Hi folks, welcome back to our discussion on continuity. Let's begin with a warm up question. We'll do one like easy and then one medium one. I'll put it on the screen here. Pause the video before I give you the answer and see if you can work it out. Here's your question. Where is the function f of x equal to 1 over x continuous? Remember CTS is just my abbreviation for continuous. Math people are the laziest people ever so we abbreviate in shorthand everything. Where's the continuous? Pause the video, see if you can figure this one out. Are you ready? Here we go. Let's talk about the answer. With any function that you know the graph of, and there's a good number at this point. Obviously there's more that we don't know than that we do. But this is one whose graph we should know. I'd like to just draw the picture, get to visualize the function you're working with. This is the function. Clearly it's not defined at zero. You can plug in zero, you get one over zero. That's bad news. Otherwise it's continuous everywhere else. There is no problem of evaluating. This is a rational function. We know that rational functions are continuous everywhere on their domain. This question about continuity really is translated into what's the domain question. Part of the reason why I like this question is now you have to be careful how you give the answer back. Basically the answer is all x except zero. You can say, you know all x, let's write it out. All x except zero, and that's okay. That's like a sentence. Looking at all these words, oh, the horror. You could write x naught equals zero. You can say like x in r. Now we're talking, this is making the math person in me happy. There's no words here. A real number, but non-zero. You can also get back if you want to use inequalities, nice union two pieces, you can get back all the numbers from negative infinity to zero, union zero to infinity. It's not a terribly difficult question, but the idea is you have to communicate it back to me in such a way that makes sense. All three of these are perfectly fine or any slight variation of them. That's the first one. Let's do another one. Mix it up here. Let's take a limit. We haven't taken a limit in a little while and you're going to get asked for a lot of limits on things that are coming up. Let's do this one. We have the limit as x goes to pi of sine of x over cosine x, most students get this wrong the first time they see it. So I want you to pause the video and work this out before I go over the answer. Try it. Are you ready? Three, two, here we go. Remember when you have a limit, this is really an exercise in thinking and problem-solving in practicing your approach. What are our techniques for limits? A lot of folks get confused with this one, but if you think about it, one, it involves Trig Series. How much trig do you remember? One of the first things we do with a limit is we just plug in when we can. The question is if our first approach is always the plug-in doesn't make sense in this case. Here's a question for you. What is sine of pi over cosine of pi? Is that defined? Is this a real thing? Let's take a little excursion off to the side here to the world of trig, have our unit circle. Good times. What does it mean sine of pi, hopefully you know this, but if you don't think about it, pi is like 180 degrees pi radians. That puts you over here. x is your cosine value, cosine of the angle and y is your sine of the angle. So over here you're at the point minus one zero. The x value is cosine of pi and the y value is sine of pi. In your head, you should picture a circle. Imagine going over the top, landing right at the x axis, and if you want sine of pi, you say, what's the y value right there, that's zero. What is the cosine value? That's minus one. So now we have the expression zero divided by negative one. Is that a real thing? Is it defined? And the answer is yes, of course it is. This is just zero. As long as your numerator can be zero and a fraction, that is not a problem. It's just when the denominator is zero, that's when you start running into trouble. Or if you have both, zero over zero. That's always undefined as well. Okay. So moving on, let's put a theorem on the board. This is a wonderful theorem, but unfortunately, it does not have a name. Like most of the important theorems in our book have names and stuff like Squeeze theorem or something like that. This one doesn't. So I'll put it down and then I'll be sad to not have a name for it. But here's what it says. We are going to use this one a lot. If f is continuous at b, and the limit as x goes to a of g(x) equals b, then the limit as x goes to a of f of g(x) equals f(b). Stare at it for a second, read it back to yourself, as with all definitions and theorems, they're abstract and weird and don't make sense at first. So if f is continuous at a point b and the limit of some other function is equal to b as x approaches a, then the limit, alright whatever.What does this mean? So i.e, we try to simplify it from its definition, from its theorem. It means if you have a composition of functions, so you have f composed with g and f is continuous, this is important, this is the big condition that allows us to work. So if you have a composition and f is continuous, then you can move the limit inside. We are going to use this a lot and I will show you an example in one second. But this is basically what it means you know, in plain English. If the outside function is continuous, then you can move the limit inside. Let us just see an example of that, to sort of appreciate this theorem. So let's take the limit as x goes to pi of sine of x plus sine of x. Nice scary function, two sine values, oh the horror, it's terrifying. However, we say, not today my friend because look, sine, this is an outside function. This is continuous. This is a nice continuous function. So what