 Welcome back to our lecture series Math 12-10 Calculus I for students at Southern Utah University. As usual, be a professor today, Dr. Andrew Misaldein. In this lecture number 11, we're going to continue to talk about limit laws that are useful for us to compute limits. But then in this lecture number 11, we're going to focus on one very important, although sometimes difficult to implement, limit law known as the squeeze theorem. The picture to the right, which we'll look at this in a little bit more detail in just a second, is an illustration of what's going on here. But before I mentioned the squeeze theorem, let me mention one preliminary to the squeeze theorem, and this is the basis of where it's going to come from. If we have two functions say f and g, such that f of x is always less than or equal to g of x whenever you're near a. So that is there's some neighborhood around the point x equals a, so that f of x is less than or equal to g of x. But a possible exception could be that at a, who knows what happens, right? It doesn't really matter because when it comes to the limit, we don't consider what happens at x equals a. We consider what happens near x equals a. So if there's a neighborhood around a, perhaps excluding a itself, so that f of x is less than or equal to g of x, then the limits will inherit this inequality. That is the limit as x approaches a of f of x will be less than or equal to the limit as x approaches a of g of x. So inequalities are preserved when it comes to the limit calculations. Now, one thing I do have to point out that if we made a slight modification here, if we had changed this to be f of x is strictly less than g of x, then we still must keep less than or equal to. Because if f, even if it's strictly less than g of x, it turns out as we get closer and closer to a and we take the limit, equality could be obtained even when the restrict inequality and all the other settings. All right. And so applying this principle twice, that is if we have one inequality and then we stack another inequality, we can sandwich a function f between two other functions and apply this principle to get the so-called squeeze theorem. Some people call it the sandwich theorem. That's a very illustrative name right there, but it makes me often too hungry when I teach a class. So we'll call it the squeeze theorem. And so the squeeze theorem will be discussed in the following way. So we have, in this case, three functions. f of x is less than or equal to g of x, which is less than or equal to h of x in some neighborhood near x equals a. So that is if this this inequality has to be true for all numbers x that are near a, although the number a itself could be excluded from that domain, just like the assumptions of the previous theorem right there. Now this right here, this inequality, we're going to often refer to as a squeeze, that g of x is squeezed between the function f of x and g and h of x there. g of x is in the middle. Now, so we have this inequality, but we also have the following observation. The limit as x approaches a of f of x is some number, we'll call it l. And likewise, the limit as x approaches a of h of x is also l. So when you look at this squeeze earlier, the limit of f is going to go off towards l. The limit of h will also go between l. So by the previous statement about inequalities, we get that the limit of g of x must be greater than or equal to l and less than or equal to l. Well, as l is the same number, the only way that could be anything is it has to be l itself. So in other words, if g is squeezed between two functions whose limit goes to zero, then the limit as x approaches a of g of x must likewise be that same number, which is l in that situation. And so this illustration over here is supposed to give you an idea of the basic, gives you the geometric picture for the basic idea behind the squeeze theorem. So we have our lower function f for which in all of these videos that we have to use the squeeze theorem, I will always denote the lower function using the color orange. We also have this upper function in our cases, h. And again, to keep things color coded, I'm going to always use the upper function. That is the upper function will always be denoted as green. And then finally, the function that gets squeezed between the upper and lower bounds, we're going to call that function g in this case, and then its color will always be yellow. All right. And so when you have these two functions, h and f, whose limit comes together at a point, the only way you can squeeze between the two is to go, well, you kind of have to pass through that bottleneck that kind of happens. It doesn't matter. The only way we can get a limit is we have to go through that common point right there. And that's the squeeze theorem. So if we can squeeze a function between two others and their limits go together, then the inner function must have that same limit. This squeeze theorem will be a very powerful tool when we have to calculate limits of functions for which algebra itself is insufficient. In previous videos, particularly in lecture 10, we saw examples of computing limits of difference quotients, and we were able to do algebraic simplification like the FOIL method, rationalizing square roots, clearing denominators and things like that. We were able to use algebraic techniques, algebraic manipulations to simplify functions, and then we could take the limit there. But there are some functions where the techniques necessary to make these calculations transcend what algebra can do for us. These functions are often referred to as transcendental functions, because again, they transcend the methods of algebra. This would include exponential and logarithmic functions, trigonometric functions, sine cosine tangent, and their inverses. And so in those situations, we often need non-algebraic tools to help us, and the squeeze theorem is going to be one of