# Ch 2.3 Limit Laws Ch 2.4 Precise Definition of a Limit (Delta-Epsilon Function) Ch 2.5 Continuity

Feb 22, 2022parabola, graph the cubic

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, and what you have in x equals three, they should match. Unless you want to graph#### function

s piecewise. I could show you that (mumbles). You're all like, "Don't go through with it, we get it." (students laughing) Hey, can't you do a hundred of these in a row? Yeah, what's the idea? We are only concerned here, right? There you go Hello, last problem. You know what I wanted to do in the end, do you remember? It was that problem that I forgot last class, remember? It's one for parts, let me do one of those.So when I look for one in this section, I know they have one, and there it is. This is number 41, it's on page, it's actually on page 128. Alright, here it is. For what values of x is f of x discontinuous? Well, f of x is equal to two plus x squared four minus x if x is less than or equal to zero and zero is less than x is less than or equal to two. And my last, x is greater than two, x. For what values of x is f of x discontinuous? Any suggestions on how to do this, any recommendations?

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## ch 2 3 limit laws ch 2 4 precise definition of a limit delta epsilon function ch 2 5 continuity...

You do this in a test, what do you recommend? I bet they all have great ideas. Check this out, it's a piecewise function. We saw this earlier today. We saw it with the

#### limit

s, right? What was our strategy? - Graphic. - Graphic. Do you all want to graph it? Yes, let's see the graph. What would it mean to be continuous? Connected, right? If it's a broken piece, then it's not going to be what? Continuing, that's all we have to do. Will that work? Now when we're done, I say we look for easier ways. Sounds good? This is how we will end the class.All we're going to do is make the graph. We know how to make a graph. Let's do the graph, let's do the graph in parts. But when we're done, let's find easier ways to say, "Wait a minute," you know what you just have to do? "You really just have to check the 'what?' Look, I don't want to think about it. For

## continuity

, we only care, aren't these smooth graphics? Yes, it's a smooth straight line, that's a straight line, that's a parabola. Actually, we are only concerned with the values of x. X equals zero, and the x equals what?Two, so some of you might find it easier to say, "Why don't you just check the zeros of these two?" X equals zero, and check two for these two, and make sure they match. If they don't match, you're not connected, right? In order for it to be connected, the value of y has to be equal to the value of y, right? But I'm going to pretend, hey, if you like to do graphics, it always works. Here it comes. Alright, two plus x squared, I'm going to make a little table. What is happening at zero? Two, give me a number less than zero.

Negative one, what is two plus negative one? All right, zero-two, one-three, and there's the parable. Whoooo, ch-ch-ch. Alright, make a little table, four minus x, replace the zero. What do you get? - Four. - Plug two, what is four minus two? - Two. - Two, okay. Straight line, right? Okay, where's the hole, where's the hole? Where is the hole in the second, where do I put the open hole, the open circle? - Zero. - Yeah, I'm going to put it right there. I want to make sure you can see that. I should really put an open circle there, right?

Open circle, closed point. I already know a value of x. Can we leave it as an answer? What is one of the answers? Check it out. Where is that broken thing? What value of x? The function is discontinuous at x equals zero. Are all agreed? That's my point, that's how fast we can figure this out. Very good, good, but we also have to review this one. - Two. - Two, so I want one more McTable. What happens at two? You connect two, what do you get? Two, you connect a three and you get a three. A-ha, but it has, I don't have an equals sign.

But this was already closed, so do I put an open circle over a closed point? What happens when you put an open circle on a closed point? What remains? - Closed. - It remains closed. I'll say it again. Look, I'm putting an open circle on something that was already closed, it stays closed. This is still sitting there. And then at three, you have until what? Three and keep going up. So everyone, what do you think? Is the function discontinuous in two? No, it's continuous. Continuity, connected continuous media. Cool? - Awesome. - Hey, in terms of l and c, everyone, if a function has

#### limit

s, if a function is not continuous, can there be a limit?I think so. So first of all, if there is a limit, do you think the functions will be continuous here? Yeah true? Can you say that this is a true statement? If a function is continuous at a point, the limit must exist at the point, right? Are all agreed? - Yes. - That is a true statement. Did you know that you can write the contrapositive? That means you can change this and put the words "no". So I'm going to make another statement. Wouldn't have to imply what? - No c. - No c. You can write it that way.

That means the function doesn't have an existing bound, so what's going on? The function is not what? Continuous. Hey, one more thing before you go. That chart, nice job on the chart, right? Does everyone agree with this? Guess what will happen later in calculus? We speak of a word, "differentiability". See, that's continuous there, but differentiability, we're talking about how it has to be nice and smooth. And the idea that the slope on the left has to be equal to the slope on the right. Everyone, can you see the slope there? Is the slope not negative? What about the slope from the right?

It's positive, so that's our next topic, which will be covered later. Is the function differentiable in two? No. (Laughter) That's a later topic, because the slope would have to equal the slope. You can see, the slope is negative, the slope is positive. But we just finished the

## continuity

. Is it continuous? It's definitely continuous, it's connected, right? It's just a corner. (Laughter) That's not smooth. do you follow me Awesome. Hey, and before you go, who can hit him? Who had a quick way to solve this problem without the graph? Could you solve this problem without the graph? - Yes. - What did you have to check? - Check zero and two. - Mark the zero.Do the numbers match? - No. - You put zero and you get two, you put zero and you get four, and he goes, "Okay, that's one of the answers." So, as he said, you connect two here and you connect two there, but do they match? Yeah, four minus two equals two, and you say, "Okay, that's it," and you're done with the problem. And that's your job. That's all you would have to show, I mean it. Okay, if you hate graphics, or just want to save some time, do it that way. Have a great one. great job.

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