Next thing we want to talk about is how limits behave when working with inequalities. And remember that if I have an inequality, I do need to take some care. Just as an example, like if I have x + 2 less than or equal to 7, I can very easily subtract 2 and you get x less than or equal to 5. But if I had something like -2x less than or equal to 7, all of a sudden little bells and whistles, and all the warnings should be kicking in. I'm about to divide by a negative, and so the sign changes. So the question is, if I have an inequality now with functions and I want to take limits to both sides, do I have to worry about changing the sign under certain circumstances or whatever? The good news is that limits are so nice, the answer is no. Like I said, it's kind of hard to do something bad here, but this is important enough that it gets a theorem. So, I abbreviate that with Thm. So, here's the statement that we're going to use, and this will be a lead to our next technique to solve some limits. If I have an inequality involving functions f(x) and g(x), so I will say f(x) is less than or equal to g(x) for x near a. Then now and the limit exists of course, okay? Then with this inequality, you can take the limit as x approaches a to both sides and nothing's going to go wrong, and you don't have to change the sign. So this is important enough that it gets a theorem. This kind of nice, it doesn't happen with division of negatives, it doesn't happen with multiplication with negatives, you gotta be careful. For limits again, they behave so nice that they play really nice with inequalities, okay? So now from that, we have what's called the Squeeze Theorem. We get another theorem here, sometimes called the sandwich theorem too, you'll see why in a second. So let me just write this down and see what it says. If f(x) is less than or equal to g(x) is less than or equal to h(x). So I have a string of many qualities, when x is near a, and the limit as x goes to a of the two outside functions are equal, Then the limit as x goes to a of g(x) is equal to L. Again, abstract the wordy definition, perhaps not super clear from just seeing it, so let's do an example to see how this works. Let's set up here, so I'll give you the inequality. So let's let 4x- 9 be less than or equal to some function, less than or equal to x squared- 4x +7. Okay, so I'm telling you that there's some function that satisfies this inequality. I want to find the limit as x approaches 4 of this unknown function. Now I don't know anything about the function, I just know that it's bounded between these two things. So now I want to play with this inequality, and in fact I want to take limits of both sides. So from the inequality I'll take a limit as x goes to 4 of 4x- 9. Limit as x goes to 4 of this function f(x), and then less than or equal to the limit as x goes to 4 of x squared- 4x + 7. This is something that you just have to get comfortable doing to inequality or an equation taking limits to both sides. Just as you're pretty comfortable I'm sure adding, multiplying, dividing, doing whatever. Maybe taking e, taking logs taking limits is the sort of calculus equivalent of all that. I'm going to take a limit to all the pieces. Now, the thing I'm after is in the middle, but I don't know anything about f(x), I told you nothing about f(x). But I know stuff by the outside, the 4x- 9, that's a line I know how to evaluate that I just plug in. So this becomes 4 times 4- 9 that's a 16- 9 that's 7, so this thing goes to 7. And this is less than or equal to x- 4 of f(x). Again, right the limit every time. Over here on this side, this becomes, this is a quadratic x squared- 4x + 7, I can just plug in, you get 16- 16 + 7, so it's also 7. So realize what you're looking at here. The thing that I don't know. The thing that I'm after, the limit is a number that is less than or equal to 7, and greater than or equal to 7. What is the only number that is both less than or equal to 7, or greater than or equal to 7? That's 7, that's what this thing is saying. It says when the two outside pieces are equal, then the one in the middle has to be that sort of an obvious statement. But anyway, so this thing is equal to 7. Whenever you use the theorem, it's always nice to sort of cite your work. So this is, we'll say by Squeeze Theorem. Anything that has a name is pretty important. Squeeze hoops, S-Q-U-E-E-Z-E theorem. So, then you get to write a Z, which is always fun. So squeeze them. So we're going to use to do that. Okay, so this is just one way to do it. Let's do another example, maybe a little harder one, to see what's going on. So this is more like the ones that we saw last time. So find the limit as x goes to 0 from the right of the square root of e to the sine of pi over x. Okay, nothing in here says squeeze theorem, but obviously since we're talking about it, it's the way to do it. But let's just go through the thought process. Let's say you see this on a test and you don't know what to do. So the first one is, can you plug in? Is that