 Let's recap the main ideas from section 1.7 in active calculus on limits, continuity and differentiability. So as the title of this section suggests, we have three interconnected ideas at work here in section 1.7. The first has to do with limits. We haven't seen these in a few sections. In this section, we're defining the concept of what's called a one-sided limit. We're going to say that the function f has a limit, l1, as x approaches a from the left, and we'll write this notation here. If we can make the value of f of x as close to l1 as we like by taking x sufficiently close to a while keeping x less than a. Now here the little minus sign in the notation means that we're approaching a only from the left of a. Informally, this would mean that as the values of x approach a from the left, the values of f are getting closer and closer to l1. And we're not concerning ourselves with what happens if we approach a from the right. Likewise, we're going to say that f has a limit, l2, as x approaches a from the right, and we're going to write this notation. If we can make the value of f of x as close to l2 as we like by taking x sufficiently close to a while having x bigger than a. And here the little plus sign in the limit notation means that we're approaching a only from the right of a. Informally, this limit would mean that as the values of x approach a from the right only, then the values of f are getting closer and closer to l2 and we do not think or consider what is happening as x approaches a from the left. So these two one sided limits are actually just the two halves of the process of finding just a limit that we saw very early on in the course. In fact, we saw this fact in the new section that a function f has a limit as x approaches a if and only if the limit is x approaches a from the left and the limit is x approaches a from the right both exist and both equal each other, namely the number l. Now this idea of the one side and limit is related to the second big idea in this section, namely that of a function being continuous. The idea of continuity is that we'd like to distinguish functions whose graphs can be drawn in a single unbroken stroke without having to stop at any point. This kind of stoppage could happen for instance if there's a hole in the graph or a jump in the graph. More precisely, we're going to define a function f to be continuous if three criteria are met simultaneously. First, f must have a limit as x approaches a. Second, f must be defined at x equals a. And third, to bring those two pieces together, the limit as x approaches a of f of x has to actually equal f of a. So in other words, the function has to be defined at this point at x of a, x equals a, so there's no hole in the graph there. The function f has to approach a single value as x approaches a, both from the left and from the right. And this value equals the value that f actually takes on when we reach x equals a. So the third and final big idea in this section is differentiability. We defined a function f to be differentiable at a point x equals a if f prime of a exists. Which means that the graph of f of x must have a tangent line at x equals a with a well-defined slope. So how could differentiability fail? Well, if f of x had no tangent line at x equals a, it could mean, for example, that the graph of f is not locally linear at x equals a. That means if you zoomed in on the graph at this point, it would look like something besides a line with a well-defined slope. This could happen, for instance, if f has a sharp corner that would remain sharp and nonlinear no matter how closely in on it we zoomed. There could be other ways for f to fail differentiability, which we'll explore in upcoming examples. But the idea of being locally linear is central to all such failures. These linked ideas of limits, continuity, and differentiability set up the framework for more advanced study of derivatives that we're about to undertake and carry out throughout the remainder of the course.