:49:03.500 --> 00:49:05.600
They have to equal each other.
00:49:05.600 --> 00:49:08.300
If they are not equal to each other, the limit does not exist.
00:49:08.300 --> 00:49:16.000
The left hand limit exists, the right hand limit exists, but the limit itself does not exist.
00:49:16.000 --> 00:49:19.000
All things are possible, you might have a left hand limit exist but the right hand limit does not exist.
00:49:19.000 --> 00:49:23.800
Anything is possible.
00:49:23.800 --> 00:49:28.300
One more time, I know you are going to get sick and tired of hearing it.
00:49:28.300 --> 00:49:31.500
I’m certainly sick and tired of hearing myself saying, but it is very important repetition.
00:49:31.500 --> 00:50:03.400
f(x), the limit, the symbol, limit as x approaches a of f(x) is asking, as x gets arbitrarily close to a, what is f(x) doing?
00:50:03.400 --> 00:50:08.300
That is it, very intuitive, use your intuition.
00:50:08.300 --> 00:50:13.100
You have the graph, you have the table of values, and you are going to learn to do this analytically.
00:50:13.100 --> 00:50:48.200
It either converges to a number, diverges to positive or negative infinity, or oscillates, or it just not does not exist.
00:50:48.200 --> 00:50:49.100
Those are the possibilities.
00:50:49.100 --> 00:50:51.200
Oscillates is the same thing.
00:50:51.200 --> 00:50:54.200
When something oscillates, it does not exist.
00:50:54.200 --> 00:50:56.600
It either converges to a number, it exists.
00:50:56.600 --> 00:51:01.400
It diverges to infinity, positive or negative, or it does not exist.
00:51:01.400 --> 00:51:08.200
Those are the possibilities for a limit and that is all.
00:51:08.200 --> 00:51:12.100
Let us round it out.
00:51:12.100 --> 00:51:22.300
When the limit as x approaches a of f(x) = l and the limit from below,
00:51:22.300 --> 00:51:41.700
the limit as x approaches a from above of f(x) = l, we say that the limit exists.
00:51:41.700 --> 00:51:49.500
The limit as x approaches a of f(x) = l.
00:51:49.500 --> 00:51:52.800
There you go, thank you so much for joining us here at www.educator.com.
00:51:52.800 --> 00:51:53.000
We will see you next time, bye.