our favorite and useful of those techniques. So consider the limit as x approaches 0 of x squared times sine of pi over x. We're going to show that that's equal to 0. Now the problem, and I mean, the way you might think of it is like, okay, I have a product two functions, there's x squared and there's sine of pi over x, for which you could try to break this up into the limit of x squared, right? And then you also get the limit of sine of pi over x. And so it's very tempting to be like, well, the limit of x squared will be 0 as x approaches 0, but notice the limit as x approaches 0 of sine of pi over x. That thing is undefined. This is what we played around with earlier in this lecture series to show you that the intuitive notion of a limit can have some problems, but the limit of sine of pi over x doesn't exist. And so what is 0 times does not exist? Well, does not exist is not a number by the name, the limit doesn't exist. So we can't factor the limit in order to calculate this one. We need a different approach to correctly demonstrate that this limit is equal to 0. And this is where the squeeze theorem comes into play. Now, whenever you have an inequality or excuse me, whenever a function involving like sine or cosine, it's important to remember the following inequality. Sine of theta, no matter what that angle is, will always be trapped between negative one and one. The largest sine could ever be as one. The smallest sine could ever be is negative one. And of course, I mean this with no transformations to its amplitude in this play. This is the standard sine function will sit between negative one and one. And this is independent of that angle function theta. So in our case, we're going to replace theta with pi over x. This, the inequality is no less true in this setting. Sine of pi over x, it always fluctuates between one and negative one. That you can never get larger than that. You can never get smaller than that. You have those bounds. Now, if we take this, if we take this inequality and we times everything by x squared, notice you'll take negative one times x squared, you'll get negative x squared. You'll take sine of pi over x, you'll get x squared sine of pi over x. And if you take one times x squared, you're going to get x squared. You'll notice now in yellow, we've recreated the function that we're trying to take the limit of. And now this function is squeezed between two other functions, negative x squared and x squared. Now I should mention that in this example, I'm using the fact that x squared is always greater than or equal to zero. So when I times an inequality by x squared, I don't change the direction of inequality. If you're using something like x cubed on the other hand, an approach similar to what we're seeing in this video could be used, but be aware that x cubed will change its sign, depending on whether you're above zero, that is you're approaching zero from above, or you're below zero. And that just means that you have to do this problem in two parts. You'll compute the right-handed limit using the squeeze theorem, and you can compute the left-handed limit using the squeeze theorem. It's not a horrible modification, but we'll keep it simple in this one with just x squared. So we now have our squeeze. We have x squared sine of pi over x is squeezed between two functions, negative x squared and x squared, for which I have confidence we can compute these limits by direct substitution. After all, by limit laws we've seen previously, the limit as x approaches zero of x squared will just be zero, just by direct substitution, zero of course is zero squared. And direct substitution also applies to negative x squared, because this will be negative zero squared. So these limits are equal to zero. So our function is sandwiched between two functions, such that these two functions both go off towards zero as x approaches zero. And so by the squeeze theorem, that then implies that the middle function will have the same limit, which was going to have to be zero in this case. And so up here I have an illustration of these functions. The green dotted function right here is y equals x squared. The orange dotted function is this one, y equals negative x squared. And then our function right here is in play, x squared times sine of pi over x. So while sine of pi over x by itself acts super bizarre when you get close to zero, it turns out throwing a factor of x squared in there dampers the amplitude of sine, so that the squeeze theorem applies and forces that the limit has to be zero. Now notice this function, x squared times sine of pi over x is undefined at x equals zero. If you plug in zero, you'll divide by zero, it's undefined. I'm not saying the function's defined at zero. I'm saying the limit is equal to zero, and this is a consequence of the squeeze theorem. Let's play around with a similar example of this. Let's compute the right hand and limit as x approaches zero from above of the square root of x times e to the sine of pi over x. And so we have the sort of the same problem here. We have this limit as x approaches zero from the right of sine of pi over x. It doesn't, it's not, this limit doesn't exist. And therefore we can't just use direct substitution to compute this limit. We're gonna need to use the squeeze theorem here. For which consider the function for a moment, f of x equals e to the sine of pi over x. So consider that, consider that function. Well, exponential functions are always positive. Irrelevant of what your base is, as long as it's a positive number to avoid imaginary numbers. If the base of an exponential is positive, the exponential expression will always be positive. And so it doesn't matter how complicated the exponent is, e to the g of x will always be greater than zero. That's gonna serve as our lower bound in just a second. Because our function f of x, we can allow for as complicated as an exponent as we want. We