Limits and Continuous Functions
PROFESSOR: OK. I promised a video about limits and continuous functions and here it is. So I'll begin with the most basic idea and with a picture instead of definition in symbols first. So the most basic idea is that I have a bunch of numbers-- let's make them positive numbers-- and I want to know, what does it mean for them to approach a limit-- capital A-- as I go out this sequence of numbers, a1, a2, a3, a4. And let me say right away, the first four numbers, the first million numbers, make no difference about the limit. So here's what it means. For example, there's a equals 7, let's say. What does it mean for these numbers to approach 7? It means that if I take any thin little space around a, above and below, the numbers can start out whatever. They can go in there, they could go out, they could come back, whatever, they could grow way big, way small. But in the end, beyond some point, eventually, they have to get in that slit and stay in there. And...