Calculus: The Limit Laws, Part I

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Limits
The Concept of the Limit
The Limit Laws Page [1 of 1]
Okay, so we're taking these limits and you can see how we can actually approach a function and see what the function is heading towards as the x's get closer and closer to whatever it is that we want the x's to get closer and closer to.
Now, in fact, once you have the limit of one function, you can put them together in all sorts of interesting ways, and the results, not surprising, can be combined. So let me just show you. Suppose we have the limit as x a of f(x), and suppose that that equals L. And suppose I have a different function, limit, as x a of some other function, let's say g(x), and that equals M. So both limits exist. They both are heading toward a target.
Now, the question is, well, what can be said about combinations of these functions? Well, these are just limit laws that I think make a lot of since. For example, suppose you want to take the limit as x a of the sum of these two functions? What would you guess that limit would be? That's right, I heard you, it's going to be just the sum of the limits. And what if I look at the difference? If I look at the difference of the two functions, then you just look at the difference of the answers. So that makes a lot of sense.
What if you take their product? So let's try to be really, really fancy. x a of f(x) and you multiply by g(x). Well then, in fact, what you get would be just the product of the limits. In fact, you could almost see the word "limit" right there. If I put a little "i" in there, "limit"! That's perfect. Okay, great!
All right, now what about taking the quotient? Well, taking the quotient, we've got to be a little bit more careful, but not too much more careful. This will equal , but we've got to make sure, of course, that we're not dividing by 0. So I want to make sure that, in fact, this is going to be where M 0, because if M = 0, then, of course, this limit wouldn't make a lot of sense, because I'd have 0 downstairs and who know what that would be. So I want to actually make sure that both limits exist in this case and the denominator 0. Otherwise, you can just plow away and do limits as you would normally think.
Oh, and by the way, here's another one. In fact, make up your own. There's a great thing; make up your own little limit law. Here's one I'll make up right now live. Suppose I give you just some number, I'll call it c for a constant, and I take the limit of this. Well, what do you think it is? Well, since c is fixed and unchanging and I know that this thing is heading toward L, then I would guess this goes to c L, and that's the right answer. So you can make up your own limit laws and have a lot of fun with them.
Anyway, here's some basic facts that I think are pretty intuitive, but now we've seen them. I'll see you at the next lecture.