Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. If LMC and K are real numbers, and the limit as X approaches C of F of X = L, and the limit of X approaches C of G of X equal M, then the limit as X approaches C of the two functions added together is really just their limits added together, L + M. Or as the limit of X approaches C of the difference of the two functions is just the difference of the two limits. We also have a constant K times the function as the limit goes to X approaches C That's just really the constant times the limit. If we multiplied the two functions as the limit of X goes to C, that's just the two limits multiplied. If we divided, we just divide the two limits where M can equal 0. The limit as X goes to C of the function to some power is just equal to that limit to some power where N is a positive integer. The limit as X goes to C of the NTH root of the function is just equal to the NTH root of the limit where if N is even, L has to be positive because the inside of an even root has to be positive. The sandwich theorem or squeeze theorem says that if we have 3 functions G of XF of X&H of X for all X in some open interval. So if we have AG of X and we have an H of X and we have an F of X. Let's change this a little bit. And we say for all X and some open interval. So let's look at an interval and call it from here to here, maybe open interval. So let's call it from here to here. Well, the G of X is smaller than the F of X, which is smaller than the H of X for all of those points within this interval and except for possibly at some Point C. And then we're going to suppose that the limit is X goes to C of G of X equals the limit of X goes to C of H of X at some point L. So let's call this Point C Would we agree that the Y value of G of X and the Y value of H of X is the same? Think about zooming it in so you can see it. But if we shrunk it back, those would be basically the same location. And if that's true, then we know that the function that was in between those two would have to also go to the same limit. So that's our sandwich theorem or our squeeze theorem. We also know that if F of X is less than or equal to G of X for all X and some open interval containing C So a similar picture as before. And it doesn't have to be the same function, they just have to be less than or equal to. So here's my G of X. Here's my F of X. We'll indicate some open interval here. So in this open interval, possibly, except at some location C, the limit of F&G both exist as X approaches C So if we call right here a Point C, do you agree that the Y value for the F of X would have to be less than or equal to the Y value for the G of X? That's what this other theorem states. Thank you and have a wonderful day.