 Consider the limit as x approaches infinity of the function e to the negative x times sine of x. Now it might make sense to just approach this one by trying to stick in the number infinity, right? You know, we can do some arithmetic with infinity at times, right? So we get e to the negative infinity times sine of infinity here. It's tempting to be like, well, e to the negative infinity, that's gonna be zero, right? As you, as x approaches negative infinity, e to the x has a horizontal asymptote right there. It makes sense to try to argue, well, this is gonna be zero times sine of infinity and anything times zero is equal to zero. Well, that is the correct limit, but that's the line of logic doesn't quite work because the issue is sine of infinity, right? Does not exist. There is no in behavior to a periodic function like sine of x. So how do we know that as x is bigger and bigger, bigger, bigger, that something isn't going wrong to throw this off, right? Because not everything times zero is equal to zero, right? Notice zero times infinity is actually an indeterminate form. It could be whatever it wants to be. And so because of that, we have to be a little bit more careful on our calculation. And this is actually a limit calculation we can do using the squeeze theorem. So we have to construct a squeeze and it's really the periodic nature of sine that's sort of forcing us to do the squeeze theorem. That's why can't we just use a straightforward calculation? Well, because sine doesn't have in behavior. But the good news is even though sine of x is kind of forcing us to do this other type of argument, sine comes with very natural boundaries which can be used to give us the nice squeeze we need. Because after all, sine of x sits between negative one and one. Using the color scheme we've introduced in previous videos with the squeeze theorem, the color yellow will represent the function we're trying to squeeze. Green will be the right bound and orange will be the lower bound here with the negative one. So if we take this inequality which holds for sine of course, we could times everything by e to the negative x because every exponential expression that has e to any power will always be positive. And if you multiply inequality by a positive quantity, well, that doesn't change the inequalities. So if you times everything by e to the negative x, you'll get negative e to the negative x right there. You'll get a e to the negative x times sine of x right there and you'll get e to the negative x times one right there. So we get these inequalities. And so this then gives us the squeeze that we want. Let's consider this term on the right. As x goes to infinity here, we're gonna see that this becomes e to the negative infinity as we've seen before. That's just a zero. And this one right here by the same reasoning, if you take the limit as x approaches infinity of negative e to the negative x, right? Well, the negative sign in front can actually come outside in which case you're gonna get negative zero which is the same thing as zero of course in the situation. So the limit on the right goes to zero. The limit on the left goes to zero. And since e to the negative x times sine of x is squeezed between them, its limit will also be zero. So the dampering of e to the negative x going towards zero that did force the limit to be zero but we can't just plug in infinity because sine of infinity is meaningless. But the fact that sine is a bounded function, if we multiply a bounded function by a function which will asymptotically approach zero then the product will also asymptotically approach zero by the squeeze theorem right here. And this function e to the negative x times sine of x is actually a very important function. It's sort of a representative of function family because if you just take this function sine or cosine you can use sines and cosines to model simple harmonic motion. But if you have like a pendulum that's spinning eventually it comes to a stop because mostly friction. And so if you allow that pendulum to swing over and over and eventually it kind of comes to a stop. And so if you wanna talk about dampered harmonic motion you use transformations of the function e to the negative times sine of x. And that's because it has the repetition that comes from sine but eventually it dampers off towards zero because of the exponential factor.