Introduction

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Oil Spills and Dispersant Chemicals

CHAPTER 1 PROJECT

The Chapter Project on page 143 looks at a hypothetical situation of pollution in a lake and explores some legal arguments that might be made.

On April 20, 2010, the Deepwater Horizon drilling rig exploded and initiated the worst marine oil spill in recent history. Oil gushed from the well for three months and released millions of gallons of crude oil into the Gulf of Mexico. One technique used to help clean up during and after the spill was the use of the chemical dispersant Corexit. Oil dispersants allow the oil particles to spread more freely in the water, thus allowing the oil to biodegrade more quickly. Their use is debated, however, because some of their ingredients are carcinogens. Further, the use of oil dispersants can increase toxic hydrocarbon levels affecting sea life. Over time, the pollution caused by the oil spill and the dispersants will eventually diminish and sea life will return, more or less, to its previous condition. In the short term, however, pollution raises serious questions about the health of the local sea life and the safety of fish and shellfish for human consumption.

The concept of a limit is central to calculus. To understand calculus, it is essential to know what it means for a function to have a limit, and then how to find the limit of a function. Chapter 1 explains what a limit is, shows how to find the limit of a function, and demonstrates how to prove that limits exist using the definition of limit.

We begin the chapter using numerical and graphical approaches to explore the idea of a limit. Although these methods seem to work well, there are instances in which they fail to identify the correct limit. We conclude Section 1.1 with a precise definition of limit, the so-called \(\epsilon\)-\(\delta\) (epsilon-delta) definition.

In Section 1.2, we provide analytic techniques for finding limits. Some of the proofs of these techniques are found in Section 1.6, others in Appendix B. A limit found by correctly applying these analytic techniques is precise; there is no doubt that it is correct.

In Sections 1.3–1.5, we continue to study limits and some ways they are used. For example, we use limits to define continuity, an important property of a function.

Section 1.6 is dedicated to the \(\epsilon\)-\(\delta\) definition, which we use to show when a limit does, and does not, exist.