 So one of the important things we want to do is to not have to recalculate every single limit every single time that we encounter it, and to do that we can rely on certain properties of limits. So let's take a look at this. Suppose I have somehow obtained the limit of f of x as x gets close to a, and maybe the limit of g of x as x gets close to a. So let's say I have a couple of limits. Well, if I have these limits then I also have the following facts. First off, the limit as x approaches a of c times f of x is c times the limit for any constant c. In other words, the limit of a function multiplied by a constant is the constant multiplied by the limit of the function. Next, I might want to take a look at the limit as x gets close to a of the sum of two functions, f plus or minus g of x. So I want to take a look at the limit of a sum, and I can actually break that apart. The limit of a sum of two functions is the same as the sum of the limit of the two functions separately. Finally, I might also look at the limit of a product, f times g of x, and the limit of a product of two functions is going to be the same as the product of the two limits. And finally, I might look at the limit of a function f over g of x, and this is going to be the quotient of the two limits. The one requirement here is that that limit of the denominator cannot be equal to zero, because limits still require us to adhere to the requirement you cannot divide by zero. Now, we actually use these properties implicitly, but let's see if we can use them in a more explicit fashion. So let's consider the following problem. So here, I have a function value, I have a limit, I have the value of another function, and I have the value of the same limit. Now, I know absolutely nothing about the functions other than these four facts, but just given this very small amount of information, I can find other things like the function values, I can find a whole bunch of limits. So let's take a look at that. So let's start off with finding f plus g of three. Now remember, this is notation that means f of three plus g of three, and it turns out I know both of those values. Here's f of three, here's g of three. So that means I can find f plus g of three. This is really a precalculus problem, but it's sort of a warm up to what we're doing. So f plus g of three is f of three plus g of three, and I know f of three and g of three, and so I'll substitute those in, and I'll compute that that value is going to be equal to 11. Well, let's take a look at this limit of f plus g of x. Now, this is the limit of a sum of two functions. So the limit of a sum of two functions is the sum of the limits. So I know that the limit of the sum is the sum of the two limits. So that's the limit as x approaches 3 of f of x plus the limit as x approaches 3 of g of x, and I know what those two values are. This limit is 11. This limit is 5. 11 plus 5 equals 16. How about that product? Well, again we have the rule that the limit of a product is the product of the limits. So if I want to find the limit of fg of x, well that's the same as the limit of f of x times g of x, and that is a limit of a product. So I can express that as the product of the two limits. And again, if only I knew what the limit of x approaches 3 was. Ah, well, here it is. And if only I knew what the limit as x approaches 3 of g of x was. Ah, well, there it is. So I can find those two limits, 11 times 5, and the limit is 55. And now we'll do a hard problem. Well, we'll find the limit of the quotient of two functions. Now, actually that isn't too hard. The limit as x approaches 3 f over g of x. Remember, this is just, this is just notation for the limit of f of x over g of x, and the limit of a quotient of two functions is the same as the quotient of the limits. And again, I know this limit, 11 and 5, and so the limit is 11 fifths.