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Chapter 2 | Limits
2.5 | The Precise Definition of a Limit Learning Objectives 2.5.1 Describe the epsilon-delta definition of a limit. 2.5.2 Apply the epsilon-delta definition to find the limit of a function. 2.5.3 Describe the epsilon-delta definitions of one-sided limits and infinite limits. 2.5.4 Use the epsilon-delta definition to prove the limit laws.
By now you have progressed from the very informal definition of a limit in the introduction of this chapter to the intuitive understanding of a limit. At this point, you should have a very strong intuitive sense of what the limit of a function means and how you can find it. In this section, we convert this intuitive idea of a limit into a formal definition using precise mathematical language. The formal definition of a limit is quite possibly one of the most challenging definitions you will encounter early in your study of calculus; however, it is well worth any effort you make to reconcile it with your intuitive notion of a limit. Understanding this definition is the key that opens the door to a better understanding of calculus. Quantifying Closeness Before stating the formal definition of a limit, we must introduce a few preliminary ideas. Recall that the distance between two points a and b on a number line is given by | a − b | . • The statement | f ( x )− L | < ε may be interpreted as: The distance between f(x) and L is less than ε. • The statement 0< | x − a | < δ may be interpreted as: x ≠ a and the distance between x and a is less than δ. It is also important to look at the following equivalences for absolute value: • The statement | f ( x )− L | < ε is equivalent to the statement L − ε < f ( x ) < L + ε . • The statement 0< | x − a | < δ is equivalent to the statement a − δ < x < a + δ and x ≠ a . With these clarifications, we can state the formal epsilon-delta definition of the limit .
Definition Let f ( x ) be defined for all x ≠ a over an open interval containing a . Let L be a real number. Then lim x → a f ( x ) = L if, for every ε >0, there exists a δ >0, such that if 0< | x − a | < δ , then | f ( x )− L | < ε .
This definition may seem rather complex from a mathematical point of view, but it becomes easier to understand if we break it down phrase by phrase. The statement itself involves something called a universal quantifier (for every ε >0), an existential quantifier (there exists a δ >0), and, last, a conditional statement (if 0< | x − a | < δ , then | f ( x )− L | < ε ). Let’s take a look at Table 2.9 , which breaks down the definition and translates each part.
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