 Hi everyone and welcome to this lesson on the properties of limits. If you think about what you have learned so far about limits, you were introduced first to graphic and numeric methods for finding limits. This lesson gets us more into the algebraic or analytic side of finding limits. What we will be talking about today are some properties of limits which you will see as we soon discuss the algebraic determination of limits. And I think these properties will very much make sense to you. So suppose that c is a constant and the limit of f of x as x approaches a equals l and the limit of g of x as x approaches a number a is equal to m both exist. And the limit of the sum of f of x and g of x as x approaches a simply is the sum of the limit of each of those individual functions, therefore we have simply l plus m. If we want the limit of the difference between the functions as x approaches a that simply will be l minus m. And finally if we have the function f of x or it could have been g of x multiplied by a constant c that limit would be equal to c times l. Again I think these will pretty much make very easy sense to you. Continuing on if we want the product of the functions f of x and the limit of that product as x approaches a that simply will be l times m. The limit of the quotient of the two functions as x approaches a simply will be l over m. And finally if we have the limit of the function raised to a power and once again this could have easily been the g function instead this limit simply would be l raised to the n power. Let's look at a few practice examples for you. In this first one we have two limits that are provided to us. The limit of f of x as x approaches a number c equal to three over two and the limit of g of x as x approaches that same number c is equal to one half. Notice in all of these properties with the two functions you are provided information on Notice how you are approaching the same number in each that does need to be the case. In letter a if we want the limit of four times the function f of x as x approaches c that simply is going to be four times the limit of the function which is three halves which in the end gives us six. In part b where we want the limit of the sum remember that's simply going to be the sum of the individual limit so we have three halves plus one half which in the end simplifies to two. If we want the limit of the product we simply are going to multiply the two individual limits and we obtain from that three fourths. And finally if we want the limit of the quotient of the two functions as x approaches c you will have three halves which was the limit of the f of x function divided by one half which was the limit of the g of x function and that simplifies to three. Let's take a look at another example. Here we have the limit of f of x as x approaches c to equal twenty seven. In letter a we want the cube root of that function so we simply will do the cube root of the given limit of f of x which comes to the cube root of twenty seven which is three. In letter b we want the limit of the function divided by eighteen so we're going to do twenty seven which was the limit of that function as x approaches c. Divide that by eighteen and we can simplify that to three halves. In letter c we want the limit of the function squared so we simply take the limit which was twenty seven and we square that. We can leave it like that or do it out unfortunately I don't have my calculator available. Now the last one is a great example of one that you would see in an assessment that you might encounter. Typically these problems are going to show up on non calculator assessments so you must be prepared to do these in your head and this is a great example of that. Here we want the limit of the function that is raised to the two-thirds power so we're going to take the twenty seven which was the limit of the function and we need to raise that to the two-thirds so remember how we do that. The denominator of the exponent tells us what root of twenty seven we are taking so we do cube root of twenty seven which is three. We still have the numerator part of the exponent and therefore in the end we have nine as our answer. Now if you take a look at the properties as they were stated and the examples we've looked at here you will notice that in each of these cases we were doing x approaches a number. It should be noted that these properties do apply in the same way if you were to be doing a limit as x approaches infinity as well. One other theorem for us to explore as we talk about properties of limits is one called the squeeze theorem. This is often used in a lot of other proofs you might encounter in the calculus. You won't necessarily have to do any problems with it here but it does help just to know about it and have it in your knowledge base. What the squeeze theorem states is if we have a function f of x the values for which lies in between the values of h of x and g of x for all x's in some open interval containing c except possibly at c itself and if the limit of h of x as x approaches c happens to be equal to the limit of g of x as x approaches c both of those limits equaling l then the limit of the function f of x which is literally squeezed in between h of x and g of x will also therefore equal l. I think it makes sense. I have for you a great demonstration applet of the squeeze theorem courtesy of the mathematics department of the University of Minnesota. You'll notice we have a blue function and a green function and you'll notice this black dot coming down this other red function and you'll notice it's sort of teeter totters right in between the two in between the green function and the blue function as we get closer and closer to the origin so this would be a case in which for the green function the limit as x approaches zero is zero same for the blue function the limit of the blue function as x approaches zero is zero and this red function that was in between as it settles in between the blue function and the green function its limit as x approaches zero is zero as well again you don't have to solve any problems related to the squeeze theorem you just need to know about it so that if you encounter it again it will sound familiar to you thanks for watching and we'll see you soon