an option? No, I have sine of pi over x can't plug in 0 downstairs. Should I be worried about this little from the right thing? No, that's just kind of bookkeeping because of the square root. Remember, it doesn't make sense to come in from the left of the square root, so that's just there, just for bookkeeping. Man, what else do we have? We have factoring, can I factor? No, no, no, it doesn't look good. Can I foil? No, there's nothing to do here. Can I do like algebra to clean this up? There isn't anything to do here, I can't combine fractions. I can't do, now, what about conjugate? Maybe I can do conjugate. Is there, I see a square root, but it's not the right form, where the last one, I do change a plus to a minus or something like that. I think this is all we had so far. The next one we're going to keep in mind is the squeeze theorem. Let's squeeze their rescue. I don't know what is we're doing, squeeze theorem. Squeeze theorem says, can I bound something? And it's used a lot when you see sine or cosine. So if you see a sine or cosine in your expression, keep this one in mind. It does, it's not a hard fast rules, exception is everything, but I see a sine or cosine. And the reason for that is, remember from trig, sine of whatever, I don't care what is in the parentheses, it doesn't matter. Sine of any expression oscillates, you're changing how many oscillations, but the actual amplitude is always between 1 and -1, always. You could put the grossest thing you want in this parentheses, the whole sign function is bounded between 1 and -1. All right, so that's hopefully a fact that we all know from trig, that this thing goes from there. Now e to a thing, e raising doesn't change any qualities also, that's nice. So this is like this is known. This is a known, but I'm not working with the function, sine of pi over x. I want to build this thing up. So now let's say, what happens now? Let's do, trying to think I want to present this. I don't want to confuse anyone. So let's go, e to the -1 less than or equal e to the sine of pi over x, e to the 1. Either the one is obviously just e, that's fine. So this is, we're going to exponentiate. So what I'm trying to do, is build up the function that I'm after, so that it appears in the middle. The next thing I'm going to do, is multiply everything, I'm going to multiply everything by square root of x. Now I'm multiplying in an inequality, so I gotta be super careful. Do I have to change the sine? No, the square root of x always positive. So the square root of x e to the -1, which is my one over her if you want, square root of x e to the sine of pi over x, less or equal to, sorry. So I've built the function I'm after in the middle. See how I built it up starting from the first inequality. The last thing I want to do now, just because that's what I'm after, is I want to take limits. So remember the theorem says, if you have an inequality you can take limits to all this stuff. So limit as x goes to 0 of square root of x times say 1 over e, less than or equal to limit, as x goes to 0 from the right, square root of x e to the sine of pi over x, less than or equal to the limit, as x goes to 0 from the right of root x times e. So the thing that I'm after appears in the middle, and it's usually complicated where I don't know what to do. But once I bound the trig function, so notice that I have a 1 here. I replaced the trig function with this upper bound, I replace a trig function with this lower bound, the minus one. The idea is that it simplifies the expression enough, in this case it actually does. Because now I've replaced the limit that I want, right? This is the thing I want with two limits, so the idea is I do two easier limits then one harder limit. The thing on the outside is if you plug in, this goes to 0, right, square root of 0 is 0, so it become 0 times 1 over e, that's just 0. Let me write this again, square root of x, e sine pi over x, Less than or equal to, here's another, one plugin, right? So I replaced tough one with just two plugins you get 0 times e, which is 0. So the thing that I want, the mystery limit from the beginning is some number, less than or equal to 0, and greater than or equal to 0. What does that have to be? That has to be 0 and we'll say by squeeze theorem. Zero by squeeze theorem, so our final answer here is 0, our final answer here is 0. This is a tougher one because if you bury this one on a page of all limits, which is sort of what you're going to see. This just becomes one of the techniques, and this technique obviously is more involved in harder than just plugin and factoring. So squeeze theorem comes up a lot ,we'll do more examples. But if nothing else seems to work and you have a sine or cosine in your question, it's probably a squeeze theorem one. All right, go over this one before we try some other ones. And try this, to keep this in mind is just one more of the techniques. There will be one more technique that we use, but these are the big six, we'll teach the other one at a later time.
