 One of the useful things we want to be able to do is to find the limits from the graph of a function or a set of functions. So for example, I might have two functions, f of x and g of x, which are half the graphs shown, and I can find the limit as, for example, x approaches 4 of the product, and the limit as x approaches 4 of the quotient. So here's my graph in red is the graph of g of x, in blue is my graph of f of x. Now remember by the properties of limits, I can find the limit of a product as the product of the limits provided that the limits themselves actually exist. So the question is, what are these individual limits? To find them, remember that if I'm graphing y equals f of x, y equals g of x, then my y-coordinates are my function values. So what I'm interested in knowing then is what happens to the y-values as x gets close to 4. So here's my graph, again, in blue is f of x, so I'm going to follow that graph as x gets close to 4 and see what happens to the y-values. So I'll follow the graph until I get close to x equals 4, which is right at this point, and it looks like if x is close to 4, then my y-values are going to be close to 1. So it appears that as x gets close to 4, the y-values get close to 1, and we can do the same thing with our values of g of x. So g of x is this graph in red here, and I'm going to follow the graph. I'm going to let x get close to 4. I'm going to follow the graph until I get close to 4. And it looks like if I'm close to 4, if x is close to 4, then my y-values are going to be close to 3. So that tells me that my y-coordinates of f of x get close to 1, my y-coordinates of g of x get close to 3, and so the limit of the product is the product of limits 1 times 3 equal to 3. Likewise, if I want to find the limit of the quotient f over g of x, well, the limit of the quotient is the same as the quotient of the limits, provided that the limits exist and the denominator limit is not equal to 0. Well, I already know what those two limits are, so I can just substitute those in as x approaches 4, my f of x gets close to 1, my g of x gets close to 3, and so the limit is going to be 1 over 3.