t is a fair question to ask, “why bother?”, when it comes to analysis. There is a
certain philosophical satisfaction in knowingwhy things work, but a pragmatic person
may argue that one only needs to know how things work to do real-life problems.
The calculus training you receive in introductory classes is certainly adequate for
you to begin solving many problems in physics, chemistry, biology, economics,
computer science, finance, engineering, or whatever else you end up doing—and
you can certainly use things like the chain rule, L’Hôpital’s rule, or integration by
parts without knowing why these rules work, or whether there are any exceptions to
these rules. However, one can get into trouble if one applies rules without knowing
where they came from and what the limits of their applicability are. Let me give
some examples in which several of these familiar rules, if applied blindly without
knowledge of the underlying analysis, can lead to disaster.
