Hello wonderful mathematics people, this is Anna Cox from Kellogg Community College if L,M,c, and K are real numbers and the limit as x approaches c of f(x) equals L and the limit of x approaches c of g(x) equal M then the limit as x approaches c of the two functions added together is really just their limits added together. L plus M or as the limit of cx 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 is the 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't equal zero 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 rootsof 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 three functions g(x), f(x), and h(x) for all x in some open interval so if we have a g(x) and we have an h(x) and we have an f(x) let's change this a little bit and we say for all x on 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(x) is smaller than the f(x) which is smaller than the h(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 as x goes to c of g(x) equals the limit as x goes to c of h(x) at some point L. So let's call this point c would we agree that the y-value of g(x) and the y-value of h(x) is the same. Um, 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(x) is less than or equal to g(x) for all x on some open interval containing c, so 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(x) here's my f(x) we'll indicate some open interval here so in this open interval, possibly except at some location c, the limit of f and 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(x) would have to be less than or equal to the y-value for the g(x) that's what this other theorem states thank you and have a wonderful day