Chapter 7
Limits and Continuity
IN THIS CHAPTER
Taking a look at limits
Evaluating functions with holes — break out the mothballs
Exploring continuity and discontinuity
As discussed in Chapter 3, you can use ordinary algebra and geometry when the things in a math or science problem aren’t changing (sort of) and when the graph of the function in question is a straight line. But you need calculus when things are changing — these changing things show up on graphs as curves. Calculus can handle such things by zooming in on the curves till they become straight (zooming in infinitely far, sort of). At that point, ordinary algebra and geometry can be used. Limits are the seemingly magical trick or tool that does this zooming-in process. It’s the mathematics of limits that makes calculus work.
Limits are fundamental for both differential and integral calculus. The formal definition of a derivative involves a limit, as does the definition of a definite integral. (If you’re a real go-getter and can’t wait to read the actual definitions, check out Chapters 9 and 14.) Now, it turns out that after you learn the shortcuts for calculating derivatives and integrals, you won’t need to use the longer ...
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