Unit 2: Functions, Graphs, Limits, and Continuity
The concepts of continuity and the meaning of a limit form the foundation for all of calculus. Not only must you understand both of these concepts individually, but you must understand how they relate to each other. They are a kind of Siamese twins in calculus problems, as we always hope they show up together.
A student taking a calculus course during a winter term came up with the best analogy that I have ever heard for tying these concepts together: The weather was raining ice - the kind of weather in which no human being in his right mind would be driving a car. When he stepped out on the front porch to see whether the ice-rain had stopped, he could not believe his eyes when he saw the headlights of an automobile heading down his road, which ended in a dead end at a brick house. When the car hit the brakes, the student's intuitive mind concluded that at the rate at which the velocity was decreasing (assuming continuity), there was no way the car could stop in time and it would hit the house (the limiting value). Oops. He forgot that there was a gravel stretch at the end of the road and the car stopped many feet from the brick house. The gravel represented a discontinuity in his calculations, so his limiting value was not correct.
Completing this unit should take you approximately 19 hours.