 There are a couple of limits that we can't handle using the basic limit theorems. So we'll introduce one more limit theorem, and this is known as the sandwich theorem. So suppose I have some interval containing a limiting point x equal to a, and suppose f of x, g of x, and h of x are functions that meet the following requirements. For all x, other than possibly a itself in our interval, f of x is less than or equal to h of x, is less than or equal to g of x. In other words, h of x is sandwiched between f of x and g of x. Furthermore, suppose that the limit as x approaches a of f of x is equal to the limit as x approaches a of g of x, and they're both equal to the same value, l. Then the sandwich theorem says that the limit as x approaches a of h of x is going to be equal to l itself. For example, suppose I want to find the limit as x approaches 0 of x times sine of 1 over x. So to apply the sandwich theorem, we'll want to find some interval containing our limiting value, and on that interval, we want to find two functions f of x and g of x, where f of x is less than or equal to x sine of 1 over x, which is less than or equal to g of x, for this interval containing x equals 0, and the limit as x approaches 0 of f of x, and the limit of x approaches 0 of g of x, is equal to the same number, l. Now, there's no good way of figuring out what these functions are. There's no formula. There's no algorithm that'll tell you what to use for these functions. The only way we can find them is through intuition and insight, and there's no way of developing these, except to do many problems involving them. However, one useful guideline is to always ask yourself, self, what do we know? And in this case, because we're dealing with the sign of something, one of the things we know is that the sign of something is always between negative 1 and 1. And so this tells us that x times the sign of something is always between minus x and x. And so that gives us a pair of functions f of x and g of x, that sandwich x sine of 1 over x in between them. But this sandwich is only useful if the limits of the two functions are the same. And here, we happen to be in luck, because the limit as x approaches 0 of minus x, and the limit as x approaches 0 of x are both equal to 0, and so we can apply the sandwich theorem. And so we can put it all together. We have negative x is less than or equal to x sine of 1 over x, which is less than or equal to x, and the limit as x approaches 0 of minus x is the limit as x approaches 0 of x, and both of these are equal to 0, so because the slices of red go to 0, then the meat also has to go to 0. And we get a limit as x approaches 0 of x sine of 1 over x of 0.