does that mean? I can bring the limit inside. It only works if the function is continuous, but lucky for us, there's lots of continuous functions that we're going to see as we go through this. So let's write this this way, sine of the limit as x approaches pi of x plus sine of x. Now I have an easier limit to work with. And this is like a pretty straightforward question because I can plug in as x goes to pi, this thing just becomes pi and then sine of x becomes sine of pi. We just did this, sine of pi of course is zero. So this becomes sine of pi, really like pi plus zero, which is sine of pi, which is zero. So when you have something that's built on compositions and it looks kind of scary, if the outside function is continuous, you can always move this in. Let's do one more example and you'll see another one. So example here, what is the limit as x goes to one of the natural log of five minus x squared over one plus x? Alright, so natural log once again is another continuous function. So it's kind of a big expression built on compositions. So this is continuous. So by the theorem with no name, we can move the limit inside. And this becomes the natural log of the limit as x goes to one of five minus x squared over one plus x. So now I have an easier limit, like I can completely ignore the natural log as I work out this limit. So what does this work out to be? This becomes the natural log. This is a rational function we can just plug in because one is inside the domain, so it becomes five minus one squared, that's four over one plus one, which is two. This of course is better known as natural log of two. So just two examples and you are going to see more of them as we go through of how to use this theorem. One more example, I'm going to use a different color for this. I'm going to call this one the classic. This one is on every single standardized tests that you might get about continuity on any calculus thing. So know how to do this one, or ones like it, and I'll put it here for us to see it. It involves piecewise functions, so piecewise classic. So find a number c such that the function f of x is continuous, and the function f is defined by piecewise, cx squared plus 2x, and x cubed minus c of x. We'll say this piece will be the one we want if x is less than two, and this piece will be the one we want if x is greater than two. So this is a classic thing. So what's happening? Let's just talk about this before we go ahead and solve this for a second. So the top piece is some quadratic degree-two thing, and it's a parabola for x less than two. So some like left side of the parabola, may not be the vertex exactly, but if we drawing a picture, at two something's happening. So it's continuous everywhere. It depends on c obviously which one I pick, but there's some part of the parabola that's going off and doing its thing. So there's some left side of the parabola. For greater than or equal to 2, there's a cubic function. This is the disco function, and it's some cubic, and it would start low, go to high, and it's doing its thing. So they don't necessarily have to, some like that, give or take. They don't have to touch. So there's two pieces, they don't have to touch. What you are trying to find is the value of c that moves these two things in alignment. You're trying to find the value of c that moves these two things in alignment, so that the function is continuous, so that there'll be no gap, there will be no jump discontinuity between the two things. So what's the value of c that makes this work? So this is a classic question because it puts a lot of stuff together. So remember the definition of continuity. If you ever stuck with a continuity question, recall the definition. I need the limit, so hopefully, you realized that the only place I have to really worry about is two, quadratic and cubic or continuous elsewhere. So I'm really trying to close this gap. So everything I really just wanted to be continuous at two, so definition means the limit as x goes to two of f of x would equal the function evaluated at two. The limit as I approach two from the right and left is got to be equal to the function evaluated at two. The function evaluated at two, hopefully we know how to evaluate piecewise functions, that's going to be the one that involves the greater than or equal. If I evaluate at two, this is going to be 2 cubed minus 2c or 8 minus 2c, and so the cubic is a polynomial. The limit from the right as I come in from the right, the overall limit here, the two-sided limit, as I come in from the right, I'm going to go to this 8 minus 2c, whatever my c is, this is 8 minus 2c right at this point. In order for these things to match, I have to approach from the left. As I approach from the left, I got to get this point down here, whatever it is, to equal 8 minus 2c. Well, what is the point from the left? That's going to be the top piece. So I need to set this equal to c while I got to plug in at two, so this is going to be c times 2 squared or 4 plus 2 times 2. But the point is you plug in a two and I need to set these equal to each other. The limit from the right has to equal the limit from the left. So now it just becomes an algebra problem solved for c. So we have, let me clean this up a little bit. 8 minus 2c is 4c plus 4. What can we do? We could subtract 4, we could add 2c and you get 4 equals 6c, and therefore c is equal to four-sixth better known as two-thirds. So that is the value of c. It notices the only value of c that makes this function continuous, it combines the two pieces so that you can draw them without picking up your pencil. So set the limit as you approach two is equal to the function evaluated at two. So these are just classic examples that you're going to see, and keep these in mind as you go through, and work on the homework and some other problems. Good job. See you the next video.