see that zero will be less than e to the sine of pi over x. All right. We also know, like I mentioned on the previous slide, that sine of pi over x is less than or equal to one. Now, since e to the x is an increasing function, that is, you know, if you look at the graph of this thing, it gets bigger, bigger, bigger as x, the y-coordinates gets bigger as the x-coordinates gets bigger here, right? So if sine is bounded between negative one and one, the biggest sine of pi over x is ever gonna get is gonna be one. And as e to the x is an increasing function, that means that the biggest that sine, excuse me, the biggest that e to the sine of pi over x could ever be, the biggest it could be is e to the first, where one is the largest the x one it could ever be. e to the first, of course, is just e. And so now we see some important bounds. We can see that this, the squeeze is now evolving. e to the sine of pi over x will be greater than or equal to zero. I mean, admittedly it was greater than zero, but we'll just say square root of n equal to, right? That's fine. Even if even equality is impossible, it's still true. So we get the e to the sine of pi over x is greater than or equal to zero. It's also less than or equal to e. Now we're gonna multiply this inequality by the square root of x, much like we saw in the previous example, the square root of x is always not negative. So when we multiply by anything, we don't have to worry about the inequality shifting directions here. So we're gonna get that zero times square root of x, of course, is zero. We're gonna get e to the sine pi over x times square root of x. That's the function we care about. That's the limit we're trying to evaluate. And then e times square root of x will be itself, right? e is just a number in this situation. Now notice what happens here, that as we take the limit as x approaches zero from the right, e to the square root of x, we can evaluate this one. This is gonna become, there should be an e right there, this is gonna become e to the square root of zero, which the square of zero is just zero, e times zero is zero, hence where we get this number right here. And then the left bound, since zero is a constant function, the limit as x approaches zero from the right will just be zero itself. And so what we see here is that the left bound goes to zero, the right bound goes to zero, and so the function we care about, which is squeezed in between them, it has to also go towards zero. So the squeeze theorem applies here, and we see that the limit as x approaches zero from the right of the square root of x times e to the sine of pi over x must be zero. And so computing the right-hand limit is really no different than a two-sided limit or a left-handed limit here. The reason we mostly need a right-hand limit is that the square root of x is not defined for negative numbers, so we can't actually approach zero from the left on this function. Let's look at one last example. It's kind of a strange one, but it also, it's a good illustration of the squeeze theorem here, sometimes called a squeezed lemma. Lemma is just a theorem that is used to support other arguments. So we're using it to compute this theorem, or compute this result here. So let's take the piecewise function where f of x, it looks like x squared when x is a rational number, but it looks like zero when x is an irrational number. And so this is a really, really curious function because it's kind of impossible to draw in the usual sense because irrational numbers are arbitrarily close to rational numbers. What I mean by that is any irrational number can be approximated by a sequence of rational numbers. In terms of the geometry, you can't really separate them. They're right next to each other. They're kind of, they're intermingled, that is. And so drawing this thing is kind of a curious thing, but in terms of a function definition, it makes sense. If you give me a number, I can decide whether I'm gonna square it or whether I'm just gonna give you zero, depending on whether it's rational or irrational. So if I took something like f of one-half, this, since one-half is rational, I'm gonna get one-fourth. If I take like f of pi, this is gonna be zero because it's irrational. If you give me like f of one, this is gonna be one squared, which is one because one is rational. But you'd want like f of seven, this will be 49 because it's rational. If you take f of the square root of two, that'll be zero since it's irrational. We can compute this function. So it makes sense to ask things about limits. Now, this is a very curious function for which this function is defined for all real numbers, but at no point is the limit defined with one exception. Turns out this function does have a limit defined at zero, and in that case, it's gonna be zero. And so let's demonstrate how this works using the squeeze theorem. So our function f of x, I'm gonna squeeze between two functions, zero and x squared, which you can see where those came from, right? x squared is gonna be the upper bound and zero is gonna be the lower bound in this situation. Notice that zero is gonna go off towards zero since it's a constant function and x squared is gonna go off towards zero as x gets closer to zero because by direct substitution, that'll look like zero squared. So our function, as weird as it might seem, is squeezed between two other functions whose limits go off towards zero. So this also shows that the limit as x approaches zero of f of x here is gonna be zero. So whenever you have a squeeze, that as you have a function sandwiched between two other functions, and at the limit on the left and the limit on the right, both go off towards some common value in this case, it was zero. Then we can infer that the middle function, the function squeezed between the other two must, its limit also exists and it must be the same shared value as the one on the left and the one on the right.